Question

This is vaguely similar to how the IQ scale was constructed. Suppose you want to set up a grading scale for [plug in your favorite candidate], and decide to have it such that the top 2% will score 80 and above, and the bottom 2% will score 20 and below. Also, assume that the distribution will be normal (Gaussian) What should the expectation (μ) and the standard deviation (σ) of the resulting normal ) of the resulting normal distribution be?

Answer 1

Let X be the scores. We are given that top 2% will score 80 and above

ie P(X80)=0.02. from tables, we know that corresponding to p=0.02, the Z is 2.0537.

ie

On rearranging, we get

  

Bottom 2% will score 20 and below

We get from the tables that teh value of Z=-2.0537

Therefore we get,

Adding (1) and (2), we get

Now substituting teh value of in (1), we get

  

The expectation (μ) =50

The standard deviation (σ)=14.6078