Statistical Concepts and Analytics Explained

Standard Data Visualizations

2025-12-23T19:32:00.001+05:30

BAR CHART

The classic bar chart uses either horizontal or vertical bars to show discrete, numerical comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs).

DEVIATION BAR CHART

When graphs display deviation relationships, they communicate how one or more set of metric values differ from a primary set of values. Deviation relationships can be effectively displayed using the following objects:

- Horizontal bars (except when combined with a time-series relationship)
- Vertical bars

DUAL AXIS BAR CHART

A Dual Axis bar chart uses either horizontal or vertical bars to show discrete, numerical comparisons among categories. It can be a combination of a bar and a line with 3 axes. One axis of the chart shows the categories and the other two axes show respective values.

STACKED BAR CHART

Stacked Bar Graphs segment the bars of multiple datasets on top of each other. They are used to show how a larger category is divided into smaller categories and what the relationship of each part has on the total amount. There are two types of Stacked Bar Graphs:

- Simple Stacked Bar Graphs

- 100% Stacked Bar Graphs

LINE CHART

Line charts are used to display quantitative values over a continuous interval or time span. They are most frequently used to show trends and relationships (when grouped with other lines). This gives the "big picture" over an interval, to see how it has developed over that period. Line graphs are drawn by first plotting data points on a Cartesian coordinate grid, then connecting a line between the points. Typically, the y-axis has a quantitative value, while the x-axis has either a category or sequenced scale.

PIE CHART

Pie charts help show proportions and percentages between categories, by dividing a circle into proportional segments. Each arc length represents a proportion of each category; the full circle represents the total sum of all the data, equal to 100%. Pie charts are used for making part-to-whole comparisons with discrete or continuous data. They are most impactful with a small data set.

BUBBLE CHART

Bubble Charts are typically used to compare and show the relationships between labeled/categorized circles, by the use of positioning and proportions. The overall picture of Bubble Charts can be used to analyze patterns/correlations. Bubble Charts use a Cartesian coordinate system to plot points along a grid where the X and Y axis are separate variables. Each point is assigned a label or category (either displayed alongside or on a legend). Each plotted point then represents a third variable by the area of its circle. Colors can also be used to distinguish between categories or to represent an additional data variable.

SCATTER PLOT

A Scatter plot can help you identify the relationships that exist between different values. By displaying a variable in each axis, you can detect if a relationship or correlation between the two variables exists. The Various types of correlations that can be interpreted are positive (values increase together), negative (one value decreases as the other increases), null (no correlation), linear, exponential and U-shaped. The strength of the correlation can be determined by how closely packed the points are to each other on the graph.

AREA CHART

Area Charts are Line Charts with the area below the line filled in with a certain color or texture. Area Graphs are drawn by first plotting data points on a cartesian coordinate grid, then joining a line between the points and finally filling in the space below the completed line. Like Line Charts, Area Charts are used to display the development of quantitative values over an interval or time period. They are most commonly used to show trends and relationships.

BOX PLOT

A box plot is a convenient way to visually display groups of numerical data through their quartiles. It shows distribution of data based on minimum, maximum, median, and percentiles. Typically used in descriptive statistics, box plots are a great way to quickly examine one or more data sets graphically. Although they may seem primitive in comparison to a histogram or density plot, they have the advantage of taking up less space, which is useful when comparing distributions between many groups or data sets.

GANTT CHART

Gantt charts (also referred to as project timelines) are bar charts that help plan and monitor project development or resource allocation on a horizontal time scale. They are essentially horizontal bar charts which provide graphical illustration of a schedule that can help users plan, coordinate, and track specific tasks in a project. The data analyzed in a Gantt chart has a defined starting and ending value; for example, Project A begins 4/15/06 and ends 5/10/06.

HILOW STOCK / CANDLESTICK

This chart control displays financial data as a series of candlesticks representing the high, low, opening, and closing values of a data series (four metrics). The top and bottom of the vertical line in each candlestick represent the high and low values for the data point, while the top and bottom of the filled box represent the opening and closing values.

HISTOGRAM

A histogram visualizes the distribution of data over a continuous interval or a certain time period. Each bar in a histogram represents the tabulated frequency at each interval/bin. The total area of the histogram is equal to the total number of datasets. Histograms help give an estimate of where values are concentrated, what the extremes are and whether there are any gaps or unusual values.

PARETO CHART

A Pareto chart is designed to help identify the cause of a quality problem or loss. It includes a Histogram that shows how often a specific problem is occurring or the different types of problems that are occurring. In general, Pareto charts allow you to display the specific areas in which improvement or investigation is necessary. It contains both a bar and a line chart. The values are represented by descending bars and the running % to total is represented by the line. It depicts the percent journey to total & also displays actual values.

POLAR CHART / RADAR CHART

Radar Charts are a way of comparing multiple quantitative variables. This makes them useful for seeing which variables have similar values or if there are any outliers amongst each variable. They are also useful for seeing which variables are scoring high or low within a dataset, making them ideal for displaying performance.

WATER FALL

A Waterfall visualization highlights the increments and decrements of the values of metrics over time. Analysts can use the widget to identify aspects of their business that are contributing to the fluctuations in the values. The visualization can also be used to perform “what-if” analyses. For e.g., % Revenue Y/Y Variance by Month. It shows how different aspects of the business positively or negatively affect the bottom line.

GAUGES

A Gauge visualization is a simple status indicator that displays a needle that moves within a range of numbers displayed on its outside edges. A real-world example of a gauge is a car's speedometer. Like the Cylinder and Thermometer widgets, this type of visualization is designed to display the value of a single metric. The needle within the gauge is a visual representation of that single metric value.

TIME SERIES

A Time Series Slider is an area graph that allows a document analyst to choose which section of the graph to view at a time. The visualization consists of two related graphs, one positioned above the other. The top graph is the controller, and contains a slider. The bottom graph is the primary graph. You use the slider on the controller to select some portion of the controller, which determines the range of data visible in the primary graph. It allows users to see a high level trend of one or more metrics and a detailed view by varying the window of the visible data. For e.g., Revenue trend by Date.

MAPS

A Map allows users to visualize the data so they can identify and analyze relationships, patterns, and trends in their data. Some of the functionalities available are:

- Displaying areas, points, and data that are color-coded based on metric values

- Using image markers, bubble markers, density maps, or color-coded areas to visualize data on the map

- Zooming/panning on the map and data

- Displaying an Information window with additional data for a marker or area

- Providing the ability to customize the Information window, such as providing additional details or metric information, including demographic content from the mapping service

- Drilling up to summary levels of data and down to detailed levels of data

HEAT MAP

A Heat Map presents a combination of colored rectangles, each representing an attribute element, that allow you to quickly grasp the state and impact of a large number of variables. Heat Maps are often used in the financial services industry to review the status of a portfolio. The rectangles contain varieties and shadings of colors, that emphasize on the status of various components. In a Heat Map, the size of each rectangle represents its relative weight and the color represents the relative change in the value of that rectangle. You can hover over each rectangle to see which attribute element the rectangle represents; and its metric values.

FUNNEL

A Funnel helps to quickly analyze various trends across several metric values. It is a variation of a stacked percent bar chart that displays data that adds up to 100%. Therefore, it can allow analysts to visualize the percent contribution of sales data. It can also show the stages in a sales process and reveal the amount of potential revenue for each stage. When the visualization is used to analyze a sales process, analysts can use the widget to drill down to key metrics such as deal size, profit potential, and probability of closing. The size of the area is determined by the series value as a percentage of the total of all values.

MICRO CHARTS

Micro chart visualizations gives the trend of a metric at a glance without having to know many additional details. The bar, sparkline, and bullet microcharts used in the microcharts convey information that an analyst can understand just by looking at the chart once. It consists of compact representations of data that allow analysts to quickly visualize trends in data. It conveys information so that a user can, at a glance, determine the trend of a metric over time or how a metric is performing compared to forecasted figures.

DATA CLOUD

A Data Cloud displays attribute elements in various sizes to depict the differences in metric values between the elements. This type of visualization is similar to a Heat Map in that they both allow an analyst to quickly identify the most significant, positive, or negative contributions. A Data Cloud widget is basically a list of attribute elements. The first metric on the template determines the font size for the attribute elements. A bigger font for an element indicates a larger metric value.

Python Packages for Data Science

2022-09-11T21:07:00.001+05:30

In order to do data analysis in Python, you should know a little bit about the main packages relevant to analysis in Python. A Python library is a collection of functions and methods that allow you to perform lots of actions without writing any code. The libraries usually contain built-in modules providing different functionalities, which you can use directly. And there are extensive libraries, offering a broad range of facilities.

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We have divided the Python data analysis libraries into three groups:
Scientific Computing Libraries

i. Pandas offers data structure and tools for effective data manipulation and analysis. It provides fast axis to structured data. The primary instrument of Pandas is a two-dimensional table consisting of column and row labels, which are called a DataFrame. It is designed to provide easy indexing functionality. ii. The Numpy library uses arrays for its inputs and outputs. It can be extended to objects for matrices, and with minor coding changes, developers can perform fast array processing. iii. SciPy includes functions for some advanced math problems, as well as data visualization. Using data visualization methods is the best way to communicate with others, showing them meaningful results of analysis.

Libraries to create graphs, charts and maps

i. The Matplotlib package is the most well-known library for data visualization. It is great for making graphs and plots. The graphs are also highly customizable.

ii. Seaborn: It is based on Matplotlib. It's very easy to generate various plots such as heat maps, time series, and violin plots.

Machine Learning algorithms:

i. The Scikit-learn library contains tools for statistical modeling, including regression, classification, clustering and so on. This library is built on NumPy, SciPy and Matplotlib.

ii. StatsModels is also a Python module that allows users to explore data, estimate statistical models, and perform statistical tests.

Statistical Tools to Kick Start with Data Science

2022-09-11T20:40:00.006+05:30

Some of the most common and convenient statistical tools like correlation, regression, box plot, line of best fit to get started with data science.

Histogram

Histograms are a statistical way of representing the frequencies of data values in particular intervals. The more traditional description is that a histogram is a chart plot of a frequency table where the height of the bars in it tell us how many data points are in each interval.

Box Plot

Box plots, sometimes referred to as 'Box and Whisker Plots', are a visual way of summarizing basic characteristics of a data set. Box Plots show the highest and lowest values (that are not outliers), the middle value and the values at the 1st quarter and 3rd quarter mark. Outliers are shown as dots after the 'whiskers'. This gives as a simple way to quickly understand the spread of the data and is great for quickly comparing two data sets.

Think of the quartiles as, if your data set was ranked from it's lowest to its highest value, Q1 would be the middle of the low values, (below the median) and Q3 is the middle of the high values (above the median). Box Plots break out a data set into 4 sets, before Q1, Q1 to the Median (Q2), the Median to Q3 and, after Q3. The interquartile range is defined as Q3 - Q1.

The lowest part of the box in the Box Plot is Q1, there is a line inside the box, the median, and the end of the box, Q3.

Outliers are typically defined as 1.5 * (Q3 - Q1) and if a data point is (1.5* the interquartile range) away from Q1 and Q3 (the edges of the box), it is considered an outlier. The whiskers of the box plot are lines from each end of the box out to the farthest data point that is not an outlier.

Scatter Plot

Scatter Plots are charts that visualize the relationship between two sets of data. Every data point for one variable is plotted against a corresponding data point for another variable and the resultant pattern looks like scattered data points.

One of the most common relationships that can be shown in Scatter Plots is the relationship between a product's price and the number of units sold. Typically when the price goes down (a sale), more units are sold and when the price goes up, people buy less of a product. There are exceptions but that tends to be the case.

The thing we want to reduce uncertainty about goes on the Y axis. We want to know what our sales will be if our price changes. The variable on the Y axis, sales, is the dependent variable. The variable on the X axis is the independent variable and it's the thing we can change, increase or decrease, to see what happens to Y.

To plot a Scatter Plot, we take each value of Y and X for a particular observation e.g., a day, a week, a month and we mark them on a graph. Done.

A Scatter Plot showing many dots following the line between 11 and 5 on a clock is a negative linear relationship e.g., when price goes up, sales go down. A Scatter Plot running from 1 to 7 on a clock shows a positive relationship, when X goes up, so does Y.

Correlation

Correlation is a numerical way of interpreting the relationship between two variables. A Regression analysis uses the 'least squares method' to fit a line through a scatter plot and is measured by R Squared.

The Correlation coefficient is the square route of R Squared, taking on the sign (+ or -) of the slope of the data. When R is high and positive, we say that there is a positive correlation e.g., when X goes up, Y goes up.

Correlation or r, measures the tightness of the scatter dots to the line of best fit and its sign tells us whether Y goes up with changes in X or Y goes down with changes in X. An r of 0 (zero) means that there is no relationship between X and Y and when r is 1, that means that there is a perfect relationship between X and Y where, when X goes up, Y goes up.

Line of Best Fit

A line of best fit is a line drawn through a scatter plot so that each point on that line minimizes the total distance to any of the scatter data points. This is traditionally called a 'Least Squares Line' and it follows the formula, y = mX + C. Imagine a line running through a scatter plot. Each point on that line will have an X and a Y value. The least squares method says that we would take each line y value and subtract it from the scatter dots Y value. Our intention is to sum all these values however, some of the values in this subtraction will be negative as some scatter dots will be below the line so, we square each value before we sum all the values. The Line of Best Fit is one where that sum of the squares is the lowest number it can be i.e., all the dots are either on or very close to the line. When this happens, our line is a good way of predicting Y values, given values of X.

The Line of Best Fit is one where that sum of the squares is the lowest number it can be i.e., all the dots are either on or very close to the line. This summation is called 'Sum of Squared Errors'. When the SSE is at its lowest, our line is a good way of predicting Y values, given values of X.

Regression

In a scatter plot with a line of best fit running through it, we can assess how well the X variable explains the changes in the Y variable using Regression e.g., What percentage of Sales is explained by changes in Price?

When we find a high 'R Squared', in percentage terms, the changes in Y are largely explained by the changes in X. If the R Squared is 95%, we say that, 95% of the variation in Sales is explained by the change in Price and the rest, the 5%, is due to error. The error referred to here is the distances between the dots in the scatter and the line of best fit.

There are many other components to a Regression table output including, confidence intervals, P Values and t ratios.

Adobe Analytics Connector - Microsoft Power BI Desktop

2017-12-12T14:22:00.001+05:30

In the month of December 2017 Microsoft Power BI is releasing a new connector for Adobe Analytics i.e., Omniture. This new connector will allow to import and analyze your Adobe Analytics data within Power BI.
Being a Beta version you have to first enable the connector from Options and Settings >> Options >> Preview Features to view the Adobe connector under Online services.

This Adobe Analytics connector can be found in the Get Data dialog, under the Online Services category.

After the connection is established, you can view and select multiple dimensions and measures within the Navigator dialog box to create a tabular output, also one can provide input parameters for the selected items.

After selecting it can be load into Power BI Desktop for any further additional data transformations and filters within the Query Editor UX, through the Edit option in this dialog.

Once you select your required Dimensions and Measures you will see the table connectors are created automatically in power BI, you will see filed created and also the editor window.

Skewness and Kurtosis in Statistics

2017-05-05T10:37:00.000+05:30

The average and measure of dispersion can describe the distribution but they are not sufficient to describe the nature of the distribution. For this purpose we use other concepts known as Skewness and Kurtosis. The symmetrical and skewed distributions are shown by curves as

Skewness

Skewness means lack of symmetry. A distribution is said to be symmetrical when the values are uniformly distributed around the mean. For example, the following distribution is symmetrical about its mean 3.

x : 1 2 3 4 5

frequency (f ) : 5 9 12 9 5

In a symmetrical distribution the mean, median and mode coincide, that is, mean = median = mode.

Several measures are used to express the direction and extent of skewness of a dispersion. The important measures are that given by Pearson. The first one is the Coefficient of Skewness:

For a symmetric distribution S_k = 0. If the distribution is negatively skewed then S_k is negative and if it is positively skewed then S_k is positive. The range for S_k is from -3 to 3.

The other measure uses the b (read ‘beta’) coefficient which is given by, where, m₂ and m₃ are the second and third central moments. The second central moment m₂is nothing but the variance. The sample estimate of this coefficient is where m₂ and m₃ are the sample central moments given by

For a symmetrical distribution b₁ = 0. Skewness is positive or negative depending upon whether m₃ is positive or negative.

Kurtosis

A measure of the peakness or convexity of a curve is known as Kurtosis.

It is clear from the above figure that all the three curves, (1), (2) and (3) are symmetrical about the mean. Still they are not of the same type. One has different peak as compared to that of others. Curve (1) is known as mesokurtic (normal curve); Curve (2) is known as leptocurtic (leading curve) and Curve (3) is known as platykurtic (flat curve). Kurtosis is measured by Pearson’s coefficient, b₂ (read ‘beta - two’).It is given by .
The sample estimate of this coefficient is where, m₄ is the fourth central moment given by m₄ =

The distribution is called normal if b₂ = 3. When b₂ is more than 3 the distribution is said to be leptokurtic. If b₂ is less than 3 the distribution is said to be platykurtic.

Measures of Dispersion in Statistics

2017-01-03T11:33:00.000+05:30

We know that averages are representatives of a frequency distribution but they fail to give a complete picture of the distribution. They do not tell anything about the scatterness of observations within the distribution.

Suppose that we have the distribution of the yields (kg per plot) of two paddy varieties from 5 plots each.

The distribution may be as follows:

Variety I 45 42 42 41 40

Variety II 54 48 42 33 30

It can be seen that the mean yield for both varieties is 42 kg. But we can not say that the performance of the two varieties is same. There is greater uniformity of yields in the first variety whereas there is more variability in the yields of the second variety. The first variety may be preferred since it is more consistent in yield performance. From the above example, it is obvious that a measure of central tendency alone is not sufficient to describe a frequency distribution. In addition to it we should have a measure of scatterness of observations. The scatterness or variation of observations from their average is called the dispersion. There are different measures of dispersion like the range, the quartile deviation, the mean deviation and the standard deviation.

Range

The simplest measure of dispersion is the range. The range is the difference between the minimum and maximum values in a group of observations for example, suppose that the yields (kg per plot) of a variety from five plots are 8, 9, 8, 10 and 11. The range is (11 - 8) = 3 kg. In practice the range is indicated as 8 - 11 kg.

Range takes only the maximum and minimum values into account and not all the values. Hence it is a very unstable or unreliable indicator of the amount of deviation. It is affected by extreme values. In the above example, if we have 15 instead of figure 11, the range will be (8 - 15) = 7 kg. In order to avoid these difficulties another measure of dispersion called quartile deviation is preferred.

Quartile Deviation

We can delete the values below the first quartile and the values above the third quartile. It is assumed that the unusually extreme values are eliminated by this way. We can then take the mean of the deviations of the two quartiles from the second quartile (median). That is,

This quantity is known as the quartile deviation (Q.D.).

The quartile deviation is more stable than the range as it depends on two intermediate values. This is not affected by extreme values since the extreme values are already removed. However, quartile deviation also fails to take the values of all deviations

Mean Deviation

Mean deviation is the mean of the deviations of individual values from their average. The average may be either mean or median. For raw data the mean deviation from the median is the least. Therefore, median is considered to be most suitable for raw data. But usually the mean is used to find out the mean deviation. The mean deviation is given by

M.D. = for raw data and M.D. = for grouped data

All positive and negative differences are treated as positive values. Hence we use the modulus symbol | |. We have to read as “modulus”. If we take as such, the sum of the deviations, will be 0. Hence, if the signs are not eliminated the mean deviation will always be 0, which is not correct.

The steps of computation are as follows :

Step 1: If the classes are not continuous we have to make them continuous.

Step 2: Find out the mid values of the classes (mid - X = x).

Step 3: Compute the mean.

Step 4: Find out for all values of x.

Step 5: Multiply eachby the corresponding frequencies.

Step 6: Use the formula.

The mean deviation takes all the values into consideration. It is fairly stable compared to range or quartile deviation. Since, the mean deviation ignores signs of deviations, it is not possible to use it for further statistical analysis and it is not stable as standard deviation which is defined as:

Standard Deviation

Ignoring the signs of the deviations is mathematically not correct. We may square the deviation to make a negative value as positive. After calculating the average squared deviations, it can be expressed in original units by taking its square root. This type of the measure of variation is known as Standard Deviation.

The standard deviation is defined as the square root of the mean of the squared deviations of individual values from their mean. Symbolically,

Standard Deviation (S.D.) or

This is called standard deviation because of the fact that it indicates a sort of group standard spread of values around their mean. For grouped data it is given as

Standard Deviation (S.D.) or

The sample standard deviation should be an unbiased estimate of the population standard deviation because we use sample standard deviation to estimate the population standard deviation. For this we substitute n - 1 for n in the formula. Thus, the sample standard deviation is written as

For grouped data it is given by

where,

C = class interval

d = (x - A) / C as given under mean.

The square of the standard deviation is known as the variance. In the analysis of variance technique, the termis called the sum of squares, and the variance is called the mean square. The standard deviation is denoted by s in case of sample, and by s (read ‘sigma’) in case of population.

The standard deviation is the most widely used measure of dispersion. It takes all the items into consideration. It is more stable compared to other measures. However, it will be inflated by extreme items as is the mean.

The standard deviation has some additional special characteristics. It is not affected by adding or subtracting a constant value to each observed value. It is affected by multiplying or dividing each observation by a constant. When the observations are multiplied by a constant, the resulting standard deviation will be equivalent to the product of the actual standard deviation and the constant. (Note that division of all observations by a constant, C is equivalent to multiplication by its reciprocal, 1/C. Subtracting a constant C is equivalent of adding a constant, - C.)

The standard deviations can be pooled. If the sum of squares for the first distribution with n₁ observations is SS₁, and the sum of squares for the second distribution with n₂ observations is SS₂, then the pooled standard deviation is given by,

Measures of Relative Dispersion

Suppose that the two distributions to be compared are expressed in the same units and their means are equal or nearly equal. Then their variability can be compared directly by using their standard deviations. However, if their means are widely different or if they are expressed in different units of measurement, we can not use the standard deviations as such for comparing their variability. We have to use the relative measures of dispersion in such situations.

There are relative dispersion in relation to range, the quartile deviation, the mean deviation, and the standard deviation. Of these, the coefficient of variation which is related to the standard deviation is important. The coefficient of variation is given by,

C.V. = (S.D. / Mean) x 100

The C.V. is a unit-free measure. It is always expressed as percentage. The C.V. will be small if the variation is small of the two groups, the one with less C.V. is said to be more consistent.

The coefficient of variation is unreliable if the mean is near zero. Also it is unstable if the measurement scale used is not ratio scale. The C.V. is informative if it is given along with the mean and standard deviation. Otherwise, it may be misleading.

How to Choose Sample Size for a Simple Random Sample

2016-10-06T07:17:00.000+05:30

Before we proceed to the concept, consider the following problem. You are conducting a survey. The sampling method is simple random sampling, without replacement. You want your survey to provide a specified level of precision.

To choose the right sample size for a simple random sample, you need to define the following inputs.

Specify the desired margin of error ME. This is your measure of precision.
Specify alpha.
For a hypothesis test, alpha is the significance level.
For an estimation problem, alpha is: 1 - Confidence level.
Find the critical standard score z.
For an estimation problem or for a two-tailed hypothesis test, the critical standard score (z) is the value for which the cumulative probability is 1 - alpha/2.
For a one-tailed hypothesis test, the critical standard score (z) is the value for which the cumulative probability is 1 - alpha.
Unless the population size is very large, you need to specify the size of the population (N).

Given these inputs, the following formulas find the smallest sample size that provides the desired level of precision.

Sample statistic	Population size	Sample size
Mean	Known	n = { z² * σ² * [ N / (N - 1) ] } / { ME² + [ z² * σ² / (N - 1) ] }
Mean	Unknown	n = ( z² * σ² ) / ME²
Proportion	Known	n = [ ( z² * p * q ) + ME² ] / [ ME² + z² * p * q / N ]
Proportion	Unknown	n = [ ( z² * p * q ) + ME² ] / ( ME² )

This approach works when the sample size is relatively large (greater than or equal to 30). Use the first or third formulas when the population size is known. When the population size is large but unknown, use the second or fourth formulas.

For proportions, the sample size requirements vary, based on the value of the proportion. If you are unsure of the right value to use, set p equal to 0.5. This will produce a conservative sample size estimate; that is, the sample size will produce at least the precision called for and may produce better precision.

Sample Problem

At the end of every school year, the state administers a reading test to a simple random sample drawn without replacement from a population of 100,000 third graders. Over the last five years, students who took the test correctly answered 75% of the test questions.

What sample size should you use to achieve a margin of error equal to plus or minus 4%, with a confidence level of 95%?

Solution: To solve this problem, we follow the steps outlined above.

Specify the margin of error. This was given in the problem definition. The margin of error is plus or minus 4% or 0.04.

Specify the confidence level. This was also given. The confidence level is 95% or 0.95.

Compute alpha. Alpha is equal to one minus the confidence level. Thus, alpha = 1 - 0.95 = 0.05.

Determine the critical standard score (z). Since this is an estimation problem, the critical standard score is the value for which the cumulative probability is 1 - alpha/2 = 1 - 0.05/2 = 0.975.

To find that value, we use the Normal Calculator. Recall that the distribution of standard scores has a mean of 0 and a standard deviation of 1. Therefore, we plug the following entries into the normal calculator: Value = 0.975; Mean = 0; and Standard deviation = 1. The calulator tells us that the value of the standard score is 1.96.

And finally, we assume that the population proportion p is equal to its past value over the previous 5 years. That value is 0.75. Given these inputs, we can find the smallest sample size n that will provide the required margin of error.

n = [ (z2 * p * q ) + ME2 ] / [ ME2 + z2 * p * q / N ]

n = [ (1.96)2 * 0.75 * 0.25 + 0.0016] / [ 0.0016 + (1.96)2 * 0.75 * 0.25 / 100,000 ]

n = (0.7203 + 0.0016) / ( 0.0016 + 0.0000072) = 449.2

Therefore, to achieve a margin of error of plus or minus 4 percent, we will need to survey 450 students, using simple random sampling.

Introduction to Normal Distribution in Statistics

2015-12-20T15:42:00.001+05:30

A continuous random variable has an infinite number of values that can be represented by an interval on the number line. It’s probability distribution is called a continuous probability distribution. In this article, we will be understanding the most important continuous probability distribution in statistics, the normal distribution.

A normal distribution is a continuous probability distribution for a random variable, x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties.

1. The mean, median and mode are equal.

2. The normal curve is bell-shaped and is symmetric about the mean.

3. The total area under the normal curve is equal to 1.

4. The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean.

5. Between m - s and m + s (in the center of the curve) the graph curves downward. The graph curves upward to the left of m - s and to the right of m + s. The points at which the curve changes from curving upward to curving downward are called inflection points.

6. A normal distribution can have any mean and any positive standard deviation. These two parameters, m and s completely determine the shape of a normal curve. The mean gives the location of the line of symmetry and the standard deviation describes how much the data are spread out.

See the line of symmetry for each? That’s the mean. However, if it is fatter, then the standard deviation is greater. That’s the difference.

Understanding Mean & Standard Deviation

Which normal curve has a greater mean?

Which normal curve has a greater standard deviation

The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12. So, curve A has a greater mean.

Curve B is more spread out than curve A, so curve B has a greater standard deviation.

The Empirical Rule

In a normal distribution with mean m and standard deviation s, you can approximate areas under the normal curve as follows:

1. About 68% of the area lies between m - s and m + s
2. About 95% of the area lies between m - 2s and m + 2s
3. About 99.7% of the area lies between m - 3s and m + 3s

Storytelling with Data - Web Analysis

2015-05-01T22:40:00.000+05:30

Why Storytelling?
Visual analysis means exploring data visually. A story unfolds as you navigate from one visual summary into another.
You and your team have sorted through and analyzed a dense data set, made industry-relevant discoveries and created data visualizations that allow you to share those insights with others—whether other team members, current or potential clients, or the community at large. Before you present your work, think about your audience and the goals you want to achieve.
Keep in mind that the people you want to communicate with may not have the same background or technical knowledge that you do. Although eager to understand, audience members may need additional explanation, simplified insights or a more slowly paced presentation to grasp your point of view. Making the data accessible to your audience is your responsibility.
A tried-and-true method of connecting with your audience is to embed your data insights within a story. The story framework will capture your audience's attention and help you meet the business objectives driving your work.

Story Framework
Before you embark on your data journey, identify the questions you want to answer. You may have a specific question in mind or a general area you'd like to explore. As you dig into your data, you may find an entirely new or unexpected story, but it helps to have a starting point.

An example of one story framework for a web analysis example on how to visualize online campaign performance.

and the output...

Statistical p-values

2014-11-14T06:47:00.001+05:30

When results of studies or research are reported, important decisions are made on the basis of these results. For example, new varieties are often tested against standard varieties to determine if the new varieties is more effective. Several methods of manufacturing may be compared to select the best technique to manufacture the best product. Several evidence may be examined to determine if there is a possible link between one activity and a result. In such kind of studies, results are summarized by a statistical test, and a decision about the significance of the result is based on a p-value. Therefore, it is important for the reader to know what the p-value is all about.
To describe how the p-value works, we'll use a common statistical test as an example, the Student's t-test for independent groups. For this test, subjects are randomly assigned to one of two groups. Some treatment is performed on the subjects in one group, and the other group acts as a control where no treatment or a standard treatment is given. For this example, suppose group one is given a new drug and group 2 is given then standard drug. Time to relief is measured for both groups. The outcome measurement is assumed to be a continuous variable which is normally distributed, and it is assumed that the population variance for the measure is the same for both groups.
For this example the sample mean for group one is 10 and the sample mean for group two is 12. The sample standard deviation for group one is 1.8 and the sample standard deviation for group two is 1.9. The sample size for both groups is 12. Entering this data into a statistical program will produce a t-statistic and a p-value. Calculated t = -2.65 with 22 degrees of freedom, and a p-value of 0.0147. This means that you have evidence that the mean time to relief for group one was significantly different than for group two.
To interpret this p-value, you must first know how the test was structured. In the case of this two-sided t-test, the hypotheses are:
Ho: u1 = u2 (Null hypothesis: means of two groups are equal)
Ha: u1 <> u2 (Alternative: means of the two groups are not equal)
A low p-value for the statistical test points to rejection of the null hypothesis because it indicates how unlikely it is that a test statistic as extreme as or more extreme than the one given by this data will be observed from this population if the null hypothesis is true. Since p=0.015, this means that if the population means were equal as hypothesized (under the null), there is a 15 in 1000 chance that a more extreme test statistic would be obtained using data from this population. If you agree that there is enough evidence to reject the null hypothesis, you conclude that there is significant evidence to support the alternative hypothesis.
The researcher decides what significance level to use i.e., what cutoff point will decide significance. The most commonly used level of significance is 0.05. When the significance level is set at 0.05, any test resulting in a p-value under 0.05 would be significant. Therefore, you would reject the null hypothesis in favor of the alternative hypothesis. Since you are comparing only two groups, you can look at the sample means to see which is largest. The sample mean of group one is smallest, so you conclude that medicine one acted significantly faster, on average, than medicine two. This would be reported in an article using a phrase like this: "The mean time to relief for group one was significantly smaller than for group two. (two sided t-test, t(22) = -2.65, p=0.015)."
P-values do not simple provide you with a Yes or No answer, they provide a sense of the strength of the evidence against the null hypothesis. Lower the p-value, the stronger the evidence.

Cautions about Regression and Correlation

2014-05-11T08:44:00.000+05:30

Correlation and regression are powerful tools for describing the relationship between two variables. When you use these tools, you must be aware of their limitations.

■ Correlation and regression lines describe only linear relationships. You can do the calculations for any relationship between two quantitative variables, but the results are useful only if the scatterplot shows a linear pattern.
■ Correlation and least-squares regression lines are not resistant. Always plot your data and look for observations that may be influential.
■Extrapolation. Suppose that you have data on a child’s growth between 3 and 8 years of age. You find a strong linear relationship between age x and height y. If you fit a regression line to these data and use it to predict height at age 25 years, you will predict that the child will be 8 feet tall. Growth slows down and
then stops at maturity, so extending the straight line to adult ages is foolish. Few relationships are linear for all values of x. Don’t make predictions far outside the range of x that actually appears in your data.
■Lurking variable. the relationship between two variables can often be understood only by taking other variables into account. Lurking variables can make a correlation or regression misleading.

You should always think about possible lurking variables before you draw conclusions based on correlation or regression.

E X T R A P O L AT I O N
Extrapolation is the use of a regression line for prediction far outside the range of values of the explanatory variable x that you used to obtain the line. Such predictions are often not accurate.

L U R K I N G V A R I A B L E
A lurking variable is a variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.

Statistical Terminology - Explained

2013-01-07T11:23:00.000+05:30

We often make use of statistical techniques in the analysis of findings. Some are highly sophisticated and complex, but those most often used are easy to understand. The most common are measures of central tendency (ways of calculating averages) and correlation coefficients (measures of the degree to which one variable relates consistently to another). There are three methods of calculating averages, each of which has certain advantages and shortcomings. Take as an example the amount of personal wealth (including all assets such as houses, cars, bank accounts and investments) owned by 13 individuals. Suppose they own the following amounts:

1. £000 (zero)
2 £5,000
3. £10,000
4. £20,000
5. £40,000
6. £40 ,000
7. £40,000
8. £80,000
9. £100,000
10. £150,000
11. £200,000
12. £400,000
13. £10,000,000

The mean corresponds to the average, arrived at by adding together the personal wealth of all 13 people and dividing the result by 13. The total is £11,085,000; dividing this by 13, we reach a mean of £852,692.31. This mean is often a useful calculation because it is based on the whole range of data provided. However, it can be misleading where one or a small number of cases are very different from the majority. In the above example, the mean is not in fact an appropriate measure of central tendency, because the presence of one very large figure, £ 1 0,000,000 skews the picture. One might get the impression when using the mean to sumnarize this data that most of the people own far more than they actually do. In such instances, one of two other measures may be used. The mode is the figure that occurs most frequently in a given set of data. In our example, it is £40,000. The problem with the mode is that it does not take into account the overall distribution of the data - i.e., the range of figures covered. The most frequently occurring case in a set of figures is not necessarily representative of their distribution as a whole and thus may not be a useful average. In this case, £40,000 is too close to the lower end of the figures. The third measure is the median, which is the middle of any set of figures; here, this would be the seventh figure, again £40,000. Our example gives an odd number of figures 13. If there had been an even number - for instance, 12 - the median would be calculated by taking the mean of the two middle cases. Like the mode, the median gives no idea of the actual range of the data measured.

Sometimes a researcher will use more than one measure of central tendency to avoid giving a deceptive picture of the average. More often, he will calculate the standard deviation for the data in question. This is a way of calculating the degree of dispersal, or the range, of a set of figures - which in this case goes from zero to £10,000,000. Correlation coefficients offer a useful way of expressing how closely connected two (or more) variables are. Where two variables correlate completely, we can speak of a perfect positive correlation, expressed as 1. Where no relation is found between two variables - they have no consistent connection at all, the coefficient is zero. A perfect negative correlation, expressed as -1, exists when two variables are in a completely inverse relation to one another. Correlations of the order of 0.6 or more, whether positive or negative, are usually regarded as indicating a strong degree of connection between whatever variables are being analysed. Positive correlations on this level might be found between, say, social class background and voting behaviour.

Multicollinearity

2011-10-02T13:01:00.000+05:30

The use and interpretation of a multiple regression model depends implicitly on the assumption that the explanatory variables are not strongly interrelated. In most regression applications the explanatory variables are not orthogonal. Usually the lack of orthogonality is not serious enough to affect the analysis. However, in some situations the explanatory variables are so strongly interrelated that the regression results are ambiguous. Typically, it is impossible to estimate the unique effects of individual variables in the regression equation. The estimated values of the coefficients are very sensitive to slight changes in the data and to the addition or deletion of variables in the equation. The regression coefficients have large sampling errors which affect both inference and forecasting that is based on the regression model. The condition of severe non-orthogonality is also referred to as the problem of multicollinearity.

The presence of multicollinearity has a number of potentially serious effects on the least squares estimates of regression coefficients. Multicollinearity also tends to produce least squares estimates that are too large in absolute value.

Remedial Measures

i) Collection of additional data: Collecting additional data has been suggested as one of the methods of combating multicollinearity. The additional data should be collected in a manner designed to break up the multicollinearity in the existing data.

ii) Model respecification: Multicollinearity is often caused by the choice of model, such as when two highly correlated regressors are used in the regression equation. In these situations some respecification of the regression equation may lessen the impact of multicollinearity. One approach to respecification is to redefine the regressors. For example, if x1, x2 and x3 are nearly linearly dependent it may be possible to find some function such as x = (x1+x2)/x3 or x = x1x2x3 that preserves the information content in the original regressors but reduces the multicollinearity.

iii) Ridge Regression: When method of least squares is used, parameter estimates are unbiased. A number of procedures have been developed for obtaining biased estimators of regression coefficients to tackle the problem of multicollinearity. One of these procedures is ridge regression. The ridge estimators are found by solving a slightly modified version of the normal equations. Each of the diagonal elements of X'X matrix are added a small quantity.

Diagnostics and Remedial Measures

2011-05-27T05:59:00.005+05:30

The interpretation of data based on analysis of variance (ANOVA) is valid only when the following assumptions are satisfied:
1. Additive Effects: Treatment effects and block (environmental) effects are additive.
2. Independence of errors: Experimental errors are independent.
3. Homogeneity of Variances: Errors have common variance.
4. Normal Distribution: Errors follow a normal distribution.
Also the statistical tests t, F, z, etc. are valid under the assumption of independence of errors and normality of errors. The departures from these assumptions make the interpretation based on these statistical techniques invalid. Therefore, it is necessary to detect the deviations and apply the appropriate remedial measures.

• The assumption of independence of errors, i.e., error of an observation is not related to or depends upon that of another. This assumption is usually assured with the use of proper randomization procedure. However, if there is any systematic pattern in the arrangement of treatments from one replication to another, errors may be non-independent. This may be handled by using nearest neighbour methods in the analysis of experimental data.

• The assumption of additive effects can be defined and detected in the following manner:

Additive Effects: The effects of two factors, say, treatment and replication, are said to be additive if the effect of one-factor remains constant over all the levels of other factors. A hypothetical set of data from a randomized complete block (RCB) design, with 2 treatments and 2 replications, with additive effects is as

Treatment                        Replication       Replication Effect
                                         I           II            I - II
A                                    190       125            65
B                                    170       105            65
Treatment Effect (A-B)    20        20
Here, the treatment effect is equal to 20 for both replications and replication effect is 65 for both treatments.
When the effect of one factor is not constant at all the levels of other factor, the effects are said to be non-additive.

Normality of Errors: The assumptions of homogeneity of variances and normality are generally violated together. To test the validity of normality of errors for the character under study, one can take help of Normal Probability Plot, Anderson-Darling Test, D'Augstino's Test, Shapiro - Wilk's Test, Ryan-Joiner test, Kolmogrov-Smirnov test, etc. In general moderate departures from normality are of little concern in the fixed effects ANOVA as F - test is slightly affected but in case of random effects, it is more severely impacted by non-normality. The significant deviations of errors from normality, makes the inferences invalid. So before analyzing the data, it is necessary to convert the data to a scale that it follows a normal distribution. In the data from designed field experiments, we do not directly use the original data for testing of normality or homogeneity of observations because this is embedded with the treatment effects and some of other effects like block, row, column, etc. So there is need to eliminate these effects from the data before testing the assumptions of normality and homogeneity of variances. For eliminating the treatment effects and other effects we fit the model corresponding to the design adopted and estimate the residuals. These residuals are then used for testing the normality of the observations. In other words, we want to test the null hypothesis H0: errors are normally distributed against alternative hypothesis H1: errors are not normally distributed. In SAS and SPSS commonly used tests are Shapiro-Wilk test and Kolmogrov-Smirnov test. MINITAB uses three tests viz. Anderson-Darling, Ryan-Joiner, Kolmogrov-Smirnov for testing the normality of data.

Homogeneity of Error Variances: A crude method for detecting the heterogeneity of variances is based on scatter plots of means and variance or range of observations or errors, residual vs fitted values, etc.

Based on these scatter plots, the heterogeneity of variances can be classified into two types:
1. Where the variance is functionally related to mean.
2. Where there is no functional relationship between the variance and the mean.
The scatter-diagram of means and variances of observations for each treatment across the replications gives only a preliminary idea about homogeneity of error variances. Statistically the homogeneity of error variances is tested using Bartlett's test for normally distributed errors and Levene test for non-normal errors.

Remedial Measures: Data transformation is the most appropriate remedial measure, in the situation where the variances are heterogeneous and are some functions of means. With this technique, the original data are converted to a new scale resulting into a new data set that is expected to satisfy the homogeneity of variances. Because a common transformation scale is applied to all observations, the comparative values between treatments are not altered and comparison between them remains valid.

Error partitioning is the remedial measure of heterogeneity that usually occurs in experiments, where, due to the nature of treatments tested some treatments have errors that are substantially higher (lower) than others.

Here, we shall concentrate on those situations where character under study is non-normal and variances are heterogeneous. Depending upon the functional relationship between variances and means, suitable transformation is adopted. The transformed variate should satisfy the following:

1. The variances of the transformed variate should be unaffected by changes in the means. This is also called the variance stabilizing transformation.
2. It should be normally distributed.
3. It should be one for which effects are linear and additive.
4. The transformed scale should be such for which an arithmetic average from the sample is an efficient estimate of true mean.

The following are the three transformations, which are being used most commonly.

a) Logarithmic Transformation
b) Square Root Transformation
c) Arc Sine or Angular Transformation

Web Analytics - An Overview

2011-03-11T23:28:00.004+05:30

Web analytics is the practice of measuring, collecting, analysing and reporting online data for the purposes of understanding how a web site is used by its visitors and how to optimise its usage. The focus of web analytics is to understand a site’s users, their behaviour and activities.

The study of online user behaviour and activities generates valuable marketing intelligence and provides:

• performance measures of the website against targets

• insights on user behaviours and needs, and how the site is meeting those needs

• optimisation ability to make modifications to improve the website based on the results

An average web analytics tool offers hundreds of metrics. All are interesting but only a few would be useful for measuring the website’s performance. Focus on what is important to get meaningful insights about your website, start your web analytics initiative by defining realistic and measurable objectives for your site.

A business will not be successful if the customers are not satisfied. The same applies to your website. You must provide a compelling customer experience that creates value for your users and persuades them to take action. Each website may have a number of different users. To create a compelling user experience you must study each user segment in detail. Create user profiles for each segment that answer:

• Who is your target market?

• Why would they visit your site?

• What do they wish to accomplish on your site?

• What are the barriers to their satisfaction?

Key performance indicators or KPIs are a simple and practical technique widely used to measure performance. They are often expressed in rates, ratios, averages, percentages. The challenge is to choose the KPIs that will drive action and challenge you to continually optimise your site to achieve your objectives. It is important to understand the difference between an interesting metric and an insightful KPI. Peterson, in his book, suggests:

KPIs should never be met with a blank stare. Ask yourself “If this number improves by 10% who should I congratulate?” and “If this number declines by 10% who do I need to scream at?” If you don’t have a good answer for both questions, likely the metric is interesting but not a key performance indicator.

How is the user activity data collected?

There are two distinct methods to collect user activity data:

• Web server log files – Web servers are capable of logging “user requests”, or a user’s movements around a website. These files can be used to perform analysis and create reports on website traffic.

• Tracking scripts inserted into web pages – With this approach a small java script is inserted into a web page and every time the page is downloaded into a user’s browser, this script executes itself, capturing information about the activity performed. Since the web page contains the tracking script, regardless of how it is served to the user, it will execute each time the page is downloaded on the user’s browser.

Standard user activity data can be enriched through:

• URL tracking parameters – Tracking parameters are added to a web page’s URL so you can collect additional information about site usage. For example. to understand what the users are searching for, you can put the keywords being searched for into the URL of the search results page. That result page’s URL will then look like this: “search_results.html?keyword=public holidays”.

• Cookies - Cookies are small packets of data deposited on the computer hard disk of the user when the person visits a website. Cookies can contain all sorts of information, such as visitor's unique identification number for that site; the last time that person visited the site and so on. Your web analytics solution can be configured to detect the cookie for identifying returning users and read its content for more advanced reporting such as recency of a visit.

• Online forms - Forms often constitute low-cost/high-value interaction points for websites. They are part of shopping carts, they facilitate many online processes such as applications, subscriptions, registrations, or they are simply used to seek feedback. Your web analytics solution can be configured to capture certain information collected from web forms through custom fields for more advanced reporting such as demographic profiling.

Qualitative data

In Web Analytics, to understand the ‘why’ behind an issue revealed by quantitative data, we turn to qualitative data. Sources of qualitative data include:

• Surveys – Online or offline surveys are one way to capture information on what customers think and how they feel.

• Web Site Testing – Testing could take place in a lab or online where participants are asked to undertake a task

How web analytics tools identify users?

Web analytics tools need a way of identifying users to be able to report on user sessions (also referred to as visits). There are different techniques to identify users such as IP addresses, user agent and IP address combination, cookies, authenticated user. Nowadays, the most common user identification technique is via cookies which are small packets of data that are usually deposited on the computer hard disk of the user when the person visits a website.

There are several types of cookies:

• First Party Cookie is served from the website being visited.

• Third Party Cookie is served by a third party organisation such as ad agencies or web analytics vendors on behalf of the website being visited.

• Session Cookie is not saved to the computer and expires at the end of the session.

Increased cookie blocking and deletion practices whereby users configure their browsers to not accept the cookie or manually remove cookies from their computers presents a challenge for web analytics tools to accurately identify users.

Test of Significance

2011-03-10T22:07:00.001+05:30

In applied investigations, one is often interested in comparing some characteristic (such as the mean, the variance or a measure of association between two characters) of a group with a specified value, or in comparing two or more groups with regard to the characteristic. For instance, one may wish to compare two varieties of wheat with regard to the mean yield per hectare or to know if the genetic fraction of the total variation in a strain is more than a given value or to compare different lines of a crop in respect of variation between plants within lines. In making such comparisons one cannot rely on the mere numerical magnitudes of the index of comparison such as the mean, variance or measure of association. This is because each group is represented only by a sample of observations and if another sample were drawn the numerical value would change. This variation between samples from the same population can at best be reduced in a well-designed controlled experiment but can never be eliminated. One is forced to draw inference in the presence of the sampling fluctuations which affect the observed differences between groups, clouding the real differences. Statistical science provides an objective procedure for distinguishing whether the observed difference connotes any real difference among groups. Such a procedure is called a test of significance.
The test of significance is a method of making due allowance for the sampling fluctuation affecting the results of experiments or observations. The fact that the results of biological experiments are affected by a considerable amount of uncontrolled variation makes such tests necessary. These tests enable us to decide on the basis of the sample results, if

i)the deviation between the observed sample statistic and the hypothetical parameter value,
or

ii)the deviation between two sample statistic,

is significant or might be attributed to chance or the fluctuation of sampling.

For applying the tests of significance, we first set up a hypothesis - a definite statement about the population parameters. In all such situations we set up an exact hypothesis such as, the treatments or variate in question do not differ in respect of the mean value, or the variability, or the association between the specified characters, as the case may be, and follow an objective procedure of analysis of data which leads to a conclusion of either of two kinds:

i)reject the hypothesis, or

ii)not reject the hypothesis

For applying any test of significance, the following steps should be followed.

i) Identify the variables to be analyzed and identify the groups to be compared

ii) State null hypothesis

iii) Choose an appropriate alternative hypothesis

iv) Set alpha (level of significance)

v) Choose a test statistic

vi) Compute the test statistic

vii) Find out the p-value

viii) Interpret the p-value

ix) Compute power of the test, if required.

Computing and interpreting p

When the data are subjected to significance testing, the resulting value is called statistic. This can be Z-, t-, chi-square statistic, , F-, etc. depending on the test used. This statistic is used to find out the p value available from tables (statistics software can automatically calculate the p value). If the p value is less then the cut off value (level of significance, i.e , alpha), it is considered that the difference between the groups is statistically significant. When p is <0.05, it indicates that the probability of obtaining the difference (between groups) purely by chance (when there is no difference) is less than 5%. If p>0.05, the difference is considered statistically non-significant and it is concluded that there is no difference between the groups or the difference is not detected.

Non- significant result can be due to two reasons:

1.There is really no difference between the groups.

2 The study is not powerful enough to detect the difference.

Hence, one should calculate the power to conclude whether there is no difference or the power is inadequate. If the power is inadequate (<80%), The conclusion is “the study did not detect the difference “rather than “there is no difference between groups”.

Statistical Terms

2011-03-09T07:07:00.003+05:30

Acceptance Error, Beta Error, Type II Error - An error made by wrongly accepting the null hypothesis when the null is really false.

Acceptance Region - Opposite of the Rejection Region. It is better to call this the "Fail to Reject Region." In the case of a two-tailed hypothesis t-test, it is shaded in light blue on the picture below. If the test statistic falls between -tcritical and tcritical then we fail to reject the null hypothesis.

Adjusted R-Squared, R-Squared Adjusted - A version of R-Squared that has been adjusted for the number of predictors in the model. R-Squared tends to over estimate the strength of the association especially if the model has more than one independent variable.

Alpha [A, a ], Chosen Significance Level - The maximum amount of chance a statistician is willing to take that they will do not accept a null hypothesis that is true (Type I Error).

Alpha Error, Type I Error - An error made by wrongly rejecting the null hypothesis when the null is really true.

Alternative Hypothesis, Research Hypothesis - An hypothesis that does not conform to the one being tested, usually the opposite of the null hypothesis. Symbolized.

Analysis of Variance (ANOVA) - A test of differences between mean scores of two or more groups with one or more variables.

Approximation Curve, Curve Fitting - the general method for using a line or curve to estimate the relationship between two associated numerical variables.

Autocorrelation - This occurs when later variables in a time series are correlated with earlier variables.

Backward Elimination - A method of determining regression equation that starts with a regression equation that includes all independent variables and then remover variables that are not useful one at a time

Best Subsets Regression - A method of determining the regression equation used with statistical computer applications that allows the user to run multiple regression models using a specified number of independent variables. The computer will sort through all of the models and display the "best" subsets of all the models that were run. "Best" is typically identified by the highest value of R-squared. Other diagnostic statistics such as R-square adjusted and Cp are also displayed to help the user determine their best choice of a model.

Bell-Shaped Curve - A symmetrical curve. Looks like the cross-section of a bell.

Best Fit, Goodness of Fit - A model that is the best model for the given data.

Beta Error, Acceptance Error, Type II Error - An error made by wrongly accepting the null hypothesis when the null is really false.

Bivariate Association/ Relationship - The relationship between two variables only.

Cp Statistic - Cp measures the differences of a fitted regression model from a true model, along with the random error. When a regression model with p independent variables contains only random differences from a true model, the average value of Cp is (p+1), the number of parameters. Thus, in evaluating many alternative regression models, our goal is to find models whose Cp is close to or below (p+1).

Cook’s Distance: Cook’s distance combines leverages and studentized residuals into one overall measure of how unusual the predictor values and response are for each observation. Large values signify unusual observations. Geometrically, Cook’s distance is a measure of the distance between coefficients calculated with and without the ith observation. Cook and Weisberg suggest checking observations with Cook’s distance > F (.50, p, n-p), where F is a value from an F-distribution.

Coefficient of Determination – In general the coefficient of determination measures the amount of variation of the response variable that is explained by the predictor variable(s). The coefficient of simple determination is denoted by r-squared and the coefficient of multiple determination is denoted by R-squared.

Coefficient of Variation – The coefficient of variation, in regression, is the standard deviation of the predictor variable divided by the mean of the predictor variable. If this value is small, your variation in the y-values (predictor values) is nearly constant. This implies that the data are ill-conditioned.

Confidence Bands (Upper & Lower) - This is the range of the responses that can be expected for all of the appropriate inputs of X's. The upper confidence band is the highest value that the ÿh value is predicted to be. The lower confidence band is the lowest value predicted that ÿh could be.

Confidence Level - This is the amount of error allowed for the model (given as a percent or a).

Confidence Intervals - A range of values to estimate a value of a population parameter. Associated with the range of values is also the amount of confidence the researcher has in the estimate. For example, we might estimate the cost of a new space vehicle to be 35 million dollars. Assume that the confidence level is 95% and the margin of error is 5 million dollars. We say that we are 95% confident that the cost is between 30 and 40 million dollars.

Confidence Interval Bounds, Upper and Lower - The lower endpoint on a confidence interval is called the lower bound or lower limit. The lower bound is the point estimate minus the margin of error. The upper bound is the point estimate plus the margin of error.

Correlation - The amount of association between two or more items. In these tutorials, correlation will refer to the amount of association between two or more numerical variables.

Correlation Coefficients, Pearson’s Sample Correlation Coefficient, r - Measures the strength of linear association between two numerical variables.

Correlation Matrix - A table that shows all pairs of correlations coefficients for a set of variables.

Correlation Ratio- A kind of correlation used when the relation between two variables is assumed to be curvilinear (i.e. not linear).

Curve Fitting, Approximation Curve - the general method for using a line or curve to estimate the relationship between two associated numerical variables.

Degrees of Freedom, df, - The number of values that can vary independently of one another. For example, if you have a sample of size n that is used to evaluate one parameter, then there are n-1 degrees of freedom.

Dependent Variable, Response Variable, Output Variable - The variable in correlation or regression that cannot be controlled or manipulated. The variable that "depends" on the values of one or more variables. In math, y frequently represents the dependent variable.

DFITS, DFFITS: Combines leverage and studentized residual (deleted t residuals) into one overall measure of how unusual an observation is. DFITS is the difference between the fitted values calculated with and without the ith observation, and scaled by stdev (Ŷi). Belseley, Kuh, and Welsch suggest that observations with DFITS >2Ö(p/n) should be considered as unusual

Dummy Variable, Indicator Variable - A variable used to code the categories of a measurement. Usually, 1 indicates the presence of an attribute and 0 indicates the absence of an attribute. Example: If the measurement variable is cost of space flight vehicle then the vehicle might be manned or unmanned. Let the dummy variable be 1 if the vehicle is manned and 2 if it is unmanned. Note: Dummy variable coding can be used for more than 2 categories.

Efficiency, Efficient Estimator - It is a measure of the variance of an estimate's sampling distribution; the smaller the variance, the better the estimator.

Error - In general, the error difference in the observed and estimated value of a parameter.

Errors, Residuals - In regression analysis, the error is the difference in the observed Y values and the predicted Y values that occur from using the regression model. See the graph below.

Error, Specification (Specification error) - A mistake made when specifying which model to use in the regression analysis. A common specification error involves including a irrelevant variable and leaving out an important variable.

F (F test statistic) - This is the test statistic for whenever conducting an analysis of variance.

Fits, Fitted Values, Predicted Values - The Fits are the predicted values found by substituting the original values for the independent variable(s) into the regression equation. The name "fit" refers to how well the observed data matches the relationship specified in the model.

Forward Selection - A frequently available option of statistical software applications. A method of determining the regression equation by adding variables to the regression equation until the addition of new variables does not appear to be worthwhile.

F-test: An F-test is usually a ratio of two numbers, where each number estimates a variance. An F-test is used in the test of equality of two populations. An F-test is also used in analysis of variance, where it tests the hypothesis of equality of means for two or more groups. For instance, in an ANOVA test, the F statistic is usually a ratio of the Mean Square for the effect of interest and Mean Square Error. The F-statistic is very large when MS for the factor is much larger than the MS for error. In such cases, reject the null hypothesis that group means are equal. The p-value helps to determine statistical significance of the F-statistic.

General Linear Model (GLM) - A full range of methods used to study linear relations between one continuous dependent variable and one or more independent variables, whether continuous or categorical. “General” means the kind of variable is not specified. Examples include Regression and ANOVA.

Heteroscedasticity - Non constant error variance. Hetero = different; scedasticity = tendency to scatter.

Hierarchical Regression Analysis - A multiple regression analysis method in which the researcher, not a computer program, determines the order that the variables are entered into and removed from the regression equation. Perhaps the researcher has experience that leads him/her to believe certain variables should be included in the model and in what order.

Homoscedasticity - Constant error variance. Homo = same; scedasticity = tendency to scatter.

Hypothesis Testing - This is the common approach to determining the statistical significance of findings.

Independent Variable, Explanatory Variable, Predictor Variable, Input Variable - The variable in correlation or regression that can be controlled or manipulated. In math, x frequently represents the independent variable.

Influential Observation - An observation that has a large effect on the regression equation. Note: Outliers and leverage points may be influential observations, but influential observations are usually outliers and leverage points.

Intercorrelation - Correlation between variables that are all independent (no dependent variables involved).

Least Squares Regression - Regression analysis method which minimizes the sum of the square of the error as the criterion to fit the data. This can refer to linear or curvilinear regression.

Leverages, Leverage Points - An extreme value in the independent (explanatory) variable(s). Compared with an outlier, which is an extreme value in the dependent (response) variable.

Linear Correlation- A relationship between the independent and dependent data, that whenever plotted forms a straight line.

Linear Regression - Typically when regression is used without qualification, the type of regression is assumed to be linear regression. This is the method of finding a linear model for the dependent variable based on the independent variable(s).

Mean Square Residual, Mean Square Error (MSE) - A measure of variability of the data around the regression line or surface.

Measurement Error (Error, Measurement) - inaccurate results due to flaw(s) in the measuring instrument.

Multicollinearity, Collinearity - The case when two or more independent variables are highly correlated. The occurrence of multicollinearity can cause difficulties in multiple regression. If the independent variables are interrelated, then it may be difficult or impossible to find the specific effect of only one independent variable.

Multiple Correlation Coefficient, R - A measure of the amount of correlation between more than two variables. As in multiple regression, one variable is the dependent variable and the others are independent variables. The positive square root of R-squared.

Multiple Correlation - Correlation with one dependent variable and two or more independent variables. Measures the combined influences of the independent variables on the dependent. gives the proportion of the variance in the dependent variable that can be explained by the action of all the independent variables taken together.

Multiple Correlation Matrices - A table of correlation coefficients that shows all pairs of correlations of all the parameters with in the sample.

Multiple Correlation Plots - A collection of scatterplots showing the relationship between the variables of interest.

Multiple R - That is the name MS Excel uses for the Multiple Correlation Coefficient, R.

Multiple Regression, Multiple Linear Regression - A method of regression analysis that uses more than one independent (explanatory) variable(s) to predict a single dependent (response) variable. Note: The coefficients for any particular explanatory variable is an estimate of the effect that variable has on the response variable while holding constant the effects of the other predictor variables. “Multiple” means two or more independent variables. Unless specified otherwise, “Multiple Regression” generally refers to “Linear” Multiple Regression.

Multiple Regression Analysis (MRA) - Statistical methods for evaluation the effects of more than one independent variable on one dependent variable.

Negative Correlation- This occurs whenever the independent variable increases and the dependent variable decreases. This is also called a negative relationship.

Nonadditivity - A statement used to describe a relation when the addition of the separate effects do not add up to the total effect.

Nonlinearity - The events are not the same as their causes.

Nonlinear Relationship - A relationship between two variables for which the points in the corresponding scatterplot do not fall in approximately a straight line. Nonlinearity may occur because there is not a defined relationship between the variables as in the first figure below, or because there is a specific curvilinear relationship. See the parabolic relationship shown in the second graph below.

Normality Plot, Normal Probability Plot - A graphical representation of a data set used to determine if the sample represents an approximately normal population. A graph from Minitab is shown below. The sample data is on the x-axis and the probability of the occurrence of that value assuming a normal distribution is on the y-axis. If the resulting graph is approximately a straight line, then the distribution is approximately normal. There are statistical hypothesis tests for normality as well.

Null Hypothesis, - This is the hypothesis that two or more variables are not related and the researcher wants to reject.

Outlier - An extreme value in the dependent (response) variable. Compared with a leverage point, which is an extreme value in the independent (explanatory) variables.

Partial Correlation - Correlation between two variables given that the linear effect of one or more other variables has been controlled. Example. r12.3 is the correlation of variables one and two given that variable three has been controlled.

Partial Correlation Coefficients - This is the square root of a coefficient of partial determination. It is given the same sign as that of the corresponding regression coefficient in the fitted regression function.

Partial Determination Coefficients- This measures the marginal contribution of one X variable when all others are already included in the model. In contrast, the coefficient of multiple determination, , measures the proportion reduction in the variation of Y achieved by the introduction of the entire set of X variables considered in the model.

Partial Regression Coefficient, Partials - In a multiple regression equation, the coefficients of the independent variables are called partial regression coefficients because each coefficient tells only how the dependent variable varies with the selected independent variable.

Pearson’s Sample Correlation Coefficient, r - Measures the strength of linear association between two numerical variables.

Population - A group of people that one whishes to describe or generalize about.

Predictor Variable, Independent Variable, Explanatory Variable, Input Variable - The variable in correlation or regression that can be controlled or manipulated. In math, x frequently represents the independent variable.

Prediction Equation - An equation that predicts the value of one variable on the basis of knowing the value of one or more variables. Note: Formally prediction equation is a regression equation that does not include an error term.

Prediction Interval - In regression analysis, a range of values that estimate the value of the dependent variable for given values of one or more independent variables. Comparing prediction intervals with confidence intervals: prediction intervals estimate a random value, while confidence intervals estimate population parameters.

Population Parameter, Parameter - A measurement used to quantify a characteristic of the population. Even when the word population is not used with parameter, the term refers to the population. Example: The population mean is a measure of central tendency of the population. The population parametric is usually unknown.

Proportional Reduction of Error (PRE) - A measure of association that calculates how much more you can reduce your error in the predication of y if you know x, then when you do not know x. Pearson’s r is not a PRE, but r-squared is a PRE.

Positive Correlation- This relationship occurs whenever the dependent variable increases as the independent variable increases

P-values, Observed Significance Level - The probability of making a Type I error. (i.e. given that the null is true, the probability of getting a data set like the one we have or one more extreme in the direction of the alternative.)

r, Correlation Coefficients, Pearson’s r - Measures the strength of linear association between two numerical variables.

R, Coefficient of Multiple Correlation - A measure of the amount of correlation between more than two variables. As in multiple regression, one variable is the dependent variable and the others are independent variables. The positive square root of R-squared.

r2 , r-squared (r-sq.), Coefficient of Simple Determination - The percent of the variance in the dependent variable that can be explained by of the independent variable.

R-squared, Coefficient of Multiple Determination - The percent of the variance in the dependent variable that can be explained by all of the independent variables taken together.

R-Squared Adjusted (R-sq. adj.), Adjusted R-Squared - A version of R-Squared that has been adjusted for the number of predictors in the model. R-Squared tends to over estimate the strength of the association especially if the model has more than one independent variable.

Range of Predictability, Region of Predictability - The range of independent variable(s) for which the regression model is considered to be a good predictor of the dependent variable. For example, if you want to predict the cost of a new space vehicle subsystem based on the weight, and all of the input data subsystem weights all range from 100 to 200 pounds. You could not expect the resulting model to provide good predictions for a subsystem that weighs 3000 pounds.

Regression Analysis, Statistical Regression, Regression - Methods of establishing an equation to explain or predict the variability of a dependent variable using information about one or more independent variables. The equation is often represented by a regression line, which is the straight line that comes closest to approximating a distribution of points in a scatter plot. When "regression" is used without any qualification it refers to “linear” regression.

Regression Artifact, Regression Effect - An artificial result due to statistical regression or regression toward the mean.

Regression Coefficient, Regression Weight - In a regression equation the number in front of an independent variable. For example, if the regression equation is Y = mx + b then m is the regression coefficient of the x-variable. The regression coefficient estimates the effect of the independent variable(s) on the dependent variable. (Compare with Partial Regression Coefficients)

Regression Constant - Unless specified otherwise, the regression constant is the intercept in the regression equation.

Regression Equation - An algebraic equation that models the relationship between two (or more) variables. If the equation is Y = a + bX + e, then Y is the dependent variable, X is the independent variable, b is the coefficient of X, and a is the intercept, and e is the error term (See Prediction Equation).

Regression Line, Trend Line - When the best fitting regression model is a straight line, that line is called a regression “line.” Ordinary Least Squares method is usually used for computing the regression line.

Regression Model - An equation used to describe the relationship between a continuous dependent variable, an independent variable or variables, and an error term.

Regression Plane - When the regression model has two independent variables, then a plane represents the relationship between the variables two-dimensional. Example: z = a + bx + cy

Regression SS (also SSR or SSregression) - The sum of squares that is explained by the regression equation. Analogous to between-groups sum of squares in analysis of variance.

Regression Toward the Mean - The type of bias described by Francis Galton, a 19th century researcher. A tendency for those who score high on any measure to get somewhat lower scores on a subsequent measure of the same thing- or, conversely, for someone who has scored very low on some measure to get a somewhat higher score the next time the same thing is measured. Knowing how much regression toward the mean there is for a particular pair of variables gives you a prediction. If there is very little regression, you can predict quite well. If there is a great deal of regression, you can predict poorly if at all.
Regression Weight, Regression Coefficient - In a regression equation the number in front of an independent variable. For example, if the regression equation is Y = mx + b then m is the regression coefficient of the x-variable. The regression coefficient estimates the effect of the independent variable(s) on the dependent variable. (Compare with Partial Regression Coefficients)

Regress On - The dependent variable is “regressed on” the independent variable(s). We will regress the cost of the space vehicle (based) on the weight of the vehicle. If x predicts y, then y is regressed on x. (i.e. Regress the dependent variable on the independent. Response variable is regressed on the explanatory variable.)

Rejection Region - The area in the tail(s) of the sampling distribution for a test statistic. The figure below shows the Rejection Region in red.

Residuals, Errors - The amount of variation on the dependent variable not explained by the independent variable.

Response Variable- Same as the independent variable.

Robust - Said of a statistic that remains useful even when one or more of the assumptions is violated.

Sample - A group of subjects selected from a larger group, the population.

Sample Statistic, Statistic - A measurement used to quantify a characteristic of the Sample. Even when the word sample is not used, the term statistic refers to the sample. Example: The sample mean is a measure of central tendency of the sample (see Population Parametric).

Sampling Error, Sampling Variability, Random Error - The estimation of the expected differences between the sample statistic and the population parameter.

Sampling Distribution - It is all possible values of a statistic and their probabilities of occurring for a sample of a particular size.

Scaling - expresses the centered observation in the units of the standard deviation of the observations.

Scatter Diagram, Scattergram, Scatter Plot - The pattern of points due to plotting two variables on a graph.

Significance - The degree to which a researcher’s finding is meaningful or important.
Significance Level - there are two types of significance levels, the observed significance level (alpha) and the chosen significance level (p-value). The lower the probability the greater the statistical significance, called alpha level.

Simple Linear Regression - A form of regression analysis, which has only one independent variable.

Slope - The rate at which the line or curve rises or falls when covering a given horizontal distance.

Spearman Correlation Coefficient (rho), Rank-Difference Correlation, rs. - A statistical measure of the amount of monotonic relationship between two variables that are arranged in rank order.

Specification error (Error, Specification) - A mistake made when specifying which model to use in the regression analysis. A common specification error involves including a irrelevant variable and leaving out an important variable.

Standard Deviation - A statistic that shows the square root of the squared distance that the data points are from the mean.

Standardized Measure of Scale - Any statistic that allows comparisons between things measured on different scales. Example: percent, standard deviations and z-scores

Standardized Regression Coefficient - Regression Coefficients which have been standardized in order to better make comparisons between the regression coefficients. This is particularly helpful when different independent variables have different units.
Standardized Regression Model - This is the regression model used after centering and scaling of the dependent variable and independent variables.

Standardized residuals - Standardized residuals are of the form (residual) / (square root of the Mean Square Error). Standardized residuals have variance 1. If the standardized residual is larger than 2, then it is usually considered large.

Standard Error, Standard Error of the Regression, Standard Error of the Mean, Standard Error of the Estimate - In regression the standard error of the estimate is the standard deviation of the observed y-values about the predicted y-values. In general, the standard error is a measure of sampling error. Standard error refers to error in estimates resulting from random fluctuations in samples. The standard error is the standard deviation of the sampling distribution of a statistic. Typically the smaller the standard error, the better the sample statistic estimates of the population parameter. As N goes up, so does standard error.

Statistical Significance - Statistical significance does not necessarily mean that the result is clinically or practically important. For example, a clinical trial might result is a statistically significant finding (at the 5% level) that shows the difference in the average cholesterol rating for people taking drug A is lower than that of those taking drug B. However, drug A may only lower the cholesterol by 2 units more than drug B which is probably not a difference that is clinically important to the people taking the drug. Note: Large sample sizes can lead to results that are statistically significant that would otherwise be considered inconsequential.

Stepwise Regression - A method of regression analysis where independent variables are added and removed in order to find the best model. Stepwise regression combines the methods of backward elimination and forward selection.

Strength of Association, Strength of Effect Index - The degree of relationship between two (or more) variables. One example is R-squared, which measures the proportion of variability in a dependent variable explained by the independent variable(s).

Studentized Residuals: The studentized residual has the form of error/standard deviation of the error. Studentized residuals have constant variance when the model is appropraite.

Transformations - This is a method of changing all the values of a variable by using some mathematical operation.

Unbiased Estimator - A sample statistic that is free from systemic bias.

Variance Inflation Factor (VIF) - A statistics used to measuring the possible collinearity of the explanatory variables.

Weighted Least Squares - A method of regression used to take into account the non constant variance. The variables are multiples by a particular number (weights). It is typical to choose weights that are the inverse of the pure error variance in the response. (Minitab, page 2-7.) This choice gives large variances relatively small weights and visa versa.

Y-intercept - is the point where a regressin line intersects the y axis.

Probability Applied to Landing Page Testing

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So how does probability apply to landing page optimization?

The random variables are the visits to your site from the traffic sources that you have selected for the test. The audience itself may be subject to sampling bias. You are counting whether or not the conversion happened as a result of the visit. You are assuming that there is some underlying and fixed probability of the conversion happening, and that the only other possible outcome is that the conversion does not happen (that is, a visit is a Bernoulli random variable that can result in conversion, or not).

As an example, let's assume that the actual conversion rate for a landing page is 2%. Hence there is a larger chance that it will not convert (98%) for any particular visitor. As you can see, the sum of the two possible outcome probabilities exactly equals 1 (2% + 98% = 100%) as required.

The stochastic process is the flow of visitors from the traffic sources used for the test. Key assumptions about the process are that the behavior of the visitors does not change over time, and that the population from which visitors are drawn remains the same. Unfortunately, both of these are routinely violated to a greater or lesser extent in the real world. The behavior of visitors changes due to seasonal factors, or with changing sophistication and knowledge levels about your products or industry. The population itself changes based on your current marketing mix. Most businesses are constantly adjusting and tweaking their traffic sources (e.g., by changing PPC bid prices and the resulting keyword mix that their audience arrives from). The result is that your time series, which is supposed to return a steady stream of yes or no answers (based on a fixed probability of a conversion), actually has a changing probability of conversion. In mathematical terms, your time series is nonstationary and changes its behavior over time.

The independence of the random variables in the stochastic process is also a critical theoretical requirement. However, the behavior on each visit is not necessarily independent. A person may come back to your landing page a number of times, and their current behavior would obviously be influenced by their previous visits. You might also have a bug or an overload condition where the actions of some users influence the actions that other users can take. For this reason it is best to use a fresh stream of visitors (with a minimal percentage of repeat visitors if possible) for your landing page test audience. Repeat visitors are by definition biased because they have voluntarily chosen to return to your site, and are not seeing it for the first time at random. This is also a reason to avoid using landing page testing with an audience consisting of your in-house e-mail list. The people on the list are biased because they have self-selected to receive ongoing messages from you, and because they have already been exposed to previous communications.

The event itself can also be more complicated than the simple did-the-visitor-convert determination. In an e-commerce catalog, it is important to know not only whether a sale happened, but also its value. If you were to tune only for higher conversion rate, you could achieve that by pushing low-margin and low-cost products that people are more likely to buy. But this would not necessarily result in the highest profits.

Statistical Methods

Landing page testing is a form of experimental study. The environment that you are changing is the design of your landing page. The outcome that you are measuring is typically the conversion rate. Landing page testing and tuning is usually done in parallel, and not sequentially. This means that you should split your available traffic and randomly alternate the version of your landing page shown to each new visitor. A portion of your test traffic should always see the original version of the page. This will eliminate many of the problems with sequential testing.

Observational studies, by contrast, do not involve any manipulation or changes to the environment in question. You simply gather the data and then analyze it for any interesting correlations between your independent and dependent variables.

For example, you may be running PPC marketing programs on two different search engines. You collect data for a month on the total number of clicks from each campaign and the resulting number of conversions. You can then see if the conversion rate between the two traffic sources is truly different or possibly due to chance.

Descriptive statistics only summarize or describe the data that you have observed. They do not tell you anything about the meaning or implications of your observations. Proper hypothesis testing must be done to see if differences in your data are likely to be due to random chance or are truly significant.

Have I Found Something Better?

Landing page optimization is based on statistics, and statistics is based in turn on probability theory. And probability theory is concerned with the study of random events. But a lot of people might object that the behavior of your landing page visitors is not "random." Your visitors are not as simple as the roll of a die. They visit your landing page for a reason, and act (or fail to act) based on their own internal motivations.

So what does probability mean in this context? Let's conduct a little thought experiment.

I have flipped the coin and covered up the result after catching it in my hand. Now imagine if I peeked at the coin without letting you see it. What would you estimate the probability of it coming up heads to be? Still 50%, right? How about me? I would no longer agree with you. Having seen the outcome of the flip event I would declare that the probability of coming up heads is either zero or 100% (depending on what I have seen).

How can we experience the same event and come to two different conclusions? Who is correct? The answer is—both of us. We are basing our answers on different available information. Let's look at this in the context of the simplest type of landing page optimization. Let's assume that you have a constant flow of visitors to your landing page from a steady and unchanging traffic source. You decide to test two versions of your page design, and split your traffic evenly and randomly between them.

In statistical terminology, you have two stochastic processes (experiences with your landing pages), with their own random variables (visitors drawn from the same population), and their own measurable binary events (either visitors convert or they do not). The true probability of conversion for each page is not known, but must be between zero and one. This true probability of conversion is what we call the conversion rate and we assume that it is fixed.

From the law of large numbers you know as you sample a very large number of visitors, the measured conversion rate will approach the true probability of conversion. From the Central Limit Theorem you also know that the chances of the actual value falling within three standard deviations of your observed mean are very high (99.7%), and that the width of the normal distribution will continue to narrow (depending only on the amount of data that you have collected). Basically, measured conversion rates will wander within ever narrower ranges as they get closer and closer to their true respective conversion rates. By seeing the amount of overlap between the two bell curves representing the normal distributions of the conversion rate, you can determine the likelihood of one version of the page being better than the other.

One of the most common questions in inferential statistics is to see if two samples are really different or if they could have been drawn from the same underlying population as a result of random chance alone. You can compare the average performance between two groups by using a t-test computation. In landing page testing, this kind of analysis would allow you to compare the difference in conversion rate between two versions of your site design. Let's suppose that your new version had a higher conversion rate than the original. The t-test would tell you if this difference was likely due to random chance or if the two were actually different.

There is a whole family of related t-test formulas based on the circumstances. The appropriate one for head-to-head landing page optimization tests is the unpaired one-tailed equal-variance t-test. The test produces a single number as its output. The higher this number is, the higher the statistical certainty that the two outcomes being measured are truly different.

Collecting Insufficient Data

Early in an experiment when you have only collected a relatively small amount of data, the measured conversion rates may fluctuate wildly. If the first visitor for one of the page designs happens to convert, for instance, your measured conversion rate is 100%. It is tempting to draw conclusions during this early period, but doing so commonly leads to error. Just as you would not conclude a coin could never come up tails after seeing it come up heads just three times, you should not pick a page design before collecting enough data.

The laws of probability only guarantee the accuracy and stability of results for very large sample sizes. For smaller sample sizes, a lot of slop and uncertainty remain.

The way to deal with this is to decide on your desired confidence level ahead of time. How sure do you want to be in your answer—90%, 95%, 99%, even higher? This completely depends on your business goals and the consequences of being wrong. If a lot of money is involved, you should probably insist on higher confidence levels.

Let's consider the simplest example. You are trying to decide whether version A or B is best. You have split your traffic equally to test both options and have gotten 90 conversions on A, and 100 conversions on B. Is B really better than A? Many people would answer yes since 100 is obviously higher than 90. But the statistical reality is not so clear-cut.

Confidence in your answer can be expressed by means of a Z-score, which is easy to calculate in cases like this. The Z-score tells you how many standard deviations away from the observed mean your data is. Z=1 means that you are 67% sure of your answer, Z=2 means 95.28% sure, and Z=3 means 99.74% sure.

Pick an appropriate confidence level, and then wait to collect enough data to reach it.

Let's pick a 95% confidence level for our earlier example. This means that you want to be right 19 out of 20 times. So you will need to collect enough data to get a Z-score of 2 or more.

The calculation of the Z-score depends on the standard deviation ([.sigma]). For conversion rates that are less than 30%, this formula is fairly accurate:

In our example for B, the standard deviation would be 10

So we are 67% sure (Z=1) that the real value of B is between 90 and 110 (100 plus or minus 10). In other words, there is a one out of three chance that A is actually bigger than the lower end of the estimated range, and we may just be seeing a lucky streak for B.

Similarly at our current data amounts we are 95% sure (Z=2) that the real value of B is between 80 and 120 (100 plus or minus 20). So there is a good chance that the 90 conversions on A are actually better than the bottom end estimate of 80 for B.

Confidence levels are often illustrated with a graph. The error bars on the quantity being measured represent the range of possible values (the confidence interval) that would be including results within the selected confidence level. Figure 1: shows 95% confidence error bars (represented by the dashed lines) for our example. As you can see, the bottom of B's error bars is higher than the top of A's error bars. This implies that A might actually be higher than B, despite B's current streak of good luck in the current sample.

Figure 1: Confidence error bars (little data)

If we wanted to be 95% sure that B is better than A, we would need to collect much more data. In our example, this level of confidence would be reached when A had 1,350 conversions and B had 1,500 conversions. Note that even though the ratio between A and B remains the same, the standard deviations have gotten much smaller, thus raising the Z-score. As you can see from Figure 2, the confidence error bars have now "uncrossed," so you can be 95% confident that B actually is better than A.

Figure 2: Confidence error bars (more data)

Testing of Statistical Hypothesis

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Understanding the Results

The null hypothesis in probability and statistics is the starting assumption that nothing other than random chance is operating to create the observed effect that you see in a particular set of data. Basically it assumes that the measured effects are the same across the independent conditions being tested. There are no differences or relationships between these independent variables and the dependent outcomes—equal until proven otherwise.
The null hypothesis is rejected if your data set is unlikely to have been produced by chance. The significance of the results is described by the confidence level that was defined by the test (as described by the acceptable error "alpha-level"). For example, it is harder to reject the null hypothesis at 99% confidence (alpha 0.01) than at 95% confidence (alpha 0.05).
Even if the null hypothesis is rejected at a certain confidence level, no alternative hypothesis is proven thereby. The only conclusion you can draw is that some effect is going on. But you do not know its cause. If the experiment was designed properly, the only things that changed were the experimental conditions. So it is logical to attribute a causal effect to them.
What if the null hypothesis is not rejected? This simply means that you did not find any statistically significant differences. That is not the same as stating that there was no difference. Remember, accepting the null hypothesis merely means that the observed differences might have been due simply to random chance, not that they must have been.
Some concepts involved in testing of hypothesis.

In applied investigations or in experimental research, one may wish to estimate the yield of a new hybrid line of corn, but ultimate purpose will involve some use of this estimate. One may wish, for example, to compare the yield of new line with that of a standard line and perhaps recommend that the new line replaces the standard line if it appears superior. This is the common situation in research. One may wish to determine whether a new method of sealing light bulbs will increase the life of the bulbs, whether a new germicide is more effective in treating a certain infection than a standard germicide, whether one method of preserving foods is better than the other so far as the retention of vitamin is concerned, which one among the six available varieties of any crop is best in terms of yield per hectare.

Using the light bulb example as an illustration, let us suppose that the average life of bulbs made under a standard manufacturing procedure is about 1400 hours. It is desired to test a new procedure for manufacturing the bulbs. Here, we are dealing with two populations of light bulbs: those made by the standard process and those made by the proposed process. From the past investigations, based on sample tests it is known that the mean of the first population is 1400 hours. The question is whether the mean of the second population is greater than or less than 1400 hours? This we have to decide on the basis of observations taken from a sample of bulbs made by second process.

In making comparisons of above type, one cannot rely on the mere numerical magnitudes of the index of comparison such as mean, variance, etc. This is because each group is represented only by a sample of observations and if another sample were drawn, the numerical value would change. This variation between samples from the same population can at best be reduced in a well designed experiment but can never be eliminated. One is forced to draw inference in the presence of the sampling fluctuations which affect the observed differences between the groups, clouding the real differences. Hence, we have to devise some statistical procedure, which can test whether those difference are due to chance factors or really due to treatment.

The tests of hypothesis are such statistical procedures which enable us to decide whether the differences are attributed to chance or fluctuations of sampling.

Sample space: The set of all possible outcomes of an experiment is called sample space. It is denoted by S. For example in an experiment of tossing two coins simultaneously, the sample space is S = {HH, HT, TH, TT}; where ‘H’ denotes the head and ‘T’ denotes the tail outcomes. In testing of hypothesis, we are concerned with drawing inferences about the population based on a random sample. Let there are ‘N’ units in a population and we have to draw sample of size ‘n’. Then the set of all possible samples of size 'n' is the sample space and any sample x=(x₁, x₂,…,x_n) is the point of the sample space.

Parameter: A function of population values is known as parameter For example, population mean (m) and population variance(σ²).

Statistic: A function of sample values say, (x₁, x₂,…,x_n) is called a statistic. For example, sample mean
(),sample variance (s²), where

A statistic does not involve any unknown parameter.

Statistical Hypothesis: A Statistical Hypothesis is an assertion or conjecture (tentative conclusion) either about the form or about the parameter of the distribution. For example

i) The normal distribution has mean 20.

ii) The distribution of process is Poisson.

iii) Effective life of a bulb is 1400 hours.

iv) A given detergent cleans better than any washing soap.

In a statistical hypothesis, all the parameters of a distribution may be specified completely or partly. A statistical hypothesis in which all the parameters of a distribution are completely specified is called simple hypothesis, otherwise, it is known as composite hypothesis. For example, in case of normal population, the hypothesis

i) Mean(μ) = 20, variance(σ²) = 5(Simple hypothesis)

ii) μ = 20, σ²>1 (composite hypothesis)

iii) μ = 20 (composite hypothesis)
Null Hypothesis: The statistical hypothesis under sample study is called null hypothesis. It is usually that the observations are the result purely of chance. It is usually denoted by H₀.

Alternative Hypothesis: In respect of every null hypothesis, it is desirable to state, what is called an alternative hypothesis. It is complementary to the null hypothesis. Or "The desirable attitude of the statistician about the hypothesis is termed as alternative hypothesis". It is taken usually that the observations are the result of real effect plus chance variation. It is usually denoted by H₁. For example, if one wishes to compare the yield per hectare of a new line with that of standard line, then, the null hypothesis:

H₀: Yield per hectare of new line (μ₁)=Yield per hectare of standard line (μ₂)

The alternative hypothesis corresponding to H₀, can be the following:

i) H₁_:m₁> m₂ (Right tail alternative)

ii) H₁_:m₁< m₂(left tail alternative)

iii) H₁_:µ₁≠ µ₂ (Two tailed alternative)

(i) and (ii) are called one tailed test and (ii) is a two tailed test. Whether one sets up a one tailed test or a two-tailed test depends upon the conclusion to be drawn if H₀ is rejected. The location of the critical region will be decided only after H₀ has been stated. For example in testing a new drug, one sets up the hypothesis that it is no better than similar drugs now on the market and tests this hypothesis against the alternative hypothesis that the new drug is superior. Such an alternative hypothesis will result in a one tailed test (right tail alternative).

If we wish to compare a new teaching technique with the conventional classroom procedure, the alternative hypothesis should allow for the new approach to be either inferior or superior to the conventional procedure. Hence the test is two-tailed.

Critical Region: It is region of rejecting null hypothesis when it is true if sample point belongs to it. Hence ‘C’ is the critical region.

Suppose that if the test is based on a sample of size 2, then the outcome set or sample space is the first quadrant in a two dimensional space and a test criterion will enable us to separate our outcome set into two complementary subsets, C and C_bar If the sample point falls in the subset C, H₀is rejected, otherwise, H₀is accepted.

The terms acceptance and rejection are, however, not to be taken in their literal senses. Thus acceptance of H₀ does not mean that H₀ has been proved true. It means only that so far as the given observations are concerned, there is no evidence to believe otherwise. Similarly, rejection of H₀ does not disprove the hypothesis; if merely means that H₀ does not look plausible in the light of given observations.

It is now known that, in order to establish the null hypothesis, we have to study the sample instead of entire population. Hence, whatever, decision rule we may employ, there is every chance of committing errors in the decision for rejecting or accepting the hypothesis. Four possible situations, which can arise in any test procedure, are given in the following table.

From the table, it is clear that the errors committed in making decisions are of two types.

Type I error: Reject H₀ when H₀ is true.

Type II error: Accept (does not reject) H₀when H₀ is false.

For example, a judge, who has to decide whether the person has committed the crime. The statistical hypothesis in this case is:

H₀: Person is innocent; H₁: Person is criminal.

In this situation, two types of errors which the judge may commit are:

Type I error: Innocent person is found guilty and punished.

Type II error: A guilty person is set free.

Since, it is more serious to punish an innocent than to set free a criminal. Therefore, Type I error is more serious than the Type II error.

Probabilities of the errors:

Probability of Type I error = P (Reject H₀ / H₀is true) = α

Probability of Type II error = P (Accept H₀ / H₁ is true) = β

In quality control terminology, Type I error amounts to rejecting a lot when it is good and Type II error may be regarded as accepting a lot when it is bad.

P (Reject a lot when it is good) = α (producer’s risk)

P (Accept a lot when it is bad) = β (consumer’s risk)

Level of significance: the probability of Type I error (α) is called the level of significance. It is also known as size of the critical region.

Although 5% level of significance has been taken as a rough line demarcation in which deviations due to sampling fluctuations alone will be interpreted as real ones in 5% of the cases. Hence, yet the inferences about the population based on samples are subject to some degree of uncertainty. It is not possible to remove this uncertainty completely, but it can be reduced by choosing the level of significance still lesser like 1%, in which chances of interpreting the deviations due to sampling fluctuations, as real one, is only one in 100 cases.

Power function of a Test: The probability of rejecting H₀when H₁is true is called power function of the test.

Power function = P(Reject H₀ / H₁is true)

= 1-P(Accept H₀ / H₁ is true)

= 1- β.

The value of this function plays the same role in hypothesis testing as the mean square error plays in estimation. It is usually used as our standard in assessing the goodness of a test or in comparing two tests of same size. The value of this function for a particular point is called the power of the test.

In testing of hypothesis, the ideal procedure is to minimize the probabilities of both type of errors. Unfortunately, for a fixed sample size n, both the probabilities cannot be controlled simultaneously. A test which minimizes the one type of error, maximizes the other type of error. For example, if there is a critical region which makes the probability of Type I error zero, will be of the form always accept H₀ and that the probability of Type II error will be one. It is, therefore, desirable to fix the probability of one of the error and choose a critical region which minimizes the probability of the other error. As Type I error is consider to be more serious than Type II error. Therefore, we fix the probability of Type I error (α) and minimize the probability of Type II error (β) and hence, maximize the power function of test.

Steps in solving testing of hypothesis problem:

i) Explicit knowledge of the nature of the population distribution and the parameters of interest, i.e., about which the hypothesis is set up.

ii) Setting up of null hypothesis H₀ and alternative hypothesis H₁ in terms of the range of parameter values, each one embodies.

iii) The choice of a suitable test statistic, say, t=t(x₁, x₂,…,x_n)called the test statistic which will best reflect on H₀ and H₁.

iv) Partition the simple space (set of possible values of test statistic, t) into two disjoint and complementary subsets C and C_bar= A (say) and framing be test, such as

(a) Reject H₀ if value t ε C

(b) Accept H₀ if value t ε A.

After framing the above test, obtain the experimental sample observations, compute the test statistic and take action accordingly.

Degrees of Freedom (df)

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Degrees of Freedom denotes the number of samples that a statistician has the freedom to choose. Degrees of Freedom is based on a concept that one could not have exercised his /her freedom to select all the samples.

The concept can be explained by an analogy:

X+Y = 10 (1)

In the above equation you have freedom to choose a value for X or Y but not both because when you choose one, the other is fixed. If you choose 8 for X, then Y has to be 2. So the degree of freedom here is 1.

X+Y+Z = 15 (2)

In the formula (2), one can choose values for two variables but not all. You have freedom to choose 8 for X and 2 for Y, If so, then Z is fixed, So the degree of freedom is 2. df is calculated by subtracting 1 from the size of each group. The methods of df calculation may vary with the test used.

“Degrees of freedom” refers to the number of scores that are free to vary. It is required when one is working on estimates of population values (sample statistics). It is often abbreviated as df. It is the number of observations in the data collection that are free to vary after the sample statistics have been calculated. For example, in calculating a sample standard deviation, we must subtract the sample mean from each of the n data observations in order to get the deviations from the mean. But once we have completed the next to the last subtraction, the final deviation is automatically determined, since the deviations prior to squaring must sum to zero. Therefore, the last deviation is not free to vary; only n-1 are free to vary. As a rule of thumb, every estimate of a population value in a formula equals 1 degree of freedom.

The practical impact of using degrees of freedom is found when one considers small samples. Consider the t distribution, for example. With small samples it gets flatter, or another way of looking at it is that more area underneath the curve (region of rejection) is farther away from the mean. This makes using degrees of freedom a more conservative and accurate, since small samples tend to underestimate the population value.

Practical Example:

Consider a situation in which the scores that make up the distribution are unknown, and I tell you to guess at the first score in the distribution of 5 numbers. Your guesses will be wild because that score could be any number. The same is true for the second score, third score, and the fourth score. All of these scores have complete freedom to vary. But, if I tell you that the first four scores in the distribution are 3,4,5, and 6; and I tell you what the mean of the distribution (5, in this case), then the last score in this distribution has to be 7. In other words, if the mean is known, the missing score is determined by your knowledge of the other four. Therefore n-1 scores are free to vary. The (-1) is often called a “restriction.” Note that by making the numerator smaller, the standard deviation becomes larger. A larger standard deviation means that the sampling distribution is flatter; and flatter means more values are farther from the mean. It’s tougher to find significance.

In conclusion, when the statistical formula is concerned with description, degrees of freedom is n. When the formula is concerned with inference, some restrictions apply. The idea is to adjust for a small sample size’s tendency to underestimate the population parameter. As n gets larger, this becomes less of a problem because the distribution becomes less flat and more normal, but we still use the sample formula to calculate the statistic.

Web Analytics Terms

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Abandonment: is the measure when a visitor exits or leaves a conversion process on a website and does not return later during the session.

A/B Testing: A method of banner ads, emails and landing pages testing by which a baseline control sample is compared to a variety of single-variable test samples.

Accuracy: The ability of a measurement to match the actual value of the quantity being measured. Accuracy is the foundation upon which your web analytics should be built. If you can't trust that your data is accurate, you can't make confident decisions. In statistical terms, accuracy is the width of the confidence interval for a desired confidence level.

Acknowledgement Page: A page displayed after a visitor completes an action or transaction. For example, a thank-you or a receipt page. Acknowledgement pages are often important in Scenario Analysis, where it is an indicator of a completed scenario.

Acquisition: The process of gaining customers through the means of different marketing strategies. For the purposes of web analytics, it often refers specifically to the process of attracting visitors to a web site.

ACT: After-Click Tracking is the recording the activity path of a visitor to a site after they have clicked on an email link.

Actionable Data: Information that allows you to make a decision or can be made use of in any way.

Ad: A link that takes a visitor to a web site when clicked on, usually graphic or text.

Ad Click: A click on an advertisement on a website which takes a user to another site.

Ad Hoc Query: A non-standard inquiry posed to a database of information as the need arises.

Ad View: A web page that presents an ad. There may be more than one ad on an ad view. Once visitors have viewed an ad, they can click on it.

Affiliate Marketing: A method of promoting web businesses in which an affiliate is rewarded for providing customers. Compensation could be made based on a value for visits, subscriptions, leads, sales, and so on.

Aggregate Data: A summary of collected information which groups data together without individual-level statistics.

Algorithm: A mathematical formula used by search engines to determine which web sites in their database to present in search results, in which order. While search engine algorithms change regularly, primary on-page factors include keyword density and source code optimization. The primary off-page factor is link popularity.

API: Application Programming Interface is a system that a computer or application supplies in order to allow requests for service to be made of it by other computer programs. APIs allow data to be exchanged between computer programs, and a standard software API method includes Open Database Connectivity (ODBC).

ASP: Active Server Pages are a set of software components that run on a web server and let developers build dynamic web pages.

Attrition: The erosion of your customer base over time. The opposite of customer retention.

Authentication: The technique by which access to Internet or intranet resources requires the user to enter a username and password as identification.

Average Lifetime Value: The average of the lifetime value of a visitor or multiple visitors during the reporting period, where each visitor's lifetime value is the total monetary value of a visitor's past orders since visitor tracking began.

Bandwidth: Measure, in kilobytes of data transferred, of the traffic on a site.

Banner Ad: An advertisement embedded on a web page usually intended to drive traffic to a different website by linking to the advertiser's site. The Interactive Advertising Bureau (IAB) has created a standard set of banner ad sizes (Medium Rectangle, Rectangle, Leaderboard, Wide Skyscraper) into a set of guidelines called the Universal Ad Package).

Benchmark: A standard by which something can be measured or judged. For example, you benchmark your Key Performance Indicators to ensure everyone in your organization is measuring performance against the same goals.

Blog/Web Logs: A self-published, managed or maintained Web diary. Usually updated daily or weekly, blogs have historically been personal, but gained notoriety after the 2004 election as an influential media outlet. Companies now use blogs to extend their brand and improve their organic search visibility.

Bounce Rate: The percentage of entrances on a web page that result in an immediate exit from the web site.

Browser: A program used to locate and view HTML documents.

Business Intelligence: While some would claim it's an oxymoron, business intelligence refers to a category of software and tools designed to gather, store, analyze, and deliver data in a user-friendly format to help organizations make more informed business decisions. Software types include dashboarding, data mining, data warehouses, and other information systems.

Campaign Analysis: A mesaure that tracks activity originating from a marketing campaign, so you can compare your campaigns and evaluate their effectiveness.

Client: The browser used by a website visitor.

Client Error: An error that occurs because of an invalid request by the visitor's browser.

Cloaking: In terms of search engine marketing, this is the act of getting a search engine to record content for a URL that is different than what a searcher will ultimately see. It can be done in many technical ways. Several search engines have explicit rules against unapproved cloaking. Those violating these guidelines might find their pages penalized or banned from a search engine's index. As for approved cloaking, this generally only happens with search engines offering paid inclusion program.

Click Fraud: A type of internet crime that occurs in pay per click online advertising when a person, automated script, or computer program imitates a legitimate user of a web browser clicking on an ad, for the purpose of generating a charge per click without having actual interest in the target of the ad's link.

Content Management System (CMS): a software platform that aids in the management of content on a Web site.

Contextual Link Ads/Inventory: To supplement their business models, certain text-link advertising networks (like Google) have expanded their network distribution to include "contextual inventory". Most vendors of "search engine traffic" have expanded the definition of Search Engine Marketing to include this contextual inventory. Contextual or content inventory is generated when listings are displayed on pages of Web sites (usually not search engines), where the written content on the page indicates to the ad-server that the page is a good match to specific keywords and phrases. Often this matching method is validated by measuring the number of times a viewer clicks on the displayed ad. These ads typically do not perform as well as traditional text ads on search engines, but the lower cost justifies the expense.

Conversion: An action that signifies a completion of a specified activity. For many sites, a user converts if they buy a product, sing up for a newsletter, or download a file. The conversion rate is the percentage of visitors who do convert. Cookie deletion can have an impact on your conversion rate because if a cookie is being systematically deleted, repeat visitor rates will be under-counted and new visitor rates will be over-counted, thus skewing the conversion rate metric by which you analyze your site's overall effectiveness.

Conversion Funnel: The series of steps that move a visitor towards a specified conversion event, such as an order or registration signup. See also abandonment.

Conversion Rate: The relationship between visitors to a web site and actions considered to be a "conversion," such as a sale or request to receive more information. This metric is often expressed as a percentage.

Cookie: A text file that transmits information to a data collection facility via a 1x1 pixel GIF image request and includes a tracking ID that is used to identify returning visitors. Contrary to some industry speculation, cookies can not be used for malicious use such as privacy tapping. See also first and third-party cookies.

Cost-per-Click (CPC): System where an advertiser pays an agreed amount for each click someone makes on a link leading to their web site. Also known as PPC or paid listings.

Cost-per-Thousand (CPM): System where an advertiser pays an agreed amount for the number of times their ad is seen by a consumer, regardless of the consumer's subsequent action. This term is heavily used in print, broadcasting and direct marketing, as well as with online banner ad sales. CPM stands for "cost per thousand," since ad views are often sold in blocks of 1,000. The M in CPM is Latin for thousand.

Crawler/Spider/Robot: Component of search engine that indexes web sites automatically. A search engine's crawler (also called a spider or robot), copies web page source code into its index database and follows links to other web pages.

CTR: Click Through Rate. A click through rate is the rate at which visitors "click through" from one website page or property to the next. A good indication of an ad's effectiveness.

Customer Segment: A powerful aspect of relationship marketing in which you target sub-section or group of customers who share a specific trait or set of behaviors. See also demographics and psychographics.

Dashboard: A web analytics dashboard provides all of your critical metrics in one place to help you understand the health or performance of your business.

Data Warehouse: is a logical collection of information gathered from many different operational databases used to create business intelligence that supports business analysis activities and decision-making tasks, primarily, a record of an enterprise's past transactional and operational information, stored in a database designed to favor efficient data analysis and reporting.

Demographics: The physical characteristics of human populations and segments of populations, often used to identify consumer markets. Demographics can include information such as age, gender, marital status, education, and geographic location. See also psychographics.

Directories: A type of search engine where listings are gathered or reviewed by humans, rather than by search engine crawlers. In directories, web sites are often reviewed, summarized in about 25 words and placed in a particular category. The largest and most popular directory site is Yahoo!

Domain: An area in the Internet specified by a URL address. The top-level domain is at the end after the dot and the second-level domain comes before it, and shows where in the top-level domain the address can be found. For example in www.xyz.com, ".com" is the top-level domain and "xyz" is the second level domain.

Domain Name: The text name that corresponds to a numeric IP address of a computer on the Internet.

Doorway/Landing/Gateway/Bridge/Jump Pages: A web page created expressly in hopes of ranking well for a term in a search engine's organic/non-paid listings and which itself does not deliver much information to those viewing it. Instead, visitors will often see only some enticement on the doorway page leading them to other pages, or they may be seamlessly redirected to a real page within the existing web site. With cloaking, visitors may never see the doorway page at all. Several search engines have guidelines against doorway pages, though they are more commonly allowed in through paid inclusion programs.

E-commerce: The act of selling goods and services online via a standalone site or through an online auction center.

Email Bounce: The number of e-mails that were sent but never reach the intended receiver.

Entry Page: The first viewed page on a visitor's path through a site.

Exit Page: The last page viewed on a visitor's path through a site.

First Party Cookie: For most business models, first-party cookies are regarded as the most reliable method to measure visitor activity. Whereas a third-party cookie is usually set by an analytics vendor, (an entity with whom the user does not have a relationship), a first-party cookie is set by the business, an organization with whom the Web site visitor has specifically chosen to do business. Because of this relationship, first-party cookies are deemed more secure by the user. Also seecookies.

Frequency: The number of times a visitor has visited a site during a reporting period. Average Frequency is the average of frequencies of all the visitors during the reporting period. Frequency is a retention metric and is part of RFM (recency, frequency, monetary) analysis.

Hit: Any request from a file or a web-server. A single page likely contains multiple hits as multiple image and text files are downloaded from the web-server.

Home Page: The main page of a web site. The home page provides visitors with an overview and links to the rest of the site

Impression: The display of an online advertisement (usually a banner ad) to a web site visitor.

Index: The collection of information (contained in a large database) a search engine has that searchers can query against. With crawler-based search engines, the index is typically copies of all the web pages they have found from crawling the web. With human-powered directories, the index contains the summaries of all web sites that have been categorized.

Inbound/Back Link: A text or graphical hyperlink from one site to another. Google and other search engines' algorithms consider a site's popularity based on the quality and quantity of inbound links from relevant third party sites to help determine search positioning.

Internet: The Internet is the publicly accessible global system of interconnected computer networks that transmit data via a standardized Internet Protocol. See alsoWorld Wide Web.

JavaScript: A scripting language developed by Netscape. While it is often used for websites, it is also used to enable scripting access to objects embedded in other applications.

Keyword: Terms entered into the search field of a web search engine. See also organic search and PPC.

KPI: Key Performance Indicators. Key Performance Indicators are typically kept in dashboards and provide customers with an understanding of how the site is performing.

Latency: The average number of days between visits for a given visitor during a reporting period. For example, those who visit on average every seven days.

Link: On a web page, text or an image that has been coded to take a browser from one page to another or from one site to another.

Log File: A file created by a web or proxy server which contains all of the access information regarding the activity on that server.

Long Tail: First coined by Chris Anderson in an October 2004 Wired magazine article to describe the niche strategy of certain business such as Amazon.com or Netflix. In relation to search engine marketing (SEM) the Long Tail refers to the keyword phrases that are highly detailed and specific and may generate low volumes of searches and traffic, but add up to generate a majority of traffic for sites with deep content or product SKUs.

LTV: Life-Time Value. Life-Time Value is a metric used to describe the value a specific customer has over the life of their relationship with you.

Meta Search Engine: A search engine that gets listings from two or more other search engines, rather than through its own efforts.

Meta Tags: Information placed in a web page not intended for users to see but instead which typically passes information to search engine crawlers, browser software and some other applications.

Meta Description Tag: Allows page authors to say how they would like their pages described when listed by search engines. Not all search engines use the tag.

Meta Keywords Tag: Allows page authors to add text to a page to help with the search engine ranking process.
Meta Robots Tag: Allows page authors to keep their web pages from being indexed by search engines, especially helpful for those who cannot create robots.txt files. The Robots Exclusion page provides official details.

Metrics: Metrics are a system of parameters or ways of quantitative assessment of a process that is to be measured, along with the processes to carry out such measurement. Metrics define what is to be measured.

Mobile Search: An evolving branch of information retrieval services that is centered on the convergence of mobile platforms and mobile handsets or other mobile devices. The services allow users to find mobile content interactively on mobile websites, and mobile content shows a media shift toward mobile multimedia.

Multivariate Testing: A process by which more than one component of a website may be tested in a live environment. It can be thought of in simple terms as numerous split tests or A/B tests performed on one page at the same time.

Navigation: The act of moving from location to location within a web site, or between web sites, accomplished by clicking on links. Navigation can also refer to the overall structure of the links on the site, comprising the paths available to the visitor.

Non-referrals: Visitors who arrive at a site by typing a domain into an address bar, using a bookmark, or clicking on an emailed URL.

Online Reputation Management (ORM): The act of monitoring, addressing or mitigating undesirable search engine results or mentions in online media for a company or product. Techniques include generating new content and creating posts on existing content.

OpenSearch: A collection of technologies that allow publishing of search results in a format suitable for syndication and aggregation. It is a way for websites and search engines to publish search results in a standard and accessible format. Originally developed by Amazon and recently adopted by Yahoo!, OpenSearch relies on abstract-based microformats (dataRSS, eRDF, FOAF, GeoRSS, hCard, hEvent, hReview, hAtom, MediaRSS, RDFa, XFN, etc.) to integrate syndicated content into search results.

Opt-in: This permission-based email communication requires customers to verify the opt-in method before their e-mail addresses can be used to communicate with them.

Organic/Natural Listings: Listings that search engines do not sell (unlike paid listings). Instead, sites appear solely because a search engine has deemed it editorially important for them to be included, regardless of payment. Paid inclusion content is also often considered "organic" even though it is paid for. This is because that content usually appears intermixed with unpaid organic results.

Organic Search: A type of search in which web users find sites having unpaid listings, as opposed to using the pay-per-click advertisement listings displayed among the search results.

Outbound Links: Links on a particular web page leading to other web pages on a different domain.

Page: A document provided by the server, including HTML, scripts, and text files. Images, sound files and video are not considered pages. Documents are defined by the system administrator, but generally include all static content, such as complete html pages. Dynamic pages are created with variables and do not exist anywhere in a static form. Forms are scripted pages which get information from a visitor that gets passed back to the server.

Page Tag: A piece of JavaScript code embedded on a web page and executed by the browser when the page is viewed. See also log files.

Page View: is generally defined as a request to load a single page of a website. On the web, a page request would result from a web surfer clicking on a link on another page that points to the page in question. See also hit.

Parameters: These are located in the URL immediately after a question mark and followed by an equal sign and a return value, known as name=value.

Path: A path is the click pattern a visitor uses as they traverse through multiple pages.

PPC: Pay Per Click or paid search uses search keywords that cost a certain amount for each customer click on that term in order to get to your site. See also organic search.

Query: A question or inquiry used to find answers about certain metrics.

Query Parameter: An individual piece of a query string consisting of a parameter name and a value for the parameter.

Reach: The size of the audience reading, viewing, hearing, or interacting with a message in a given period of time. Reach can be understood as either an absolute number, or a fraction of a population.

Rear-View Mirror Metrics: Metrics that measure what has occurred. For example campaign response metrics are such metrics that tell you how a campaign performed.

Recency: The number of days since a visitor's most recent visit during a reporting period.

Referrals: The location that visitors come from, particularly the sites, search engines or directories. Relationship Marketing: Relationship marketing is a type of marketing that traces its roots to direct response marketing. It emphasizes building long-term relationships with customers rather than individual transactions. It requires understanding customer needs as they go through life cycles of interacting and purchasing from organizations, and requires that marketers accurately determine customer intent in order to provide them the right message at the right time.

RFM Analysis: Recency, Frequency, Monetary analysis.

ROI: Return on Investment

Robot: An automated process that performs mundane, repeatable tasks to provide information. Search engine robots or bots provide such functions, cataloging the internet for searchers to find information.

RSS: Really Simple Syndication is a type of web syndication used by news sites and weblogs which provides summaries of information with links to the complete content.

Sampling: In statistics, the selection of individual observations intended to yield knowledge about a population, especially for the purposes of statistical inference.

Search Engine: A search engine is a program that helps you find information on the web.

Search Engine: Any service generally designed to allow users to search the web or a specialized database of information. Web search engines generally have paid listings and organic listings. Organic listings typically come from crawling the web, though often human-powered directory listings are also optionally offered. Top tier search engines include Google, MSN, Teoma and Yahoo!

Search Engine Marketing (SEM): The act of marketing a web site via search engines, whether this be improving rank in organic listings (search engine optimization), purchasing paid listings (PPC management) or a combination of these and other search engine-related activities (i.e. affiliate programs, shopping feeds or link development).

Search Engine Optimization (SEO): The act of altering a web site so that it does well in the organic, crawler-based listings of search engines. In the past, has also been used as a term for any type of search engine marketing activity, though now the term search engine marketing is more commonly used as an umbrella term.

Search Terms: The words (or phrase) a searcher enters into a search engine's search box. Also used to refer to the terms a search engine marketer hopes a particular page will be found for. Also called keywords, query terms or query.

Segment: A grouping of customers, defined by website activity or other data, which can be used to target them effectively.

Social Media Marketing (SMM): A form of internet marketing which seeks to achieve branding and marketing communication goals through the participation in various social media networks (MySpace, Facebook, LinkedIn), social bookmarking (Digg, Stumbleupon), social media sharing (Flickr, YouTube), review/ratings sites (ePinions, BizRate), blogs, forums, news aggregators and virtual 3D networks (SecondLife, ActiveWorlds). Each social media site can be optimized to generate awareness or traffic.

Social Media Optimization (SMO): A set of methods for generating publicity through social media, online communities and community websites. Methods of SMO include adding RSS feeds, adding a "Digg This" button, blogging and incorporating third party community functionalities like Flickr photo slides and galleries or YouTube videos. Social media optimization is a form of search engine marketing.

Spam: Any search engine marketing method that a search engine deems to be detrimental to its efforts to deliver relevant, quality search results. Some search engines have written guidelines about what they consider to be spamming, but ultimately any activity a particular search engine deems harmful may be considered spam, whether or not there are published guidelines against it. Examples of spam include the creation of nonsensical doorway pages designed to please search engine algorithms rather than human visitors, or a heavy repetition of search terms on a page to increase keyword density. .

Spider: An automated software program that gathers pages from the Internet.

Submission: The act to submitting a URL for inclusion into a search engine's index. Unless done through paid inclusion, submission generally does not guarantee listing. In addition, submission does not help with rank improvement on crawler-based search engines unless search engine optimization efforts have been implemented. Submission can be done manually (i.e., you fill out an online form and submit) or automated, where a software program or online service may process the forms behind the scenes.

Suffix: The last part of a domain that can be used to identify the type of organization or location of a site.

Third-party cookie: Hosted web analytics services track visitor behavior by inserting a small piece of tracking code onto each page of a site. Because the cookie is served by an analytics vendor rather than your own site, the cookie is considered third-party.

Traffic: On the web, traffic refers to the amount of data sent and received by visitors to a website.

URL: A Uniform Resource Locator is a means of identifying an exact location on the Internet.

User Agent: Fields in an extended web server log file identifying the browser and platform used by a visitor.

User Session: A period of activity (all hits) for one user of a website. A unique user is determined by the IP address or cookie. Typically, a user session is terminated when a user is inactive for more than 30 minutes.
Unique Visitors: refers to a measure captured by web analytics solutions that track the interaction a single user has with a website over time.

Viral Marketing: Any marketing technique that induces Web sites or users to pass on a marketing message to other sites or users, creating a potentially exponential growth in the message's visibility and effect.

Visitor: Similar to unique visitor, visitor refers to an individual that visits a website. A visitor or unique visitor can have multiple visits.

Visitor Session: Interaction by a site visitor. The session ends when the visitor leaves the site.

Visit: A visit is an interaction a unique visitor has with a website over a specified period of time or activity. If a visitor has left a site or has not executed a click within 30 minutes, the visit session will terminate.

W3C: World Wide Web Consortium develops interoperable technologies (specifications, guidelines, software, and tools) to lead the Web to its full potential.

Web 2.0: The use of World Wide Web technology and web design that aims to facilitate creativity, information sharing, and, most notably, collaboration among users. These concepts have led to the development and evolution of web-based communities and hosted services, such as social-networking sites, wikis, blogs, and folksonomies.

Web Analytics: The measurement of data as it relates to an Internet site, including the behavior of visitors, the amount of traffic, the conversion rates, web server performance, user experience, and other information in order to understand and proof of results and continually improve the results of a site towards a set of objectives.

Website: A website is a collection of web pages, on particular domain name or sub-domain on the World Wide Web on the Internet. Usually it is made up of a set of web pages created using HTML and accessible via HTTP.

What if: A type of analysis that allows an end-user to pose hypothetical situations against their data to model or predict outcomes.

World Wide Web: Also called the web, this is a global information space which people can communicate via computers connected to the Internet. Some people use "internet" and "the web" interchangeably, even though the web is a service that operates over the internet.

XML: Extensible Markup Language is a World Wide Web Consortium (W3C) recommended general-purpose markup language for creating special-purpose markup languages, capable of describing many different kinds of data.

XML Feeds: A form of paid inclusion where a search engine is "fed" information about pages via XML, rather than gathering that information through crawling actual pages. Marketers can pay to have their pages included in a spider based search index either annually per URL or on a CPC basis based on an XML document representing each page on the client site. New media types are being introduced into paid inclusion, including graphics, video, audio, and rich media. These feeds are commonly used for Shopping Feeds.

Zero Latency: Latency is a time delay between the moment something is started, and the moment one of the effects of that event begins. When there is no time lapse between the event and the effect, it's called zero latency. In analytics, this term is used to describe instantaneous receipt of data and the ability to analyze and act on that data.

Zero-page Visit: A visit that included no page views. This is possible if a visit consisted of at least one request for a non-page file (such as a graphic) but no page files (such as .htm, .asp, .jsp, or .cfm.)

Statistics Formulas

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Notation

Capitalization
In general, capital letters refer to population attributes (i.e., parameters); and lower-case letters refer to sample attributes (i.e., statistics). For example,
- P refers to a population proportion; and p, to a sample proportion.
- X refers to a set of population elements; and x, to a set of sample elements.
- N refers to population size; and n, to sample size.
Greek vs. Roman Letters
Like capital letters, Greek letters refer to population attributes. Their sample counterparts, however, are usually Roman letters. For example,
- μ refers to a population mean; and x, to a sample mean.
- σ refers to the standard deviation of a population; and s, to the standard deviation of a sample.
Population Parameters
- μ refers to a population mean.
- σ refers to the standard deviation of a population.
- σ2 refers to the variance of a population.
- P refers to the proportion of population elements that have a particular attribute.
- Q refers to the proportion of population elements that do not have a particular attribute, so Q = 1 - P.
- ρ is the population correlation coefficient, based on all of the elements from a population.
- N is the number of elements in a population.
Sample Statistics
- x refers to a sample mean.
- s refers to the standard deviation of a sample.
- s2 refers to the variance of a sample.
- p refers to the proportion of sample elements that have a particular attribute.
- q refers to the proportion of sample elements that do not have a particular attribute, so q = 1 - p.
- r is the sample correlation coefficient, based on all of the elements from a sample.
- n is the number of elements in a sample.
Simple Linear Regression
- Β0 is the intercept constant in a population regression line.
- Β1 is the regression coefficient (i.e., slope) in a population regression line.
- R2 refers to the coefficient of determination.
- b0 is the intercept constant in a sample regression line.
- b1 refers to the regression coefficient in a sample regression line (i.e., the slope).
- sb1 refers to the refers to the standard error of the slope of a regression line.
Probability
- P(A) refers to the probability that event A will occur.
- P(A|B) refers to the conditional probability that event A occurs, given that event B has occurred.
- P(A') refers to the probability of the complement of event A.
- P(A ∩ B) refers to the probability of the intersection of events A and B.
- P(A ∪ B) refers to the probability of the union of events A and B.
- E(X) refers to the expected value of random variable X.
- b(x; n, P) refers to binomial probability.
- b*(x; n, P) refers to negative binomial probability.
- g(x; P) refers to geometric probability.
- h(x; N, n, k) refers to hypergeometric probability.
Counting
- n! refers to the factorial value of n.
- nPr refers to the number of permutations of n things taken r at a time.
- nCr refers to the number of combinations of n things taken r at a time.
Set Theory
- A ∩ B refers to the intersection of events A and B.
- A ∪ B refers to the union of events A and B.
- {A, B, C} refers to the set of elements consisting of A, B, and C.
- {∅} refers to the null set.
Hypothesis Testing
- H0 refers to a null hypothesis.
- H1 or Ha refers to an alternative hypothesis.
- α refers to the significance level.
- Β refers to the probability of committing a Type II error.
Random Variables
- Z or z refers to a standardized score, also known as a z score.
- zα refers to the standardized score that has a cumulative probability equal to 1 - α.
- tα refers to the t score that has a cumulative probability equal to 1 - α.
- fα refers to a f statistic that has a cumulative probability equal to 1 - α.
- fα(v1, v2) is a f statistic with a cumulative probability of 1 - α, and v1 and v2 degrees of freedom.
- Χ2 refers to a chi-square statistic.
Special Symbols
- Σ is the summation symbol, used to compute sums over a range of values.
- Σx or Σxi refers to the sum of a set of n observations. Thus, Σxi = Σx = x1 + x2 + . . . + xn.
- sqrt refers to the square root function. Thus, sqrt(4) = 2 and sqrt(25) = 5.
- Var(X) refers to the variance of the random variable X.
- SD(X) refers to the standard deviation of the random variable X.
- SE refers to the standard error of a statistic.
- ME refers to the margin of error.
- DF refers to the degrees of freedom.

Formulas
Parameters
- Population mean = μ = ( Σ Xi ) / N
- Population standard deviation = σ = sqrt [ Σ ( Xi - μ )2 / N ]
- Population variance = σ2 = Σ ( Xi - μ )2 / N
- Variance of population proportion = σP2 = PQ / n
- Standardized score = Z = (X - μ) / N
- Population correlation coefficient = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }

Statistics
Unless otherwise noted, these formulas assume simple random sampling.
- Sample mean = x = ( Σ xi ) / n
- Sample standard deviation = s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ]
- Sample variance = s2 = Σ ( xi - x )2 / ( n - 1 )
- Variance of sample proportion = sp2 = pq / (n - 1)
- Pooled sample proportion = p = (p1 * n1 + p2 * n2) / (n1 + n2)
- Pooled sample standard deviation = sp = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ]
- Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] }

Simple Linear Regression
- Simple linear regression line: ŷ = b0 + b1x
- Regression coefficient = b1 = Σ [ (xi - x) (yi - y) ] / Σ [ (xi - x)2]
- Regression slope intercept = b0 = y - b1 * x
- Regression coefficient = b1 = r * (sy / sx)
- Standard error of regression slope = sb1 = sqrt [ Σ(yi - ŷi)2 / (n - 2) ] / sqrt [ Σ(xi - x)2 ]

Counting
- n factorial: n! = n * (n-1) * (n - 2) * . . . * 3 * 2 * 1. By convention, 0! = 1.
- Permutations of n things, taken r at a time: nCr = n! / (n - r)!
- Combinations of n things, taken r at a time: nCr = n! / r!(n - r)! = nPr / r!

Probability
- Rule of addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Rule of multiplication: P(A ∩ B) = P(A) P(B|A)
- Rule of subtraction: P(A') = 1 - P(A)

Random Variables
In the following formulas, X and Y are random variables, and a and b are constants.
- Expected value of X = E(X) = μx = Σ [ xi * P(xi) ]
- Variance of X = Var(X) = σ2 = Σ [ xi - E(x) ]2 * P(xi) = Σ [ xi - μx ]2 * P(xi)
- Normal random variable = z-score = z = (X - μ)/σ
- Chi-square statistic = Χ2 = [ ( n - 1 ) * s2 ] / σ2
- f statistic = f = [ s12/σ12 ] / [ s22/σ22 ]
- Expected value of sum of random variables = E(X + Y) = E(X) + E(Y)
- Expected value of difference between random variables = E(X - Y) = E(X) - E(Y)
- Variance of the sum of independent random variables = Var(X + Y) = Var(X) + Var(Y)
- Variance of the difference between independent random variables = Var(X - Y) = E(X) + E(Y)

Sampling Distributions
- Mean of sampling distribution of the mean = μx = μ
- Mean of sampling distribution of the proportion = μp = P
- Standard deviation of proportion = σp = sqrt[ P * (1 - P)/n ] = sqrt( PQ / n )
- Standard deviation of the mean = σx = σ/sqrt(n)
- Standard deviation of difference of sample means = σd = sqrt[ (σ12 / n1) + (σ22 / n2) ]
- Standard deviation of difference of sample proportions = σd = sqrt{ [P1(1 - P1) / n1] + [P2(1 - P2) / n2] }

Standard Error
- Standard error of proportion = SEp = sp = sqrt[ p * (1 - p)/n ] = sqrt( pq / n )
- Standard error of difference for proportions = SEp = sp = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
- Standard error of the mean = SEx = sx = s/sqrt(n)
- Standard error of difference of sample means = SEd = sd = sqrt[ (s12 / n1) + (s22 / n2) ]
- Standard error of difference of paired sample means = SEd = sd = { sqrt [ (Σ(di - d)2 / (n - 1) ] } / sqrt(n)
- Pooled sample standard error = spooled = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ]
- Standard error of difference of sample proportions = sd = sqrt{ [p1(1 - p1) / n1] + [p2(1 - p2) / n2] }

Discrete Probability Distributions
- Binomial formula: P(X = x) = b(x; n, P) = nCx * Px * (1 - P)n - x = nCx * Px * Qn - x
- Mean of binomial distribution = μx = n * P
- Variance of binomial distribution = σx2 = n * P * ( 1 - P )
- Negative Binomial formula: P(X = x) = b*(x; r, P) = x-1Cr-1 * Pr * (1 - P)x - r
- Mean of negative binomial distribution = μx = rQ / P
- Variance of negative binomial distribution = σx2 = r * Q / P2
- Geometric formula: P(X = x) = g(x; P) = P * Qx - 1
- Mean of geometric distribution = μx = Q / P
- Variance of geometric distribution = σx2 = Q / P2
- Hypergeometric formula: P(X = x) = h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
- Mean of hypergeometric distribution = μx = n * k / N
- Variance of hypergeometric distribution = σx2 = n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ]
- Poisson formula: P(x; μ) = (e-μ) (μx) / x!
- Mean of Poisson distribution = μx = μ
- Variance of Poisson distribution = σx2 = μ
- Multinomial formula: P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n1 * p2n2 * . . . * pknk )

Linear Transformations
For the following formulas, assume that Y is a linear transformation of the random variable X, defined by the equation: Y = aX + b.
- Mean of a linear transformation = E(Y) = Y = aX + b.
- Variance of a linear transformation = Var(Y) = a2 * Var(X).
- Standardized score = z = (x - μx) / σx.
- t-score = t = (x - μx) / [ s/sqrt(n) ].

Estimation
- Confidence interval: Sample statistic + Critical value * Standard error of statistic
- Margin of error = (Critical value) * (Standard deviation of statistic)
- Margin of error = (Critical value) * (Standard error of statistic)

Hypothesis Testing
- Standardized test statistic = (Statistic - Parameter) / (Standard deviation of statistic)
- One-sample z-test for proportions: z-score = z = (p - P0) / sqrt( p * q / n )
- Two-sample z-test for proportions: z-score = z = z = [ (p1 - p2) - d ] / SE
- One-sample t-test for means: t-score = t = (x - μ) / SE
- Two-sample t-test for means: t-score = t = [ (x1 - x2) - d ] / SE
- Matched-sample t-test for means: t-score = t = [ (x1 - x2) - D ] / SE = (d - D) / SE
- Chi-square test statistic = Χ2 = Σ[ (Observed - Expected)2 / Expected ]

Degrees of Freedom

The correct formula for degrees of freedom (DF) depends on the situation (the nature of the test statistic, the number of samples, underlying assumptions, etc.).

- One-sample t-test: DF = n - 1
- Two-sample t-test: DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
- Two-sample t-test, pooled standard error: DF = n1 + n2 - 2
- Simple linear regression, test slope: DF = n - 2
- Chi-square goodness of fit test: DF = k - 1
- Chi-square test for homogeneity: DF = (r - 1) * (c - 1)
- Chi-square test for independence: DF = (r - 1) * (c - 1)

Sample Size

Below, the first two formulas find the smallest sample sizes required to achieve a fixed margin of error, using simple random sampling. The third formula assigns sample to strata, based on a proportionate design. The fourth formula, Neyman allocation, uses stratified sampling to minimize variance, given a fixed sample size. And the last formula, optimum allocation, uses stratified sampling to minimize variance, given a fixed budget.

- Mean (simple random sampling): n = { z2 * σ2 * [ N / (N - 1) ] } / { ME2 + [ z2 * σ2 / (N - 1) ] }
- Proportion (simple random sampling): n = [ ( z2 * p * q ) + ME2 ] / [ ME2 + z2 * p * q / N ]
- Proportionate stratified sampling: nh = ( Nh / N ) * n
- Neyman allocation (stratified sampling): nh = n * ( Nh * σh ) / [ Σ ( Ni * σi ) ]
- Optimum allocation (stratified sampling): nh = n * [ ( Nh * σh ) / sqrt( ch ) ] / [ Σ ( Ni * σi ) / sqrt( ci ) ]

Use of Statistical Analysis?

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One sample t-test

One sample t-test, tests whether a sample mean significantly differs from a hypothesized value

One sample median test

One sample median test is used to test whether a sample median differs significantly from a hypothesized value.

Binomial test

Binomial test is used to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value.

Chi-square goodness of fit

A chi-square goodness of fit test is used to test whether the observed proportions for a categorical variable differ from hypothesized proportions. For example, let's suppose that we believe that the general population consists of x% Hispanic, y% Asian, z% African American and a% White folks. We want to test whether the observed proportions from our sample differ significantly from these hypothesized proportions.

Two independent samples t-test

An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups.

Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test is a non-parametric analog to the independent samples t-test and can be used when you do not assume that the dependent variable is a normally distributed interval variable (you only assume that the variable is at least ordinal).

Chi-square test

A chi-square test is used when you want to see if there is a relationship between two categorical variables. Remember that the chi-square test assumes that the expected value for each cell is five or higher.

Fisher's exact test

The Fisher's exact test is used when you want to conduct a chi-square test but one or more of your cells has an expected frequency of five or less. Remember that the chi-square test assumes that each cell has an expected frequency of five or more, but the Fisher's exact test has no such assumption and can be used regardless of how small the expected frequency is.

One-way ANOVA

A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories) and a normally distributed interval dependent variable and you wish to test for differences in the means of the dependent variable broken down by the levels of the independent variable.

Kruskal Wallis test

The Kruskal Wallis test is used when you have one independent variable with two or more levels and an ordinal dependent variable. In other words, it is the non-parametric version of ANOVA and a generalized form of the Mann-Whitney test method since it permits two or more groups.

Paired t-test

A paired (samples) t-test is used when you have two related observations (i.e., two observations per subject) and you want to see if the means on these two normally distributed interval variables differ from one another.

Wilcoxon signed rank sum test

The Wilcoxon signed rank sum test is the non-parametric version of a paired samples t-test. You use the Wilcoxon signed rank sum test when you do not wish to assume that the difference between the two variables is interval and normally distributed (but you do assume the difference is ordinal).

One-way repeated measures ANOVA

You would perform a one-way repeated measures analysis of variance if you had one categorical independent variable and a normally distributed interval dependent variable that was repeated at least twice for each subject. This is the equivalent of the paired samples t-test, but allows for two or more levels of the categorical variable. This tests whether the mean of the dependent variable differs by the categorical variable.

Factorial ANOVA

A factorial ANOVA has two or more categorical independent variables (either with or without the interactions) and a single normally distributed interval dependent variable.

Friedman test

You perform a Friedman test when you have one within-subjects independent variable with two or more levels and a dependent variable that is not interval and normally distributed (but at least ordinal). We will use this test to determine if there is a difference in the reading, writing and math scores. The null hypothesis in this test is that the distribution of the ranks of each type of score (i.e., reading, writing and math) are the same.

Factorial logistic regression

A factorial logistic regression is used when you have two more categorical independent variables but a dichotomous dependent variable.

Correlation

A correlation is useful when you want to see the relationship between two (or more) normally distributed interval variables. By squaring the correlation and then multiplying by 100, you can determine what percentage of the variability is shared.

Simple linear regression

Simple linear regression allows us to look at the linear relationship between one normally distributed interval predictor and one normally distributed interval outcome variable.

Non-parametric correlation

A Spearman correlation is used when one or both of the variables are not assumed to be normally distributed and interval (but are assumed to be ordinal). The values of the variables are converted in ranks and then correlated.

Simple logistic regression

Logistic regression assumes that the outcome variable is binary (i.e., coded as 0 and 1).

Multiple regression

Multiple regression is very similar to simple regression, except that in multiple regression you have more than one predictor variable in the equation.

Analysis of covariance

Analysis of covariance is like ANOVA, except in addition to the categorical predictors you also have continuous predictors as well.

Multiple logistic regression

Multiple logistic regression is like simple logistic regression, except that there are two or more predictors. The predictors can be interval variables or dummy variables, but cannot be categorical variables. If you have categorical predictors, they should be coded into one or more dummy variables.

Discriminant analysis

Discriminant analysis is used when you have one or more normally distributed interval independent variables and a categorical dependent variable. It is a multivariate technique that considers the latent dimensions in the independent variables for predicting group membership in the categorical dependent variable.

One-way MANOVA

MANOVA (multivariate analysis of variance) is like ANOVA, except that there are two or more dependent variables. In a one-way MANOVA, there is one categorical independent variable and two or more dependent variables.

Multivariate multiple regression

Multivariate multiple regression is used when you have two or more variables that are to be predicted from two or more predictor variables.

Canonical correlation

Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables. For each set of variables, it creates latent variables and looks at the relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed.

Factor analysis

Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables. All variables involved in the factor analysis need to be interval and are assumed to be normally distributed. The goal of the analysis is to try to identify factors which underlie the variables. There may be fewer factors than variables, but there may not be more factors than variables.

Results of Statistical Analysis

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Results of statistical analyses can indicate the precision of the results, give further description of the data or demonstrate the statistical significance of comparisons. Where statistical significance is referred to in text, the reference should be included in such a way as to minimize disruption to the flow of the text. Significance probabilities can either be presented by reference to conventional levels, e.g. (P < 0.05) or, more informatively, by stating the exact probability, e.g. (P = 0.023). An alternative to including a large number of statements about significance is to include an overall covering sentence at the beginning of the results section, or some other suitable position. An example of such a sentence is:- 'All treatment differences referred to in the results are statistically significant at least at the 5% level unless otherwise stated.'

Descriptive Statistics

When simply describing a set of data with summary statistics, useful statistics to present are the mean, the number of observations and a measure of the variation or "scatter" of the observations, as well as the units of measurement. The range or the standard deviation (SD) is useful measures of the variation in the data. The standard error (SE) is not relevant in this context, since it measures the precision with which the mean of the data estimates the mean of a larger population. If there are a large number of variables to be described the means, SDs etc. should be presented in a table. However if there are only one or two variables, these results can be included in the text.

For example:-'The initial weights of 48 ewes in the study had a mean of 34.7 kg and ranged from 29.2 to 38.6 kg." or

'The mean initial weight of ewes in the study was 34.7 kg (n = 48, SD = 2.61)".

Analyses of Variance

In most situations, the only candidates from the analysis of variance table for presentation are the significance probabilities of the various factors and interactions and sometimes the residual variance or standard deviation. When included, they should be within the corresponding table of means, rather than in a separate table.

In general, authors should present relevant treatment means, a measure of their precision, and maybe significance probabilities. The treatment means are the primary information to be presented, with measures of precision being of secondary importance. The layout of the table should reflect these priorities; the secondary information should not obscure the main message of the data. The layout of tables of means depends, as is shown below, on the design of the experiment, in particular on:-

* whether the design is balanced i.e. equal numbers of observations per treatment;

* whether the treatments have a factorial structure.

* for factorial designs, whether or not there are interactions.

Measures of Precision

The measure of precision should be either a standard error of a mean (SE), or a standard error of the difference between two means (SED), or a least significant difference (LSD). In the latter case, the significance level used should be stated, e.g. 5% LSD. The SED and LSD are usually only suitable for balanced designs. For balanced designs 5% LSD ( 2 ( SED and SED = (2 ( SE ( (one and a half) x SE.

Only one of these three statistics is necessary and it is important to make it clear which is being used. Preference is for the standard error (SE). It is the simplest. We can always multiply by 11/2 or 3 to give the SED or 5% LSD, and we can use the SE for both balanced and unbalanced situations. Measures of precision are usually presented with one more decimal place than the means. This is not a strict rule. For example a mean of 74 with a standard error of 32 is fine, but a mean of 7.4, with a standard error of 0.3, should have the extra decimal place and be given as 0.32.

Some researchers like to include the results of a multiple comparison procedure such as Fisher's LSD. These are added as a column with a series of letters, (a, b, c, etc) where treatments with the same letter are not significantly different. Often, though, these methods are abused. The common multiple comparison procedures are only valid when there is no "structure" in the set of treatments, e.g. when a number of different plant accessions or sources of protein are being compared.

In addition a single standard error or LSD is given in the balanced case, individual standard errors in the unbalanced case. The results from a multiple comparison procedure are additional to, and not a substitute for, the reporting of the standard errors.

Single Factor Experiments

The most straightforward case is a balanced design with simple treatments. Here each treatment has the same precision, so only one SE (or SED or LSD) per variable is needed. In the table of results, each row should present means for one treatment; results for different variables are presented in columns. The statistical analysis results are presented as one or two additional rows: one giving SEs (or SEDs or LSDs) and the other possibly giving significance probabilities. If the F-probabilities are given we suggest that the actual probabilities be reported, rather than just the levels of significance, e.g. 5% (or 0.05), 0.01 or 0.001. In particular, reporting a result was "not significant", often written as "ns", is not helpful. In interpreting the results, it is sometimes useful to know if the level of significance was 6% or 60%. If specified contrasts are of interest, e.g. polynomial contrasts for quantitative treatments, their individual F-test probabilities should be presented with or instead of the overall F-test.

For unbalanced experiments each treatment mean has a different precision. Then the best way of presenting results is to include a separate column of standard errors next to each column of means and also a column containing the number of observations. If the number of observations per group is the same for the different variables, then only one column of numbers should be presented (usually the first column). Otherwise a separate column will be needed for each variable. If there is little variation in the number of observations per treatment, and therefore little variation in the standard errors, it is sometimes possible to use an "average" SE or SED and present results as if the experiment was balanced. This should only be done if it does not distort the results, and it should be clearly stated that this procedure has been used. Alternatively, if the numbers per group are not too unequal, another method is to present the residual standard deviation for each variable instead of a column of individual standard errors. If the groups are of very unequal size, the above compromises should not be used and the number of variables presented in any one table should be reduced.

Factorial Experiments

For factorial experiments there is usually more statistical information to present. Also the means to be presented will depend on whether or not there are interactions which are both statistically significant and of practical importance.

This section discusses two-factor experiments, but the recommendations can be easily extended to more complex cases. It also assumes a balanced experiment with equal numbers of observations for each treatment. However, the recommendations can be combined in a fairly straightforward manner with those above for the unbalanced case.

If there is no interaction then the "main effect" means should be presented. For example a 3*2 factorial experiment on sheep nutrition might have three "levels" of supplementation (None, Medium and High) and two levels of parasite control (None and Drenched), giving six treatments in total. There are five main effect means: three means for the levels of supplementation, averaged over the two levels of parasite control, and two means for the levels of parasite control. In this example there would also be two SEs and two significance probabilities for each variable, corresponding to the two factors.

If there are interactions which are statistically significant and of practical importance, then main effect means alone are of limited use. In this case, the individual treatment means should be presented. For a balanced design, there is now only one SE per variable (except for split-plot designs), but three rows giving F-test probabilities for the two main effects and the interaction. Additional rows for F-test probabilities can be used for results of polynomial contrasts for quantitative factors or other pre-planned contrasts.

Regression Analysis

The key results of a linear regression analysis are usually the regression coefficient (b), its standard error, the intercept or constant term, the correlation coefficient (r) and the residual standard deviation. For multiple regression there will be a number of coefficients and SEs, and the coefficient of determination (R²) will replace r. If a number of similar regression analyses have been done the relevant results can be presented in a table, with one column for each parameter.

If results of just one or two regression analyses are presented, they can be incorporated in the text. This can either be done by presenting the regression equation as in :-

'The regression equation relating dry matter yield (DM, kg/ha) to amount of phosphorus applied (P, kg/ha) is DM = 1815 + 32.1P (r = 0.74, SE of regression coefficient = 8.9).”

or presenting individual parameters as in:-

'Linear regression analysis showed that increasing the amount of phosphorous applied by 1 kg/ha increased dry matter yield by 32.1 kg/ha (SE = 8.9). The correlation coefficient was 0.74.'

It is often revealing to present a graph of the regression line. If there is only one line to present on a graph, the individual points should also be included. This is not always necessary and tends to be confusing, with more than one line. Details of the regression equation(s) and correlation coefficient(s) can be included with the graph if there is sufficient space. If this information would obscure the message of the graph, then it should be presented elsewhere.

Error Bars on Graphs and Charts

Error bars displayed on graphs or charts are sometimes very informative, while in other cases they obscure the trends which the picture is meant to demonstrate. The decision on whether or not to include error bars within the chart, or give the information as part of the caption, should depend on whether they make the graph clearer or not.

If error bars are displayed, it must be clear whether the bars refer to standard deviations, standard errors, least significant differences, confidence intervals or ranges. Where error bars representing, say, standard errors are presented, then one of the two methods below should be used.

(a) the bar is centred on the mean, with one SE above the mean and one SE below the mean. i.e. the bar has a total length of twice the SE.

(b) the bar appears either completely above or completely below the mean, and represents one SE.

If the error bar has the same length for all points in the graph, then it should be drawn only once and placed to one side of the graph, rather than on the points. This occurs with the results of balanced experiments.

Statistical Averages

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Summary Statistics

After the data have been properly checked for its quality, the first and foremost analysis is usually for the descriptive statistics. The general aim is to summarize the data, iron out any peculiarities and perhaps get ideas for a more sophisticated analysis. The data summary may help to suggest a suitable model which in turn suggests an appropriate inferential procedure. The first phase of the analysis will be described as the initial examination of the data or initial data analysis. It has many things in common with explanatory data analysis which includes a variety of graphical and numerical techniques for exploring data. Thus explanatory data analysis is an essential part of nearly every analysis. It provides a reasonably systematic way of digesting and summarizing the data with its exact form naturally varies widely from problem to problem. In general, under initial and exploratory data analysis, the following are given due importance.

Measures of Central Tendency

One of the most important aspects of describing a distribution is the central value around which the observations are distributed. Any arithmetical measure which is intended to represent the center or central value of a set of observations is known as measure of central tendency.

The Arithmetic Mean (or simply Mean)

Suppose that n observations are obtained for a sample from a population. Denote the values of the n observations by x₁, x_2.....x_n; x₁ being the value of the first sample observation, x₂ that of second observation and so on. The arithmetic mean or mean or average denoted byis given by

The symbol S ( read as ‘sigma’ ) means sum the individual values x₁ ,x₂,...,x_n of the variable, X. Usually the limits of the summations are not written, since it is always understood that the summation is over all n values. Hence we can write

The above formula enables us to find the mean when values x₁, x₂ ,....,x_n of n discrete observations are available. Sometimes the data set are given in the form of a frequency distribution table then the formula is as follows:

Arithmetic Mean of Grouped Data

Suppose that there are k classes or intervals. Let x₁, x₂ ,..., x_kdenote the class mid-points of these k intervals and let f₁, f₂, ..., f_k denotes the corresponding frequencies of these classes. Then the arithmetic mean

Properties of the arithmetic mean

(a) The Sum of the deviations of a set of n observations x₁ , x₂,..., x_n from their mean is zero. Let d_i as deviation of x_i fromthen

(b) If x₁ ,x₂,...,x_n are n observations,is their mean and d_i = x_i - A is the deviation of x_i from a given number A, then

(d) If in a frequency distribution all the k class intervals are of the same width c, and d_i = x_i - A denote the deviation of x_i from A, where A is the value of a certain mid-point and x₁, x₂ ,..., x_kare the class mid-points of the k-classes, then d_i = c u_i where u_i = 0, ± 1, ± 2,..... and

The Median

The median of a set of n measurements or observations x₁ , x₂ ,..., x_n is the middle value when the measurements are arranged in an array according to their order of magnitude. If n is odd, the middle value is the median. If n is even, there are two middle values and the average of these values is the median. The median is the value which divides the set of observations into two equal halves, such that 50% of the observations lie below the median and 50% above the median. The median is not affected by the actual values of the observations but rather on their positions.

The Median of Grouped Data

The formula of median of grouped data is as

The Mode

The mode is the observation which occurs most frequently in a set. In grouped data mode is worked out as

The mode can be determined analytically in the case of continuous distribution. For a symmetrical distribution, the mean, median and mode coincide. For a distribution skewed to the left ( or negatively skewed distribution ), the mean, the median and the mode are in that order (as they appear in the dictionary ) and for a distribution skewed to the right ( or positively skewed distribution) they occur in the reverse order, mode, median and mean. There is an empirical formula for a moderately asymmetrical skewed distribution, it is given by Mean - Mode = 3 (Mean - Median)

The Geometric Mean

There are two other averages, the geometric mean and harmonic mean which are sometimes used. The Geometric Mean ( GM ) of a set of observations is such that its logarithm equals the arithmetic mean of the logarithms of the values of the observations. GM = (x₁ x₂..... x_n)^1/n

log GM = 1/n (å log x_i) or in frequency distribution, log GM = 1/n (å f_i log x_i)

In case of frequency distribution,

The geometric mean can be obtained only if the values assumed by the observation are positive( greater than zero).

Harmonic mean

The Harmonic Mean ( HM ) of a set of observations is such that its reciprocal is the arithmetic mean of the reciprocals of the values of the observation

The harmonic mean is rarely computed for a frequency distribution.

Weighted Mean
If there are n observations, x₁, x₂, x_3,…,x_n with corresponding weights w₁, w₂, w_3,…,w_n, then the weighted mean is given by,

In computing the mean, we take the frequency of a class as its weight. That is

Hence, it is a special case of weighted mean. The three means are related by

A.M. ³ G.M. ³ H.M.

Important characteristics of a good average

Since an average is a representative item of a distribution it should possess the following properties :

1. It should take all items into consideration.

2. It should not be affected by extreme values.

3. It should be stable from sample to sample.

4. It should be capable of being used for further statistical analysis.

Mean satisfies all the properties excepting that it is affected by the presence of extreme items. For example, if the items are 5, 6, 7, 7, 8 and 9 then the mean, median and mode are all equal to 7. If the last value is 30 instead of 9, the mean will be 10, whereas median and mode are not changed. Though median and mode are better in this respect they do not satisfy the other properties. Hence mean is the best average among these three.

When to use different averages

The proper average to be used depends upon the nature of the data, nature of the frequency distribution and the purpose.

If the data is qualitative one, only mode can be computed. For example, when we are interested in knowing the typical soil type in a locality or the typical cropping pattern in a region we can use mode. On the other hand, if the data is quantitative one, we can use any one of the averages

If the data is quantitative, then we have to consider the nature of the frequency distribution. When the frequency distribution is skewed (not symmetrical) the median or mode will be proper average. In case of raw data in which extreme values, either small or large, are present, the median or mode is the proper average. In case of a symmetrical distribution either mean or median or mode can be used. However, as seen already, the mean is preferred over the other two.

When we are dealing with rates, speed and prices we use harmonic mean. If we are interested in relative change, as in the case of bacterial growth, cell division etc., geometric mean is the most appropriate average.