Wednesday, December 09, 2009

A statistical puzzle about averages I

I wrote this a few years ago for a departmental newsletter. For some reason the second part (with the answer) never got published. I stumbled across it almost by accident the other day and thought I'd share it. I'll publish the canonical answer in due course.

Professor Quack knows (from the UK census) that the average family has an arithmetic mean of 1.8 children. He also knows that (due to a bizarre mix-up in enrolment) that his Psychology For Everyone class is attended by a random sample of 50 people from the UK population. As part of an in-class demonstration of sampling theory he records the number of siblings of each student and calculates the average (using the standard formula for the arithmetic mean). 

To his dismay he discovers that the students in his class have an arithmetic mean of 1.2 siblings. He later repeats the demonstration with two more random samples of the UK population (again with n = 50) and obtains values of 1.1 and 1.3 siblings.

Profesor Quack consults two of his colleagues: Professor A and Professor B.

Professor A replies thus:

“You are right to be dismayed. Bias has somehow entered either your sampling procedure or your calculation of the mean. The true mean number of siblings should be 1.8 - 1 = 0.8.”

Professor B interjects thus:

“Nonsense! All is as it should be. The expected number of siblings in a random sample is most certainly not 0.8. Rather, one would expect the average student to have more than 0.8 siblings, just as you have observed.”

1) Who is correct?  Professor A or Professor B?

2) Why?

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