A very short post today!

Finally one of the major music streaming services is available in Australia (I have complained about the limited options for music streaming down under here before). Spotify has now launched in Australia. I have signed up to investigate the service, so it is a bit early to give an opinion on how good it is, but I know it is very popular in Europe and the US.

I would like to see other services opening up in Australia, including Slacker, Rhapsody and Pandora, but given that all of these have been around for longer that Spotify, I will not be holding my breath.

Last month the IMF published their latest Global Financial Stability Report. A colleague, who knows I rarely approve of pie charts*, drew my attention to the charts on page 27 of Chapter 3 of the report, which I have reproduced here (click on the image to enlarge). 

Here the authors of the report have decided to attempt some graphical improvisation, taking the pie chart and extending it. Over time some inspired new chart designs have been developed, but these have been rare. More often the result is inferior to using an established technique. While I do not wish to discourage innovation, the results should always be tested before being foisted on an unsuspecting audience.

The aim of this pair of charts is to illustrate the dwindling supply of “safe assets” in the form of highly rated sovereign debt as a result of the global financial crisis. For example, at the end of 2007, 68% of advanced economies boasted a AAA Standard & Poor’s credit rating (left hand chart, outer red arc) but  by January 2012 this proportion had fallen to 52% (left hand chart, inner red sector).

The heart of each chart is a conventional pie chart showing the current distribution of country ratings. Taken in isolation, either one of these would be a reasonable chart. But moving beyond a single pie chart, comparing the Advanced Economies chart to the Emerging Markets chart is not so easy. Edward Tufte’s adage from The Visual Display of Quantitative Information comes to mind: “the only worse design than a pie chart is several of them”. The crime against charting here is made particularly egregious with the choice of a colour scheme for ratings that is not consistent across the two charts!

If that wasn’t bad enough, the design comes right off the rails with the outer charts. These are a form of annular pie chart, but the alignment of each segment is shifted in an attempt to make the pre-crisis figure more readily comparable to the post-crisis figure for each rating. The result is highly confusing: it takes a while to work out exactly what is going on. Messing with the alignment of the outer chart also makes it harder to compare one rating to another. Even the decision to position the 2012 data in the middle and the 2007 data on the outside is a mistake. My eye expects a flow from the centre of the circle outwards rather than from outside in. An informal, if statistically insignificant, survey suggests that I am not the only one with this expectation.

The aim of any data visualisation is to provide easy access to the information. Understanding the IMF report’s chart is just too much work. A simple table of figures would have been easier to understand. But there are also more conventional charts that would do a better job. The chart below is an example of the “small multiples” technique. This involves a grid of similar charts which are readily compared as certain parameters are varied. In this case, scanning the charts horizontally reveals changes through time and vertically the differences between advanced economies and emerging markets.

Sovereign ratings from before the crisis (2007) to now (2012)

Some space could have been saved by restricting the vertical axis to a 0% to 70% range, but with the full 0% to 100% range the proportions for each rating are more readily grasped.

The small multiples chart is a vast improvement on the IMF original, and is a good illustration of the fact that choosing the right chart makes it far easier to visualise the patterns in your data.

* One of the few pie charts I do approve of is this one (I have seen this one in many places, but I am not sure of the original source).

Another post and another Law, but this time no mathematics is involved.

Imagine you are running a team of salespeople and, as a highly motivated manager, you are working on strategies to improve the performance of your team. After a close study of your team’s client call reports you realise that the high performers in the team consistently meet with their clients more frequently than the poor performers. Eureka! You now have a plan: you set targets for the number of times your team should meet with clients each week. Bonuses will depend upon performance against these targets. Confident that your new client call metric is highly correlated with sales performance, is objective and easily measurable, you sit back and wait.

Six months later, it is time to review the results. Initially you are pleased to discover that a number of your poor performers have achieved very good scores relative to your new targets. Most of the high performers have done well also, although you are a little disappointed that your best salesperson came nowhere near the “stretch target” you set. You then begin to review the sales results and find them very puzzling: despite the high number of client meetings, the results for most of your poor performers are worse than ever. Not only that, your top salesperson has had a record quarter. After you have worked out whether you can wriggle out of the commitment you made to link bonuses to your new metric, you would do well to reflect on the fact that you have fallen victim to Goodhart’s Law.

According to Goodhart’s Law, the very act of targeting a proxy (client meetings) to drive a desired outcome (sales performance) undermines the relationship between the proxy and the target. In the client meeting example, the relationship clearly broke down because your team immediately realised it was straightforward to “game” the metric, recording many meetings without actually doing a better job of selling. Your highest performer was probably too busy doing a good job to waste their clients’ time with unnecessary meetings.

The Law was first described in 1975 by Charles Goodhart in a paper delivered to the Reserve Bank of Australia. It had been observed that there was a close relationship between money supply and interest rates and, on this basis, the Bank of England began to target money supply levels by setting short-term interest rates. Almost immediately, the relationship between interest rates and money supply broke down. While the reason for the breakdown was loosening of controls on bank lending rather than salespeople gaming targets, the label “Goodhart’s Law” caught on.

Along with its close relatives Campbell’s Law and the Lucas Critique, Goodhart’s Law has been used to explain a broad range of phenomena, far removed from its origins in monetary policy. In 18th century Britain, a crude form of poll tax was levied based on the number of windows on every house. The idea was that the number of windows would be correlated with the number of people living in the house. It did not take long for householders to begin bricking up their windows. A more apocryphal example is the tale of the Soviet-era nail factory. Once central planners set targets for the weight of nail output, artful factory managers met their target by making just one nail, but an enormous and very heavy nail.

Much like the Law of Unintended Consequences, of which it is a special case, Goodhart’s Law is one of those phenomena that, once you learn about it, you cannot help seeing it at work everywhere.

Here is a quick quiz. If you visit the Wikipedia page List of countries by GDP, you will find three lists ranking the countries of the world in terms of their Gross Domestic Product (GDP), each list corresponding to a different source of the data. If you pick the list according to the CIA (let’s face it, the CIA just sounds more exciting than the IMF or the World Bank), you should have a list of figures (denominated in US dollars) for 216 countries. Ignore the fact that the European Union is in the list along with the individual counties, and think about the first digit of each of the GDP values. What proportion of the data points start with 1? How about 2? Or 3 through to 9?

If you think they would all be about the same, you have not come across Benford’s Law. In fact, far more of the national GDP figures start with 1 than any other digit and fewer start with 9 than any other digit. The columns in the chart below shows the distribution of the leading digits (I will explain the dots and bars in a moment).

Distribution of leading digits of GDP for 216 countries (in US$)

This phenomenon is not unique to GDP. Indeed a 1937 paper described a similar pattern of leading digit frequencies across a baffling array of measurements, including areas of rivers, street addresses of “American men of Science” and numbers appearing in front-page newspaper stories. The paper was titled “The Law of Anomalous Numbers” and was written by Frank Benford, who thereby gave his name to the phenomenon.

Benford’s Law of Anomalous Numbers states that that for many datasets, the proportion of data points with leading digit n will be approximated by

log₁₀(n+1) – log₁₀(n).

So, around 30.1% of the data should start with a 1, while only around 4.6% should start with a 9. The horizontal lines in the chart above show these theoretical proportions. It would appear that the GDP data features more leading 2s and fewer leading 3s than Benford’s Law would predict, but it is a relatively small sample of data, so some variation from the theoretical distribution should be expected.

As a variation of the usual tests of Benford’s Law, I thought I would choose a rather modern data set to test it on: Twitter follower numbers. Fortunately, there is an R package perfectly suited to this task: twitteR. With twitteR installed, I looked at all of the twitter users who follow @stubbornmule and recorded how many users follow each of them. With only a relatively small follower base, this gave me a set of 342 data points which follows Benford’s Law remarkably well.

;

Distribution of leading digits of follower counts

As a measure of how well the data follows Benford’s Law, I have adopted the approach described by Rachel Fewster in her excellent paper A Simple Explanation of Benford’s Law. For the statistically-minded, this involves defining a chi-squared statistic which measures “badness” of Benford fit. This statistic provides a “p value” which you can think of as the probability that Benford’s Law could produce a distribution that looks like your data set. The follower-count for @stubbornmule is a very high 0.97, which shows a very good fit to the law. By way of contrast, if those 342 data points had a uniform distribution of leading digits, the p value would be less than 10^-15, which would be a convincing violation of Benford’s Law.

Since so many data sets do follow Benford’s Law, this kind of statistical analysis has been used to detect fraud. If you were a budding Enron-style accountant set on falsifying your company’s accounts, you may not be aware of Benford’s Law. As a result, you may end up inventing too many figures starting with 9 and not enough starting with 1. Exactly this style of analysis is described in the 2004 paper The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data by Durtshi, Hillison and Pacini.

By this point, you are probably asking one question: why does it work? It is an excellent question, and a surprisingly difficult and somewhat controversial one. At current count, an online bibliography of papers on Benford Law lists 657 papers on the subject. For me, the best explanation is Fewster’s “simple explanation” which is based her “Law of the Stripey Hat”. However simple it may be, it warrants a blog post of its own, so I will be keeping you in suspense a little longer. In the process, I will also explain some circumstances in which you should not expect Benford’s Law to hold (as an example, think about phone numbers in a telephone book).

In the meantime, having gone to the trouble of adapting Fewster’s R Code to produce charts testing how closely twitter follower counts fit Benford’s Law, I feel I should share a few more examples. My personal twitter account, @seancarmody, has more followers than @stubbornmule and the pattern of leading digits in my followers’ follower counts also provides a good illustration of Benford’s Law.

One of my twitter friends, @stilgherrian, has even more followers than I do and so provides an even larger data set.

Even though the bars seem to follow the Benford pattern quite well here, the p value is a rather low 5.5%. This reflects the fact that the larger the sample, the closer the fit should be to the theoretical frequencies if the data set really follows Benford’s Law. This result appears to be largely due to more leading 1s than expected and fewer leading 2s. To get a better idea of what is happening to the follower counts of stilgherrian’s followers, below is a density* histogram of the follower counts on a log₁₀ scale.

There are a few things we can glean from this chart. First, the spike at zero represents accounts with only a single follower, accounting around 1% of stilgherrian’s followers (since we are working on a log scale, the followers with no followers of their own do not appear on the chart at all). Most of the data is in the range 2 (accounts with 100 followers) to 3 (accounts with 1000 followers). Between 3 and 4 (10,000 followers), the distribution falls of rapidly. This suggests that the deviation from Benford’s Law is due to a fair number users with a follower count in the 1000-1999 range (I am one of those myself), but a shortage in the 2000-2999 range. Beyond that, the number of data points becomes too small to have much of an effect.

Histogram of follower counts of @stilgherrian’s followers

Of course, the point of this analysis is not to suggest that there is anything particularly meaningful about the follower counts of twitter users, but to highlight the fact that even the most peculiar of data sets found “in nature” is likely to yield to the power of Benford’s Law.

* A density histogram scales the vertical axis to ensure that the histogram covers a region of area one rather than the frequency of occurrences in each bin.

On Saturday night I found myself in Melbourne at the first live performance in 30 years of the reggae band Pressure Drop.

The last time Pressure Drop played I probably couldn’t have told you what reggae was. Although I would certainly have heard Eddy Grant’s Electric Avenue on the radio, I was a New Romantic at heart and was more inclined to listen to the likes of Adam and the Ants. At this point, in the interests of full disclosure, I should admit that Pressure Drop’s concert was the second of the weekend for me: I also saw Adam Ant at the Enmore Theatre the night before.

Since my New Romantic period, I have learned to appreciate reggae and dub, but that doesn’t explain why I came to be in the audience at the Caravan Music Club. The real reason was that Pressure Drop’s guitarist is Bill Mitchell, macroeconomist, Modern Monetary Theory (MMT) pioneer and the man behind Billy Blog.

Bill and I first came into contact in 2008 when we both blogged about alternative Olympic medal tallies (my post is here and Bill’s is here). I then became very interested in Bill’s expositions on the importance a country’s monetary system has for understanding the possibilities for fiscal policy. Bill’s ideas were the inspiration of many of my posts, such as Blame Greece’s Debt Crisis on the Euro and Park the Debt Truck and I eventually came to know Bill in person after attending a couple of CofFEE conferences.

Bill’s academic interests extend from the workings of money, to concerns about full employment and equity (my only podcast to date featured Bill explaining his idea of a job guarantee, which is underpinned by an understanding of MMT). Since reggae has its roots in concerns about social justice, it came as no real surprise to learn that Bill had once played reggae. Then last year the band started rehearsing again and these rehearsals culminated in the release of a new CD, aLIVE 2011. I was keen to hear the results and I suspect I was one of the first to order a copy.

Pressure Drop

Although Bill is based in Newcastle these days, Pressure Drop had always been a Melbourne band, so it was only natural that their return performance should be in Melbourne. I had been planning a trip to Melbourne for some time and fortuitously it coincided with the night of the concert. So it was that I found myself out at Oakleigh RSL on Saturday night.

Bill Mitchell on guitar with Pressure Drop

Before Pressure Drop and after I beat my brother at a game of pool, there was a two piece support act: Ross Hannaford (of Daddy Cool fame) and Bart Willoughby, the drummer from No Fixed Address. I would hate to imagine how many years Brian has being playing guitar, but he is certainly able to coax beautiful sounds from those strings, apparently with no effort at all.

Once Pressure Drop came to the stage, it wasn’t long before dancing began. Bill had plenty of reverb on his guitar, but not so much as to hide the fact that he can play as well as he can blog (more than can be said for me). The band had a house full of enthusiastic supporters and they did not disappoint. The concert was great fun for all of us on the dance floor and it looked as though the band enjoyed themselves just as much. Don’t be surprised if Bill trades his professorial chair for a Fender on a permanent basis.

With the iPhone in hand, I managed to get a few photographs (the band has posted more here) and even a couple of brief video clips which I have crudely spliced.

I’ll be keeping an eye out now to see whether Pressure Drop decide to take the band on the road and play in Sydney. I can’t be sure when I’ll be in Melbourne next.

UPDATE: here is a video of the entire concert (1:40).

Just over a year ago, I wrote about the digital “crypto-currency” Bitcoin. It has been an eventful year for Bitcoin.

Designed to provide a secure yet anonymous, decentralised means for making payments online, the first Bitcoins were virtually minted in 2009. By early 2011, Bitcoin had begun to attract attention. Various sites, including the not-for-profit champion of rights online, the Electronic Frontier Foundation (EFF), began accepting Bitcoins as payment. But when Gawker reported that Bitcoins could be used to buy drugs on “underground” website Silk Road, interest in the currency exploded and within a few days, the price of Bitcoins soared to almost $30.

This kind of attention was unwelcome for some, and shortly afterwards EFF announced that they would no longer be accepting Bitcoins, fearing that this would be construed as an endorsement of the now controversial currency. Around the same time, the first major theft of Bitcoins was reported and the Bitcoin exchange rate fell sharply.

Bitcoin Exchange Rate

More recently, another high-profile theft has caused ructions in the Bitcoin economy, prompting e-payments provider and PayPal competitor, Paxum, to abandon the Bitcoin experiment, which in turn forced one of the larger Bitcoin “exchanges” to shut down. The anonymity of Bitcoin is a design feature, but it also makes it almost impossible to trace thieves once they have their virtual hands on Bitcoins.

How much damage this does to the fledgling currency remains to be seen, but it certainly makes for a volatile currency. The free-floating Australian dollar is a reasonably volatile real-world currency but, as is evident in the chart below, Bitcoin volatility is an order of magnitude higher. That in itself is reason enough for any online business to think twice about accepting Bitcoins.

Rolling 30 day volatility (annualised)

Whatever its future, Bitcoin is a fascinating experiment and, even if it does not survive, digital currencies of one form or another are surely here to stay.

Data sources: Bitcoin charts, Bloomberg.

Recently, a colleague drew my attention to the “Bristol Pound“, an example of a “local currency“.   Ah yes, I said, that’s been around for a few years now. Embarrassingly, I later realised I was thinking about the “Brixton Pound“. Having attended many concerts at the legendary Brixton Academy (Nick Cave, Ministry and the Sugarcubes among them), I really should have known the difference between Bristol and Brixton!

There are now a number of local currencies in Britain. The first to appear in recent years was the  ”Totnes Pound“, launched in March 2007. According to their website, the benefits of the Totnes Pound are:

• To build resilience in the local economy by keeping money circulating in the community and building new relationships
• To get people thinking and talking about how they spend their money
• To encourage more local trade and thus reduce food and trade miles
• To encourage tourists to use local businesses

The aims of the Brixton Pound, the Bristol Pound and the other local currencies are essentially the same. As far as I can tell, the take up of these currencies to date has been modest, but the Bristol Pound represents an interesting new development. Not only does it have a far slicker website, but it also offers payment by mobile phone. Perhaps most significantly, according to the FAQ, ”Business members that pay business rates to Bristol City Council will be able to pay in Bristol Pounds.”

A key tenet of “Modern Monetary Theory” is that the value of fiat money is not underpinned by gold or any other commodity; rather its value derives from the government levying tax in that currency. Since almost everyone has to pay tax at some point, this creates a base level of demand for the currency. So, perhaps the fact that the Bristol City Council is supporting the Bristol Pound will enhance its take-up prospects. It would be even more interesting if the council decided that they would only accept Bristol Pounds as payment for rates.

Following on from the last post on the Hottest 100, I received a few tweets from @mjdart demanding a deeper dive into the data. One of his questions was

Of artists charting in at least 5 yrs, are Oz artists higher represented?

I decided to broaden the questions to look at artists with at least five tracks in Hottest 100s (so artists with two tracks in one year and one track in three other years would be in). On this criterion, Australia still comes out on top.

Number of artists with at least 5 “Hot” tracks

In the last post, I complained that 2010 data is currently missing from Wikipedia. It seems that this is because Wikipedia is yet to get permission from the ABC. I have decided to risk the wrath of Auntie and have posted the full chart in the table at the bottom of the post. Having pieced it together, I have updated my original chart to include 2010.

As you can see, 2010 was a good year for Australian artists. It also turns out that the artist with the most Hottest 100 tracks is also Australian: Powderfinger. Here are all the artists with at least 10 Hottest 100 tracks.

UPDATE:

@mjdart asked another question which I thought I should also answer:

@stubbornmule If you assign 100 pts for #1 thru 1 pt for #100 each year, is Oz proportion higher/lower? eg Oz filling out top or bottom 50?

To answer this, I assigned 100 points for the top spot, 99 points for the second and so on down to one point for last place. I summed the score for each country and then scaled it by dividing by 50.5. This odd choice arises because 100 + 99 + 98 + … + 2 + 1 = 5050 and so dividing by 50.5 would give a score of 100 for a country that managed to win every spot in the top 100. This makes the score directly comparable to a simple count of places in the top 100. So, how does this weighted score compare to a simple count? The answer, evident in the chart below is not much! So, each country’s artists must be fairly evenly spread through the top 100 over time.

Finally, here is the complete listing of the 2010 Hottest 100, including country of origin. If you are feeling brave, you may wish to update Wikipedia. Just keep in mind, the list may be deleted again if the ABC does not provide permission for the list to be published!

Another year, another Australia Day. Another Australia Day, another Triple J Hottest 100. And that, of course, means an excellent excuse to  set R to work on the chart data.

For those outside Australia, the Hottest 100 is a chart of the most popular songs of the previous year, as voted by the listeners of the radio station Triple J. The tradition began in 1991, but initially people voted for their favourite song of all time. From 1993 onwards, the poll took its current form* and was restricted to tracks released in the year in question.

Since the Hottest 100 Wikipedia pages include country of origin**, I thought I would see whether there is any pattern in whose music Australians like best. Since it is Australia Day, it is only appropriate that we are partial to Australian artists and they typically make up close to half of the 100 entries. Interestingly, in the early 90s, Australian artists did not do so well. The United Kingdom has put in a good showing over the last two years, pulling ahead of the United States. Beyond the big three, Australia, UK and US, the pickings get slim very quickly, so I have only included Canada and New Zealand in the chart below.

Number of Hottest 100 tracks by Country

If you have excellent eyesight, you may notice that 2010 is missing from the chart. For some reason, this is the only year which does not include the full chart listing on the Wikipedia page. There is a link to a list on the ABC website, but unfortunately it does not include the country of origin. Maybe a keen Wikipedian reading this post will help by updating the page.

I make no great claims for the sophistication or the insight of this analysis: it was really an excuse to learn about using the XML package for R to pull data from tables in web pages.

require(XML)
require(ggplot2)
require(reshape2)

results <- data.frame()
col.names <- c("year", "rank", "title", "artist", "country")

# Skip 2010: full list is missing from Wikipedia page
years <- c(1993:2009, 2011)

for (year in years) {
    base.url <- "http://en.wikipedia.org/wiki/Triple_J_Hottest_100,"
    year.url <- paste(base.url, year, sep="_")
    tables <- readHTMLTable(year.url, stringsAsFactor=FALSE)
    table.len <- sapply(tables, length)
    hot <- cbind(year=year, tables[[which(table.len==4)]])
    names(hot) <- col.names
    results <- rbind(results, hot)
}

# Remap a few countries
results$country[results$country=="Australia [1]"] <- "Australia"
results$country[results$country=="England"] <- "United Kingdom"
results$country[results$country=="Scotland"] <- "United Kingdom"
results$country[results$country=="Wales"] <- "United Kingdom"
results$country[results$country=="England, Wales"] <-"United Kingdom"

# Countries to plot
top5 <- c("Australia", "United States", "United Kingdom",
  "Canada", "New Zealand")

# Create a colourful ggplot chart
plt <- ggplot(subset(results, country %in% top5),
    aes(factor(year), fill=factor(country)))
plt <- plt + geom_bar() + facet_grid(country ~ .)
plt <- plt + labs(x="", y="") + opts(legend.position = "none")

Created by Pretty R at inside-R.org

UPDATE: there is a little bit more analysis in this follow-up post.

* Since the shift to single year charts, there have been two all-time Hottest 100s: 1998 and 2009.

** There are some country combinations, such as “Australia/England”, but the numbers are so small I have simply excluded them from the analysis.

Spending couple of weeks down the south coast of New South Wales, spotting dolphins and echidnas, has slowed down my blogging. Fortunately, regular contributor James Glover has once more come to the rescue with a guest post. This time his topic is wheelchairs and air-travel.

Perhaps you’ve heard of a recent court case in which a wheelchair user, Sheila King, took Jetstar to court (and lost) on the basis of the Disabilities Discrimination Act? If you are a wheelchair user and you book a flight on one of our airline carriers then a fairly obvious thing won’t happen. Unlike say a bus you won’t be able to board the aircraft in your chair and be strapped in for the journey. What actually happens is that when making the booking you tick a box (or tell the booker on the phone) that you are in a wheelchair. If there are seats available for wheelies when you get to the airport you will give up your chair and be made to use a specially designed “wheelchair” (its a chair, it has wheels) that is designed to be fit the narrow corridor of most planes which I am sure you are aware of – their narrowness, for you, only apparent when the person ahead of you is blocking the aisle loading 3 pieces of carry on luggage into the overhead lockers while chatting to their new friends in the seat they are meant to occupy. We all suffer this situation. These “wheelchairs” are not designed to be used without help, they are more like children’s toy carts and cannot be operated by the user as the wheels are very small and low down. For a wheelchair user to be taken out of their wheelchair in a public place can be quite discombobulating. Many wheelchair users develop a personal relationship with their chair – it is after all a place you spend many of your waking hours.

Digression. The very first time I was in a wheelchair outside the confines of a hospital ward (it was a hospital wheelchair but is the exact same model I now own, like I said it is personal) I was being pushed by none other than the proprietor of this very website! Without going into the details let’s just say it was a pretty dramatic event and we both learned a valuable lesson in wheelchair use and the wheelchair repair workshop at the hospital was kept busy. But I digress.

So here is the thing. According to Google about 1% of the population uses wheelchairs. And a Jetstar plane has about 200 seats so they expect to get about 2 wheelchair users on average per flight. So what is the problem with only allowing this same number on each flight, as some airlines do? Well the problem is that statistically wheelchair users don’t travel in pairs and sometimes there will be less than 2 users and sometimes there will be more. Just as if you toss 10 coins sometimes there will be fewer than 5 heads (the average or expected number) and sometimes there will be more. Only on average will there be 5. In fact it is a simple problem to work out the probability of there being, say, n wheelchair users, given the average of 1% on a 200 seat plane. This is called the Binomial Distribution. If you have access to Excel then the function Binomdist(n,200,1%) will tell you this probability. Before I give you some numbers I admit that the overall population average may not be the same as the average flying on planes. It may be less than 1% due to wheelchair users being put off flying. But maybe on some routes it is higher: but I am guessing the annual “snowbird” migration of retired people from the northern United States to Florida at the start of Winter would track above the 1% rate.

So here are the Binomial probability figures.

For example, assuming a 1% chance of any given passenger a 200 hundred seat plane being in a wheelchair, the probability that there will be exactly 4 wheelchair passengers wanting seats is 9%. To work out the probability of a passenger being denied a seat on their preferred flight, we will assume that we’re dealing with an airline where more than two wheelchair passenges book on a flight, then at least on passenger will have to change their travel plans. From the table above, the chance of the flight only having 0, 1 or 2 wheelchair passengers totals 68%, so there’s a 32% chance that there will be at least one wheelchair passenger who cannot fly. For any one wheelchair passenger, there is a (n-1)/(n+1) chance of being bumped if n other wheelchair passengers book on the flight. Weighting that by the probably that there are n passengers and adding it up for all n>1 gives a probability of 27%. As a frequent flyer in a wheelchair, you can expect to miss out on a seat quite regularly! [Note: these calculations have been updated: the editor's "corrections" were undone. Ed.]

I am quite fortunate now that I no longer need to travel in my wheelchair. But as I still use a walking stick I wait for everyone else to get off the plane. You sit there, looking behind you to see if everyone else has left. But there are always these strange people who seem to sit there at the back of the plane and wait for 10 minutes or more, after everyone has disembarked, before even moving. You wonder why the airline staff don’t just hurry them off? I assume they aren’t disabled because they are sitting at the back of the plane. If airlines really had a problem with the extra time that getting wheelies off the plane then they could make this up by just moving these people along.

When I first read about this case my initial response was that being disabled and traveling is a bit of challenge anyway and you just get on with it. But the more I thought about it I wondered if the airlines just took it for granted that wheelchair users would change their plans to fit in with the rules. I am glad Sheila King took the issue up!