The following is a draft of an article I have submitted for publication in CHANCE Magazine, a publication of the American Statistical Association. With their encouragement, I am publishing it here to solicit comments from readers (and possibly corrections).
The Inspection Paradox is Everywhere
The inspection paradox is a common source of confusion, an occasional source of error, and an opportunity for clever experimental design. Most people are unaware of it, but like the cue marks that appear in movies to signal reel changes, once you notice it, you can’t stop seeing it.
A common example is the apparent paradox of class sizes. Suppose you ask college students how big their classes are and average the responses. The result might be 56. But if you ask the school for the average class size, they might say 31. It sounds like someone is lying, but they could both be right.
The problem is that when you survey students, you oversample large classes. If there are 10 students in a class, you have 10 chances to sample that class. If there are 100 students, you have 100 chances. In general, if the class size is x, it will be overrepresented in the sample by a factor of x.
That’s not necessarily a mistake. If you want to quantify student experience, the average across students might be a more meaningful statistic than the average across classes. But you have to be clear about what you are measuring and how you report it.
By the way, I didn’t make up the numbers in this example. They come from class sizes reported by Purdue University for undergraduate classes in the 2013-14 academic year. https://www.purdue.edu/datadigest/2013-14/InstrStuLIfe/DistUGClasses.html
From the data in their report, I estimate the actual distribution of class sizes; then I compute the “biased” distribution you would get by sampling students. The CDFs of these distributions are in Figure 1.
Going the other way, if you are given the biased distribution, you can invert the process to estimate the actual distribution. You could use this strategy if the actual distribution is not available, or if it is easier to run the biased sampling process.
Figure 1: Undergraduate class sizes at Purdue University, 2013-14 academic year: actual distribution and biased view as seen by students.
The same effect applies to passenger planes. Airlines complain that they are losing money because so many flights are nearly empty. At the same time passengers complain that flying is miserable because planes are too full. They could both be right. When a flight is nearly empty, only a few passengers enjoy the extra space. But when a flight is full, many passengers feel the crunch.
Once you notice the inspection paradox, you see it everywhere. Does it seem like you can never get a taxi when you need one? Part of the problem is that when there is a surplus of taxis, only a few customers enjoy it. When there is a shortage, many people feel the pain.
Another example happens when you are waiting for public transportation. Busses and trains are supposed to arrive at constant intervals, but in practice some intervals are longer than others. With your luck, you might think you are more likely to arrive during a long interval. It turns out you are right: a random arrival is more likely to fall in a long interval because, well, it’s longer.
To quantify this effect, I collected data from the Red Line in Boston. Using their real-time data service, I recorded the arrival times for 70 trains between 4pm and 5pm over several days.
Figure 2: Distribution of time between trains on the Red Line in Boston, between 4pm and 5pm.
The shortest gap between trains was less than 3 minutes; the longest was more than 15. Figure 2 shows the actual distribution of time between trains, and the biased distribution that would be observed by passengers. The average time between trains is 7.8 minutes, so we might expect the average wait time to be 3.8 minutes. But the average of the biased distribution is 8.8 minutes, and the average wait time for passengers is 4.4 minutes, about 15% longer.
In this case the difference between the two distributions is not very big because the variance of the actual distribution is moderate. When the actual distribution is long-tailed, the effect of the inspection paradox can be much bigger.
An example of a long-tailed distribution comes up in the context of social networks. In 1991, Scott Feld presented the “friendship paradox”: the observation that most people have fewer friends than their friends have. He studied real-life friends, but the same effect appears in online networks: if you choose a random Facebook user, and then choose one of their friends at random, the chance is about 80% that the friend has more friends.
The friendship paradox is a form of the inspection paradox. When you choose a random user, every user is equally likely. But when you choose one of their friends, you are more likely to choose someone with a lot of friends. Specifically, someone with x friends is overrepresented by a factor of x.
To demonstrate the effect, I use data from the Stanford Large Network Dataset Collection (http://snap.stanford.edu/data), which includes a sample of about 4000 Facebook users. We can compute the number of friends each user has and plot the distribution, shown in Figure 3. The distribution is skewed: most users have only a few friends, but some have hundreds.
We can also compute the biased distribution we would get by choosing choosing random friends, also shown in Figure 3. The difference is huge. In this dataset, the average user has 42 friends; the average friend has more than twice as many, 91.
Figure 3: Number of online friends for Facebook users: actual distribution and biased distribution seen by sampling friends.
Some examples of the inspection paradox are more subtle. One of them occurred to me when I ran a 209-mile relay race in New Hampshire. I ran the sixth leg for my team, so when I started running, I jumped into the middle of the race. After a few miles I noticed something unusual: when I overtook slower runners, they were usually much slower; and when faster runners passed me, they were usually much faster.
At first I thought the distribution of runners was bimodal, with many slow runners, many fast runners, and few runners like me in the middle. Then I realized that I was fooled by the inspection paradox.
In many long relay races, teams start at different times, and most teams include a mix of faster and slower runners. As a result, runners at different speeds end up spread over the course; if you stand at random spot and watch runners go by, you see a nearly representative sample of speeds. But if you jump into the race in the middle, the sample you see depends on your speed.
Whatever speed you run, you are more likely to pass very slow runners, more likely to be overtaken by fast runners, and unlikely to see anyone running at the same speed as you. The chance of seeing another runner is proportional to the difference between your speed and theirs.
We can simulate this effect using data from a conventional road race. Figure 4 shows the actual distribution of speeds from the James Joyce Ramble, a 10K race in Massachusetts. It also shows biased distributions that would be seen by runners at 6.5 and 7.5 mph. The observed distributions are bimodal, with fast and slow runners oversampled and fewer runners in the middle.
Figure 4: Distribution of speed for runners in a 10K, and biased distributions as seen by runners at different speeds.
A final example of the inspection paradox occurred to me when I was reading Orange is the New Black, a memoir by Piper Kerman, who spent 13 months in a federal prison. At several points Kerman expresses surprise at the length of the sentences her fellow prisoners are serving. She is right to be surprised, but it turns out that she is the victim of not just an inhumane prison system, but also the inspection paradox.
If you arrive at a prison at a random time and choose a random prisoner, you are more likely to choose a prisoner with a long sentence. Once again, a prisoner with sentence x is oversampled by a factor of x.
Using data from the U.S. Sentencing Commission, I made a rough estimate of the distribution of sentences for federal prisoners, shown in Figure 5. I also computed the biased distribution as observed by a random arrival.
Figure 5: Approximate distribution of federal prison sentences, and a biased distribution as seen by a random arrival.
As expected, the biased distribution is shifted to the right. In the actual distibution the mean is 121 months; in the biased distribution it is 183 months.
So what happens if you observe a prison over an interval like 13 months? It turns out that if your sentence is y months, the chance of overlapping with a prisoner whose sentence is x months is proportional to x + y.
Figure 6 shows biased distributions as seen by hypothetical prisoners serving sentences of 13, 120, and 600 months.
Figure 6: Biased distributions as seen by prisoners with different sentences.
Over an interval of 13 months, the observed sample is not much better than the biased sample seen by a random arrival. After 120 months, the magnitude of the bias is about halved. Only after a very long sentence, 600 months, do we get a more representative sample, and even then it is not entirely unbiased.
These examples show that the inspection paradox appears in many domains, sometimes in subtle ways. If you are not aware of it, it can cause statistical errors and lead to invalid inferences. But in many cases it can be avoided, or even used deliberately as part of an experimental design.
I discuss the class size example in my book, Think Stats, 2nd Edition, O’Reilly Media, 2014, and the Red Line example in Think Bayes, O’Reilly Media, 2013. I wrote about relay races, social networks, and Orange Is the New Black in my blog, “Probably Overthinking It”. http://allendowney.blogspot.com/
John Allen Paulos presents the friendship paradox in Scientific American, “Why You're Probably Less Popular Than Your Friends”, 18 January 2011. http://www.scientificamerican.com/article/why-youre-probably-less-popular/
The original paper on the topic might be Scott Feld, “Why Your Friends Have More Friends Than You Do”, American Journal of Sociology, Vol. 96, No. 6 (May, 1991), pp. 1464-1477. http://www.jstor.org/stable/2781907
Amir Aczel discusses some of these examples, and a few different ones, in a Discover Magazine blog article, “On the Persistence of Bad Luck (and Good)”, 4 September 4, 2013.
The code I used to generate these examples is in these Jupyter notebooks:
Allen Downey is a Professor of Computer Science at Olin College of Engineering in Massachusetts. He is the author of several books, including Think Python, Think Stats, and Think Bayes. He is a runner with a maximum 10K speed of 8.7 mph.