The Inspection Paradox

THE BUS STOP SCENARIO

Take the case of a bus that arrives, on average, every 20 minutes. It’s not a perfect rule—sometimes the bus arrives early and sometimes it’s late. But, when you calculate all the arrival times, it averages out to three times an hour, or every 20 minutes. Now, picture yourself emerging from a side street to the bus stop, with no idea when the bus last arrived. The question that naturally arises is: how long should you expect to wait for the next bus?

Your initial thought might be, ‘Well, if it’s 20 minutes on average, then I should expect to wait around 10 minutes’. This would be halfway between the average intervals and would indeed be the case if the bus arrivals were perfectly spaced out. However, if you find yourself waiting longer than this, you might start to feel like the world is against you. The question then arises: are you just unlucky, or is something else at play?

This is where we introduce the concept of the Inspection Paradox.

UNRAVELLING THE INSPECTION PARADOX

The Inspection Paradox is a statistical phenomenon that reveals how our expected wait times can differ from the average times we calculate, due to the randomness of our inspections or experiences.

To illustrate this, let’s look deeper into the bus scenario. The bus schedule is not as straightforward as it might seem. Remember, the bus arrives every 20 minutes on average, but not at precise 20-minute intervals. Variability changes things.

UNPREDICTABILITY IN THE BUS SCHEDULE

Consider a situation where half of the time the bus arrives at an interval of 10 minutes, and the other half at an interval of 30 minutes. The overall average remains at 20 minutes, but your experience at the bus stop will differ. If you show up at the bus stop at a random time, it’s statistically more probable that you will turn up during the longer 30-minute interval than the shorter 10-minute interval.

This variation has significant implications for your expected wait time. If you land in the 30-minute interval, you can expect to wait around 15 minutes, half of that interval. If you find yourself in the 10-minute interval, you’ll only wait around 5 minutes on average. However, you’re three times more likely to hit the 30-minute gap, which means your expected wait time skews closer to 15 minutes than 5 minutes. On average, your expected wait time becomes 12.5 minutes, contrary to the intuitive answer of 10 minutes. This is calculated as follows: (3 × 15 + 1 × 5)/4 = 50/4 = 12.5 minutes.

IMPLICATIONS OF THE INSPECTION PARADOX

This surprising realisation is the crux of the Inspection Paradox. It essentially states that when you randomly ‘inspect’ or experience an event without knowing its schedule or distribution beforehand, it often seems to take longer than the average time. This isn’t due to some cosmic force giving you a hard time; it’s simply how probability and statistics operate in the randomness of real life.

Understanding the Inspection Paradox can fundamentally change how you interpret your everyday experiences. It’s not about bad luck but rather about understanding that your perception of averages can be skewed by variability around the average.

EVERYDAY INSTANCES OF THE INSPECTION PARADOX

Once you’re aware of the Inspection Paradox, you might start noticing it in various aspects of your everyday life.

EDUCATION INSTITUTION: AVERAGE CLASS SIZE

Consider an educational institution that reports an average class size of 30 students. Now, if you were to randomly ask students from this institution about their class size, you might find that your calculated average is higher than the reported 30.

Why does this happen?

The Inspection Paradox is at play here. If the institution has a range of small and large classes, you’re more likely to encounter students from larger classes in your random sample. This leads to a bigger average class size in your interview sample compared to the actual average class size.

Say, for example, that the institution has class sizes of either 10 or 50, and there are equal numbers of each. In this case, the overall average class size is 30. But in selecting a random student, it is five times more likely that they will come from a class of 50 students than from a class of 10 students. So for every one student who replies ‘10’ to your enquiry about their class size, there will be five who answer ‘50’. So the average class size thrown up by your survey is 5 × 50 + 1 × 10, divided by 6. This equals 260/6 = 43.3. The act of inspecting the class sizes thus increases the average obtained compared to the uninspected average. The only circumstance in which the inspected and uninspected averages coincide is when every class size is equal.

LIBRARY STUDY TIMES

Consider another scenario where you visit a library and conduct a survey asking the attendees how long they usually study. You might notice that the reported study times are generally higher than you might have expected. This can happen not because of any over-reporting but because the sample of students you survey is skewed towards those who spend longer times studying in the library. The reason is that the longer a student stays in the library, the higher the chance you’ll find them there during your random survey. Short-term visitors are less likely to be part of your sample, skewing the average study time upwards.

THE RESTAURANT AND THE SUPERMARKET

You might think about the implications for other scenarios, such as restaurant wait times or queue lengths at supermarkets. For the reasons we have learned about, we might expect our individual experience of waiting to be that little bit longer than a calculation of the unobserved average.

THE PARADOX IN OTHER REAL-LIFE SCENARIOS

Potato Digging

Why do you often accidentally cut through the biggest potato when digging in your garden? It’s because larger potatoes take up more space in the ground, increasing the likelihood of your shovel hitting them.

Downloading Files

Consider the frustration when your internet connection breaks during the download of the largest file. It’s because larger files take longer to download, increasing the window of time for potential connection issues to arise.

CONCLUSION: A NEW LENS

Understanding the Inspection Paradox equips you with a new lens through which to look at the world. It helps explain why your experiences might often differ from average expectations. It’s simply the laws of probability and statistics unfolding in a world full of randomness. With this knowledge, you can navigate the world with more informed expectations and a greater appreciation for statistical realities.