At this point, I could contain myself no more. I spoke up and said "Yeah! My biological father was just like that–a complete deadbeat, had sex with my mother without being married, who left before I was even born. He has never volunteered child support, never paid a dime, and deflects all my attempts to contact him." The parents, who were surprised at first that I had joined in the conversation, were nodding emphatically at my statement, as if to say "See?"

I continued: "It was really hard when I found out about him, and to this day it eats at me that there is a father out there who cares nothing for his own flesh and blood." The parents continued nodding, and the rest of the waiting room quieted down, listening.

At this point, I got up, put my bag on my shoulder and walked over to the family. I put my hand on the teenage girl’s shoulder. "And the worst part is…" I put on my straightest sad face, and sighed woefully. "He’s Catholic."

The parents’ faces, the dead silence in the room, and the shocked faces of listeners nearby as I walked out was most satisfying.

My name is Dan Hughes, and I am a hopeful UWM student for the upcoming semester. I say hopeful because the cost I pay for having put off college for almost 10 years after graduating high school causes quite a bit of hardship. While I’ve tried my best and have saved up enough to cover a month or two of expenses when I move out there to attend school, it is becoming increasingly apparent with the job search that I may not be employed enough in time for the school semester to start. I am faced with having to withdrawal because I cannot fully afford the semester.

I was looking at what ways I could possibly be able to support myself AND attend college, and I came across Wisconsin statute 36.27(3)a that states the Board of Regents can remit nonresident tuition to "a number of needy and worthy nonresident students upon the basis of merit, to be shown by suitable tests, examinations or scholastic records and continued high standards of scholastic achievement." As a struggling man in this economy who has supported himself meagerly while obtaining a 4.0 in college and participating in extra activities, I’d like to claim that I might qualify as one of these students. The federal government recognizes my need and has awarded me a pell grant of around $5000, and my loans cover about $10000. I still need a couple more thousand to pay the tuition on top of paying the multitudes of bills one needs to cover to live.

I have attempted to pursue this possibility since the beginning of May. Since then I have been in contact with the Bursar’s office, who directed me to Accounting, who directed me to Financial Aid, who directed me back to Accounting. I contacted the Board of Regents themselves, whose secretary, Jane Radue, said that this is something the individual colleges do, and directed me to the residency specialist and the bursar’s office. The residency specialist said that she only determines residency, and that I should try Financial Aid. I contacted the Board again, and was redirected to Financial Aid. I’m currently in the process of trying to get them again, but they haven’t responded to my request on the website.

I hope you can sense the growing frustration. As the time gets closer for school to start, I have to decide whether I can make it or not. With the non-resident tuition, I’m going to have to say no, and give up on college for now. (The in-state colleges where I am from, Pennsylvania, are far more expensive than even Wisconsin’s out of state tuition!) However, if I can somehow find a way to apply and gain the remission, not only would that cut my costs, but that will allow me to receive a portion of my loan back to cover my living expenses, making the job search easier for me. Unfortunately, this back and forth is seriously putting a damper on my figuring out what to do.

Each department–other than the Board, who recognizes my claim, but said the school has to make the decision–has mistakenly said that only Graduate students can get remission. When I point out that the remission exceptions listed in the law are not limited to Graduates (Graduates aren’t mentioned until section G, actually, with needy, worthy, international, and athletes listed first), that is where I get directed to the next department. I feel like no one knows what to do here. What I do know, is that I am at least entitled to have someone with the power to make these decisions review my case, according to the law. I simply can’t seem to find who that is!

It is even listed in the Accounting office’s own procedures at: https://www4.uwm.edu/bfs/procedures/acct/upload/2-5-02.pdf, where it lists: I. Graduates, II. Athletes, III. Extreme Circumstances, and IV. Needy and Worthy as eligible for remission.

I am desperate. I’ve already signed a lease for an apartment on Oakland Avenue, blocks from the campus. I’m in the process of moving. But as the job hunt keeps coming up empty, without a remission and the relief that my loan will give me, I will have to give up on college for now. I did not quit my job and go to school to stop at an Associate’s degree. I don’t come from a rich family–far be it, I came from a military family who lived under the poverty line for many years. I am the first of the kids in my generation to get ANY degree, much less make it to Junior status. I’d beg for help, but I’m not sure who I can beg to get it.

Please, I need help. I need direction. I will be successful, whether I find a good enough full-time job to support myself or by going to college. I cannot commit to the college though if I cannot be sure that I’ll be able to pay it off as well as my expenses at a lower salary.

Yours,

~Dan Hughes

In response to “The Protestor” by Sharon Olds

What would I do, if I were asked to kill?
Would I do it, or would I hold back still?
My dad was a man who served his country
But he never had to kill anyone.
That I know of.
I should ask him that.
Would I be so brave as to say, “Take me to jail?”
Would my efforts succeed, or would they fail?
I am so two-sided, claiming that I’d never kill.
Don’t I cheer when the murderer dies?
Don’t I support military action?
Do I say “we should kill, but don’t make me do it?”
Don’t we need people like me around
Even during war, to educate the town?
Or is that the “woman’s” job?
The draft only called men.
Was that the right thing to do?
Is it fair to ask that I get left behind too?
All these thoughts of what I should and shouldn’t do.
I say I’d never kill. I would never kill.
But if you try to hurt me, or my kin.
I’ll kill you. I’ll kill you. I swear I’ll kill you.
I’d rather you’d be dead than me.

“It’s ten dollars, Mommy, remember?” Jay mentioned helpfully from the back seat. She had let him count the one-dollar bills before they left the house. She smiled at him, nodding, and got out of the car. She walked around to his door, unstrapped him from the booster seat, and then locked the door behind them.

She worked out numbers in her head. She had counted on getting two gallons of gas, enough to make it to the county library and back. Jay enjoyed their trips to the library—she would work on one of the computer terminals there, researching jobs and sending out her resume, sparse as it may be, and Jay would be free to roam the library, grabbing any books that interested him. He’d often find one, and sit right on the floor next to the shelf he found it on, and read it through. Then he’d reshelve it, get up, and go find another one. Twice she had found him outside the children’s area: once reading a book about using worms to create a compost heap, and once again reading a history book on chess. He said he saw the game on TV once, and wanted to know how it was played.

But with the price over five dollars, and the Bronco already dangerously near empty, there was a chance it would not be enough. She had ran out of gas once before, but the fee her insurance company charged her for running gas out to her was far more than she was willing to risk. The library would have to wait.

Willow continued walking up to the Arco convenience store, however, because she did not want to waste the gas coming to the station in the first place. She did have ten dollars, after all. Most food items were less expensive at the WinCo, but they were not so much less expensive than the Arco that it warranted the drive out there—not to mention the gas needed. So, Willow decided she would spend some of her money on food for herself and Jay, put a couple dollars in the tank to get them back home, and try again next week.

She knew only to get a few basics—this month had three paycheck Fridays in it, so that amounted to an extra paycheck that, after catching up with some of the overdue bills, usually left her with a little extra money. Next Friday, she’d be able to do a full grocery shop down at the WinCo. For now, she picked up three bags of frozen mixed vegetables (despite Jay’s protests of lima beans), a half-gallon of milk, and a box of Cheerios. Jay marched alongside her, dutifully punching numbers into his trusty Casio.

She looked back at her son. “So, where are we at?”

“Well, the veggies were a dollar and twenty-nine cents each,” he said, pointing at the objects in the basket she carried. “The milk was two dollars and twenty-nine cents, and the Cheerios are two dollars and forty-nine cents.” He nodded to himself. “That is eight dollars and sixty-five cents.”

Willow patted his head and began to praise him—and then she bumped into something, toppled over and landed square on her bottom. The basket landed with a clatter, and one of the bags of frozen vegetables tore open onto the floor. Jay was at her shoulder in an instant, stroking her shoulder like she did for him when he was hurt. “Are you okay, Mommy?”

Willow glanced around herself, disoriented by the fall, and then looked up at what had knocked her over. An older man, tall with greying hair, looked at her from behind thick bifocals, a rough hand offered to her. She took it, and was helped to her feet. “I am so sorry, young miss. I came around that corner, and boom!—” he clapped his hands for effect, “—I ran right into you!”

“You should look where you’re going, Mister,” Jay admonished, his hands on his hips and bent so that his little face pushed up as high as he could go.

Willow started to shush him, but the old man laughed. “Yes, yes, I suppose so. Here, let me help you.” He waved over the cashier and had her go and retrieve a new bag of vegetables out of the freezer. When it was delivered, he apologized again. “Glad to see you’re okay, young lady. I suppose it’s time I had these old Coke bottles checked out!” He tapped on his glasses, and Jay chortled with delight. The old man smiled, and then turned to get back in line.

“So, where were we?” Willow said to Jay.

Jay, who had gotten distracted by his mother’s fall, jumped when he realized his Casio had automatically turned off. He quickly typed in the numbers again to get the total, and told her. Then, he looked thoughtful, punched a few more keys, and said, “That leaves only one dollar and thirty-five cents.”

Willow pursed her lips and said nothing. She let him work it out.

“We can’t buy two gallons of gas with one dollar and thirty-five cents.” A single tear came from his eye. “I guess that means we won’t go to the library again this weekend.”

His tear nearly brought one from Willow’s eyes. She patted him on the head. “I’m sorry sweetie, but the gas prices are just too high. I was looking forward to going too, you know.”

As the first tear dangled from his chin, a second one started to glide from his other eye. “It’s okay, Mommy,” he said, his voice breaking. “I won’t be sad.”

Willow had to clear her eyes. She pulled Jay towards her leg, and he wiped his tears on her jeans. She again wished that these things would stop happening, that her son did not have to be constantly punished for something as simple as the current price per gallon of gas. She remained there just letting Jay pretend to not cry into her thigh.

“Ma’am?” a young voice said behind her. Willow turned and realized that the rest of the line had already paid and left; she was holding up the rest of the line. Embarrassed, she shuffled over, Jay still attached to her leg, and put her basket up on the counter. The cashier gave Jay a sympathetic look, and while Willow smiled thankfully, she could not quell a flash of irritation—the girl was hardly old enough to have children.

You were fifteen, a voice said to her. Willow thoroughly quashed it out of her mind. She handed the dollar bills to the cashier, received the change, and with Jay’s hand in one hand and the bag of food in the other, thanked her and left.

Willow strapped Jay in—or rather, Jay did it, and Willow checked to make sure it was done properly—and then tossed the bag into the passenger seat. She got into the car herself, put her seat belt on, turned the ignition, and as the car sputtered to life, she recalled the change in her hand—she had forgotten to have the cashier apply it to the gas pump. Sighing in frustration, Willow unbuckled and went to turn the car back off again—and froze.

The gas needle was on F.

She blinked. She looked back at Jay, who was still rubbing tears from his eyes. She looked back at the change in her hand and wondered if the cashier made a mistake. Of course not, she thought, that wouldn’t have put the gas into the car! She looked around the parking lot, and saw parked at the pump in front of her a silver Mustang. It was facing towards her, and she could clearly see the old man from earlier in the driver’s seat. He waved at her, smiled with a thumbs-up sign, and then pulled away.

Willow put her head on the steering wheel and cried. It was a few minutes before she could see clearly enough to drive home.

Talk probably was not the right word. More often than not, Willow would stand at the gate watching the children while she overheard the conversation taking place between the adults behind her. Her Jay was incredibly smart, a genius, and she took pleasure in listening to the other parents touting their children’s accomplishments, accomplishments that Jay had already surpassed a year or more prior. Willow would nod in agreement when everyone else did, offer a supportive comment now and again, enough to not seem aloof, but she rarely spoke of herself and Jay.

Today the conversation was leaning towards the planning of some kind of extracurricular get-together for the children over the weekend. The mom suggesting it opened her hands wide to include everyone at the table in the invitation; she even made a couple warm nods towards the other parents standing nearby, Willow included. “They have this amazing system of slides and tunnels and ball pits,” the mom was saying. “It is all supervised by trained staff, so we can sit at the coffee shop inside and get to know one another better.” Many of the adults nodded assent.

Willow liked the idea, but she had not heard of this commercial play place before now, and that meant it probably was not nearby; she said as much.

“Oh, it’s quite close,” the mom replied. “It’s right in Folsom, just about a half hour ride off the El Dorado Freeway.”

Willow felt disappointed inside, but did not let it show. “Well, in that case, I’ll see about making my way up there.” The mom took that for an affirmative and started selling the merits of the plan to another pair of mommies and daddies who seemed apprehensive of joining.

The teacher, Mrs. Canucci, blew the whistle, summoning the children into a queue on the side of the gate opposite Willow. Willow was surrounded by a much less orderly clustering of parents. Mrs. Canucci opened the gate and called each child’s name as each parent stepped forward. Willow observed as the ceremonial distribution of the students dissolved any social contact between the adults, each group of child and parent swiftly walking towards the parking lot. She chuckled at the thought of a soup kitchen handing out kids instead of soup.

Then she heard her child’s name: “Ranger Morris?” Jay perked up and galloped out the gate and into arms of his mother as she greeted him by his nickname. “Hello, Jay, did you have a great day in school? What was your score today?”

Jay answered by stopping right in the walkway to plop down his backpack and unzip it. “I got a 100 today!” he said excitedly, burrowing through the papers in his pack. “Here, I’ll show you—”

“No, dear, that’s okay, I believe you. Let’s move out of the way so the other kids can go home.” Jay blinked and looked around, seemingly surprised that he had picked that spot to stop. For a bright kid, Willow thought, it’s strange what gets past him. It’s as if he picks and chooses what merits being paid attention to. Willow took her son’s hand in hers and began to walk down the sidewalk in the opposite direction of the parking lot.

“So I heard the other kids talking,” Jay began, “about going to FunWorld this weekend.”

Willow sighed. “How’d you know it’s called FunWorld? We’ve never been there.”

“’Cause Brandon said that he went to a place like that once, and he said it was in Folsom. What else could it be?”

Of course, Willow thought. He was probably listening to us the whole time while he played. Nothing ever was spoken around him that he did not somehow file away.

“We’re not going, are we?” At this, Willow missed a step, and Jay took that to confirm his statement. “It’s okay, mommy,” he reassured her, “I won’t be sad. I know gas is expensive right now.”

Willow was not sure to be proud or surprised. He had hit the nail right on the head; with gas approaching five dollars a gallon—and her stuck with that blasted Bronco—a trip to Folsom and back would take up about a quarter tank, about ten to fifteen dollars easily. That was several gallons of milk, six or seven loaves of bread, or enough pasta to feed them both for a week or two. Jay had asked how money worked, so she shared with him her spending budget each week. She would usually start with thirty or forty dollars once the bills were paid; they would always go shopping together, Jay trotting alongside her with a solar-powered Casio keeping track of what they had bought so far.

It was Jay who mentioned that the gas for the Bronco was taking up a chunk of their budget, and it was Jay who suggested that they walk to school together. She smiled at his brilliance while still feeling guilty for needing to keep him away from a fun activity. There was no use hiding it from him now, though. “I’m sorry, Jay. We will find something else fun to do together, I promise.”

Jay skipped along beside her. “It’s okay, mommy.” Her heart broke just a little, that he had to say that so often. “We don’t have enough money in the budget.”

They walked and skipped in silence for a minute or two.

Then, Jay asked, “Mommy?”

“Yes, son?”

“Do you think there’s enough in the budget for an ice cream cone? They’re only a dollar.”

Willow smiled. “I think we might be able to swing that. How much longer ‘til we get home?”

Jay looked at the plastic watch on his wrist. “We’ve been walking for 6 minutes, so that makes…” He stopped skipping and closed his eyes to do the math in his head. “…39 minutes of walking left.”

“Well, then, if you aren’t too tired when we get home, we’ll walk down to the parlor for a cone.” Jay sprang back into step with a victorious cry, encouraging Mommy to walk just a little faster.

She gave her life for me.
And probably would again.

My life is a coursing stream, fueled by a fresh spring,
Sturdy banks absorbing the pressure
Guiding its path while it rushes on,
Clueless to the erosion caused in its rampancy.
My Dad is not the man who begat me,
But the man who taught me what a man was.
Marching into my life without missing a beat,
He taught me what a father was.
It would be the only definition I knew for half my life—
The one with his image next to it, his name in the caption.
He was Dad. He became Father.
Together, with Mom and Me, We became Family.

The title was not his, but he earned it.
He wore it so proudly.

My life is a bubbling brook, pouring over the rough earth,
The echoes of fresh water soothing weary travelers.
The flow tempo washes away bad dreams
And carries slumbering minds away in peace.
My Father constantly played the music of his time
Grumbled and griped at the crap I listened to.
He made me listen closely to the words,
And put myself in the shoes of others.
How would those cops feel listening to their deaths?
How would those bitches feel listening to their abuse?
How about the boys while their daddies use drugs?
THIS is the bullshit you pay them to write?

This is not what it means to be a MAN.
Why would you glorify such pain, violence, and objectification?

My life is a river in an ever-growing storm,
Encouraged to rush over the banks and run on the land.
Torrents of water, freedom to run, freedom to flow
Can be devastating in the wrong lands.
I was the smartest, and my parents loved me for it.
Awards, achievements, aced tests and accolades,
For fifteen years they chuckled at the accomplishments
That had become so commonplace.
There was no puzzle I could not solve.
Especially the one where my birth records
Were missing vital information,
And the excuses for it made no sense.

My dark brown eyes gazed helplessly into
My Mother’s pleading blues and my Dad’s sad, soft greens.

My life is a polluted lake, fed by a river
Chemicals and garbage lay suspended in the murk,
No one cares to clean it, and everyone’s doing it.
The water just keeps swirling ‘round.
I questioned all I knew, the dictionary was being rewritten.
What was my name? Who is my father? What did I do?
Wrestling with rejection, depression, and pain
I said I would be fine.
I lived in others’ happiness to replace my own,
Needing to be told how great a person I was.
I neglected myself, I gained weight,
I was nothing without attention and compliments.

I had my mind, and I had my loving parents.
Why could I not see that I had everything others wanted?

My life is a stagnant pond, uninviting.
People walk by without a glance
The water is stale and tasteless
Even the amphibious fauna has found a new home.
My grades suffered, I graduated without esteem or notice.
I dropped out of college and worked at minimum wage.
I worked until I got bored, then did something else.
I lost myself in fantastical worlds and fictional histories.
I sabotaged my relationships, victimized myself
I married a girl who told me what I needed to hear
In spite of the history surrounding her.
I pretended to be surprised when she hid her affairs from me.

I was self-destructive and hurt others,
I called those who gave up on me the worst kind of friends.

My life is a waterway, once dirty
But the rains overflow the pond and connect it with the rivers,
The animals and plants eat the impurities
People lend a helping hand and the bottom becomes visible.
It took years to release the fury pent up inside me
To stop living for others and live for myself.
I don’t need to find my birth father, I don’t need to punish my exwife.
I looked back and realized what I had known all along.
You make your own honor, you create your own image.
You construct who you are, you choose how you act.
All this I learned two decades ago,
When my Father was my Father.

An epiphany, now, seeing all that he’s done—
My Father IS my Father.

My life is a wide river, peaceful and swift
Providing relaxation to the those who sit on its banks
And smooth sailing to the travelers on their way by,
Key to commerce, trade, and travel.
I have the mind I left behind, and I do good things with it.
I apologize to those who kept me, and let go those who move on.
I form my own opinions, and I will not be bullied into subservience.
Though conscious of my past, I am proud of me.
I have a love based on how good I am, not how good she said I was.
I quit my job and give up my independence
To return to school and put my mind to work.
I better my mind so that I might better the world.

I cherish my friends, and they cherish me,
Because I am worth cherishing, and I see it in them.
And if I follow the path my parents showed me,
I can learn how to be a man just like my Father.

The long hours spent with you go for naught
Finding no solace in your cozy flaxen plains,
Your seductive feathery comfort,
Or your potentials of dreams unrealized.

And yet, even after our unsuccessful repose,
I return to you, eyes drooping, yearning to dream,
To attempt the endeavor once more
Ignoring your broken promises for slumber.

I lay my head and free my mind.
I wait.

As I have waited in years gone by,
As I will wait in years yet to come.
For your soft waves to find me, and roll me over
So that I might float on a sea of fantasy.

In statistics, getting numbers to say what they mean is a very difficult task.  A number with no context is next to meaningless.  If I told you I got a 30 on my statistics exam, you’d probably assume that I didn’t do so well.  The context is that the score was a 30 out of 30 (go me!).  Alternatively, if I told you I got a 30 percent, then you know I did badly because the word “percent” sets the expectation of a total maximum of 100.  Context helps us identify what the heck the numbers mean.

A z-score is a statistical maneuver that helps us learn just how relevant the number really is.  We do that by calculating where the value lies in relation to all of the other values in the data set. 

The standard deviation tells us how widely spread the numbers are from the average value, or the mean.  (We’re assuming you know how to calculate the standard deviation from previous lessons, but really, they just kind of give it to you from here on out.)  In a normal distribution, a majority of all the values, 68%, will be within 1 standard deviation of the mean.  Within 2 standard deviations, that jumps to 95%.  This means if the average test score is a 75, and the standard deviation is 10, that means that score of 80 you got is nothing special—80 is within 1 standard deviation of 75.  However, if you got a 100, you are now outside two standard deviations.  Only 5% of your classmates were that far away from the mean!  (Because the other 95% are within 2 standard deviations.)

STOP!  SURPRISE!  I totally just tricked you into doing z-scores.  All we wanted to know was how many standard deviations away we were from the mean.  We took a look at the difference between your score and the mean (80 – 75 = 5, 100 – 75 = 25), then we see how many standard deviations were in the difference by dividing it (5 ÷ 10 = 0.5, 25 ÷ 10 = 2.5).  We know from above that if the score is within 1 standard deviation, you’re in the majority.  Within 2, you’re in a smaller group, and so on.  That final number is the z-score!

So, the formula makes sense: Subtract the mean from the main value, then divide by the standard deviation:

X score    –    Mean
Standard Deviation

Or, in statistic-ese:

for a population, or for a sample

Where x is the value, μ (greek mu) and M is the population and sample mean respectively, and σ (greek sigma) and s are the population and sample standard deviation respectively.  You also may see
(pronounced “x bar”) instead of M for the sample mean—it’s the same thing.

So, if we use the formula to find the z-scores of the above examples, we simply plug the numbers in and solve:

Those are z-scores.  With these, we know that the test score that has a z-score of .5 is very common (within one standard deviation), and the one with a z-score of 2.5 is very rare (more than 2 standard deviations away).  BAM!

Finding x when you already have z

One curveball the stat teacher threw at us was finding x by giving us the Z.  In other words, instead of seeing the test score and finding out how many standard deviations away it is, she told us how many standard deviations away it is (the z-score) and asked us to find x.  She gave us the following formula:

While it is easy enough to memorize a second formula and plug in the given mean, standard deviation, and z-scores, I heard some confusion about this.  So, let me quell the fears by showing you something neat.  First, subtract M from both sides:

Then, divide by s:

Look familiar?  That’s because it is exactly the same formula!  It was just solved for X, since that was what we were looking for.  Therefore, those of you who have formula-memorization anxiety, never fear, because if you just remember the one formula, you can easily solve for any variable within it.

Proportions, or, the Area Under the Curve

So, now you have a z-score.  Yay?  Fortunately, there is much more we can learn from a z-score than just how rare something is.  Let’s take a look at our previous examples, two test scores of 80 and 100, where the mean is 75 and the standard deviation is 10.  Let’s also assume that the data is normally distributed—cause you can’t do much of anything unless it is.

So, a normal curve:

Then we plot on it the z-score we found for the first test, an 80, which was 0.5:

Now, with this, we can find out what proportion of our group here happened to get a score higher than you.  In essence, we want to know, what are the odds someone has a higher z-score than you?  Let’s fill in the area we are looking for. 

What we need to do is find the proportion that corresponds to the red area in the graph (also known as the “area under the curve”).  To do this, you will use a Z distribution table.  A Z distribution table lists a lot of z-scores and their corresponding proportions.  Each one is a little different, but I will explain the one we use for this class.  Ours has four columns:

A) the list of z-scores

B) Proportion in Body

C) Proportion in Tail

D) Proportion Between Mean and Z

It can be kind of confusing to choose between them.  Fortunately, the book gives us three specific points to keep in mind:

1) The BODY always corresponds to the LARGER part of the distribution WHETHER it is on the right side or the left side.  Similarly, the TAIL is always the SMALLER section.

2) Because the normal distribution is symmetrical, the proportions on the right-hand side are exactly the same as the corresponding proportions on the left-hand side.  For example, the proportion in the right-hand-tail beyond Z = 1 is the same as the proportion in the left-hand tail beyond Z = –1.

3) Although the z-score values will change signs (+ and –) from one side to the other, the proportions are ALWAYS positive.  Thus, column C in the table always lists the proportion in the tail whether it is the right-hand tail or the left-hand tail.

There.  So, since the red area we want to figure out is the SMALLER half of the graph, we will call it the tail.  Therefore, we will look up 0.5 as the z-score in the table and look at the column with the Tail proportions, column C.  When we do, the Proportion in Tail in column C that we find corresponds with the z-score of 0.5 is .3085.  This means, when the values are normally distributed like this, there is a 30.85% chance of finding someone who has a score higher than 80, or a z-score higher than 0.5.

Let’s do the same thing, but for the score of 100.  Again, with a mean of 75 and a standard deviation of 10, we find the z-score to be 2.5.  Let’s draw it and fill it in:

Yipes!  That’s a really tiny part of the curve!  Fortunately, that makes sense, because we know that values occurring more than 2 standard deviations away from the mean are really rare and shouldn’t happen all that often anyways.  So again, we are going to refer in the table to the z-score value of 2.5 and find the corresponding Tail proportion.  We find the proportion to be .0062.  This means that there is a very very very tiny chance of finding anyone with a higher score!  (A 0.62% chance, specifically.)

In review: finding proportions is simple.  Using the z-score as your dividing point, you can find the proportions of the curve on EITHER side (the larger is the body, the smaller is the tail) using the z distribution chart, referring to the appropriate column.

And now, for something tricky

Using the same situation as above, let’s find out the proportion of students who got a B on the test, defined as between 80 and 90.  Instead of just one z-score as the starting point and then continuing off to one side, we need two z-scores, one each for the starting and stopping point.  Let’s calculate them now for 80 and 90.  Remember, 75 is the mean, and 10 is the standard deviation:

Well, we did kind of know the z-score for 80 already, but whatever.  The z-scores of 80 and 90 are .5 and 1.5 respectively.  Let’s plot them on our curve.

Unfortunately, we don’t have a “From z=.05 to z=1.5” column in our book to tell us what the proportion is here.  We have to get creative.  There are a number of different ways to solve this problem.  Mainly we do it by breaking up the pieces to get just the parts we want.  How you break it up is up to you.  I’ll give you two examples.

First: One total we can get is the total value from the Mean (which is 0) to the 1.5 z-score.  We do this by finding the 1.5 in the z-score table, and using column D, Proportion Between Mean and Z.  This gives us the proportion of the original area we want (in red) plus the part shaded in black below, which is .4332.

However, we DON’T WANT the black part.  So we’re going to subtract just the black part from the .4332.  What is the proportion of just the black part?  Well, it starts at the mean (zero) and goes to 0.5.  So, I’m going to look up the z-score of 0.5 and look up the Proportion Between Mean and Z.  According to the book, that is .1915.  Now we subtract:

Proportion of 0 to 1.5 – Proportion of 0 to 0.5 = Proportion of 0.5 to 1.5

.4332 – .1915 = .2417

Proportion of 0.5 to 1.5 = .2417

Second:  The other value we can get is the value from 0.5 to the end, and then subtract out the part we don’t want:

If we look up the z-score 0.5 and the corresponding Proportion in Tail, we get .3085.  That is both the red and black parts.  Now, we look up the z-score 1.5 to get THAT Proportion in Tail, which is .0668.  Now subtract:

Proportion of Tail starting at 0.5 – Proportion of tail starting at 1.5 = Proportion of 0.5 to 1.5

.3085 – .0668 = .2417

Proportion of 0.5 to 1.5 = .2417

Using either method, we arrive at the answer: A proportion of students equal to .2417 got B’s on this test.

Distribution of Sample Means and the Standard Error

This one was slightly more confusing, but we’ll try our best to explain it.  So far, we’ve been dealing with a single data set: either a single sample or the one whole population.  The sample mean is just that, the mean of the items in that ONE SAMPLE.  The basic idea behind statistics is that if you experiment with enough samples out of a population, you should start getting overall scores that match the population as a whole.  The sample mean for each experiment will likely be different, but get enough sample means, and you should get an idea of what the population mean is.  We should be able to say that certain scores are expected as we do more and more experiments. 

The question is CONFIDENCE.  Confidence tells you how often a statement will be true.  When given a problem, the wording will usually look like “Given the average SAT score of 500, what are the range of SAT scores that we can say with 80% confidence will be the sample mean for any given sample.”  Alternatively, it might say, “Given the average SAT score of 500, what are the range of SAT scores that will show up as sample means at least 80% of the time?”  In this case, instead of comparing a specific value to the mean to find the z-score, we are comparing the sample mean itself to the whole population mean.  So, instead of the deviation of actual values compared to the mean, we have to find the potential change in the sample mean compared to the population mean.  This is called the standard error.

Standard Error =

We use the standard error instead of the standard deviation when we are comparing the sample mean to the population mean.  Otherwise, the z-score formula is nearly the same.  We find the difference between the sample mean (M) and the population mean (µ), and then divide by the standard error instead of the standard deviation:

z =

Let’s do that example and see where we get:  “Given the average SAT score of 500 (µ), what are the range of SAT scores that will show up as sample means at least 80% of the time.  The sample size is 25 (n = 25) and the standard deviation is 100. (σ = 100)” 

First, let’s set up the normal curve:

We are looking for the values that are common enough that they occur at least 80% of the time when you take a sample from this population.  We know that values closest to the mean are the most common, going in both directions from there.  Therefore, we want to choose the 80% of values that are closest to the mean.  We do this by choosing the 40% closest values to the mean on either side:

This area contains all of the SAT sample means that will occur 80% of the time.  The next step is to find the z-scores associated with these areas.  Since we are going from the mean outwards in each direction, we will use the Z distribution table and take a look at the Proportion Between Mean and Z column.  We want to find the value that is closest to 40%, or .4000.  As we look down that column, we find that .4000 falls between two values: .3997 and .4015.  We will choose one closest to .4000, which is .3997.  The z-score associated with that value is 1.28. 

Now, this 1.28 is only for one side, from the mean out to 40%.  Since we know that the proportions are the same on both sides, that means the other 40% goes out to –1.28.  This gives us the boundaries of our total 80 percent: from –1.28 to +1.28.

But, we’re not quite done yet.  We still need to figure out what the SAT scores are that go with each Z score.  Starting with the z formula that we have for this:

z =

If we try and fill in what we know—the average SAT score is 500 (µ = 500), the z-score (-1.28 and 1.28), and a standard deviation of 100 (σ = 100), we see that we do NOT have the standard error.  We better go find that first:

Now, let’s plug in that σ = 100 and n = 25:

NOW we can plug in the z-score (-1.28 and 1.28), the average SAT score (µ = 500), and the standard error (20) into the Z formula and solve:

Multiply both sides by 20:

Then add 500:

One of the range boundaries is an SAT score of 525.6.  Let’s go find the other score for the other z-score:

The other score is 474.4.  Therefore, we know that we are 80% confident that the value of the sample mean in a sample of 25 students with a population average of 500 and a standard deviation of 100 will be between 474.4 and 525.6.

Whew.

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