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In the Chapter 3 of Think Stats by Allen Downey book. The following is the excerpt from the book:

First, I compute the distribution as observed by students, where the probability associated with each class size is “biased” by the number of students in the class.

def BiasPmf(pmf, label):
  new_pmf = pmf.Copy(label=label)

  for x, p in pmf.Items():
    new_pmf.Mult(x, x)

  new_pmf.Normalize()
  return new_pmf

For each class size, x, we multiply the probability by x, the number of students who observe that class size. The result is a new Pmf that represents the biased distribution.

  1. What is the meaning of the bias here?
  2. Is the author skewing the value by a certain factor and thus, creating an artificial "bias"?

If there's a better forum to discuss questions regarding the book material - please let us know.

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This exercise is illustrating that the 'average class size' as calculated by the Dean is not the same as the average class size as experienced by a student. The difference can be viewed this way: the Dean's average is the expected class size when picking a class at random, while the student's average is the expected class size when picking a student at random and asking the size of that student's class. To create a student-centric distribution of class sizes, you need to replicate each class size x as many times as the number of students in the class (again, x). In other words:

for x, p in pmf.Items():
    new_pmf.Mult(x, x)

This is the sense in which we are 'biasing' class sizes: to give a student's-eye view of the class distribution. It's a legitimate approach, not anything artificial. Both versions of the average class size are correctly calculated; the difference is in the sampling scheme.

(Note this exercise is a little muddled: what if a student takes more than one class? The code for creating the biased distribution is making the simplifying assumption that each student takes just one class.)

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  • $\begingroup$ Thank you. I am having a hard time internalizing this though. Why is there a multiplication happening? $\endgroup$ – neowulf33 Sep 18 '17 at 19:16
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    $\begingroup$ @neowulf33 Example: if a college had just two classes, one with 2 students and one with 3 students, the Dean's data set would be {2, 3}, one value per class. But in the student's view the data set would be {2, 2, 3, 3, 3}, one value per student, i.e., 2 is replicated twice, 3 is replicated thrice. Say the Dean's pmf is p(x). The software achieves replication by replacing each p(x) by x * p(x), and then renormalizes the result. [ Recall Mult(x, c) means 'multiply the probability for value x by the factor c'.] There are other ways to compute the student pmf, e.g. just create a new data set. $\endgroup$ – grand_chat Sep 18 '17 at 19:56

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