The Inspection Paradox is Everywhere

From a brilliant blog by Allen Downey:

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 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.

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