Inspection Paradox

Weekly I/O#49

Why is my bus always late? Inspection paradox: Averaging final results can be different than averaging individual responses.

Article: The Inspection Paradox is Everywhere

If 10 buses come every 60 minutes, how long do we have to wait on average? An educated guess could be 3 minutes because we have to wait half of the average time between two buses (think about we can arrive anytime between two buses), and 1 bus comes every 6 minutes on average. However, we typically must wait longer than 3 minutes and sometimes even longer than 6 minutes. This phenomenon is called Waiting-Time Paradox or Inspection Paradox.

Think about another example first. If you ask college students how big their classes are, the average result might be 100 students. But if you ask the college what's their average class size, it might be only 30. If no one lies, what causes this strange difference? When you survey students and average individual responses, you will overestimate. Because if there are 100 students in one class, you will have a higher chance to sample that class compared to a 10 students class. More specifically, a class with size x will be overrepresented in the whole sample by a factor of x.

Here's another example called the friendship paradox. Why do most people have fewer friends than their friends have? If you choose a random Facebook user and then choose one of their friends randomly, the chance is about 80% that the friend has more friends. This is because when we choose a random user, every user is equally likely to be selected. However, when we pick their friend, people with more friends are more likely to be in this random user's friend group. In other words, one is less likely to be friends with someone who has fewer friends.

Now, can you guess why the bus is always late? Buses should come at constant intervals theoretically. However, in practice, some intervals are longer, and some are shorter. Therefore, similar to the two examples above, we are more likely to arrive at the bus stop during a longer interval because, unsurprisingly, a random arrival is more likely to fall in a long interval.

Inspection paradox (or other Sampling bias) is counterintuitive but ubiquitous. This also reminds me of a joke Bill Burr used in You People Are All The Same.

"Stats are so fucking stupid, you know? Not that they're stupid. It's the way people apply 'em. You already have your mind made up, and then you go to i', you start memorizing a bunch of shit, then you just… blaaah! Just throw it up at people. This guy tried to get me to go scuba diving. I go, "I'm not going. I don't wanna get eaten by a shark." He's like, "Well, actually, 90% of shark attacks actually happen in shallow water." It's like, no shit. That's where the people are. You know? It's called the beach. 90% of people are frolicking along the coastline. It's not like there are people swimming to Europe. "Let's go to Iceland, you pussies!" Right?"

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