Question

Imagine all US universities were to give an overly difficult (i.e. unfair) standardized statistics quiz. Now imagine you were to create two different distributions based on samples of all US students’ score: (1) a distribution of individual scores based on one randomly selected student from every US university; and (2) using an average computed from the scores of 5 randomly selected students per university.

How would these distributions look? Would they be unimodal and symmetric (i.e. normal)? How would they look similar or different?

Answer 1

(1)

Distribution of individual scores based on one randomly selected student from every US university will be unimodal, not symmetric (i.e., not normal). It will be positively skewed: many students achieving marks near the bottom end of the score range. The highest marks will be obtained by a few meritorious students.There will be a higher frequency of low scores and a lower frequency of high scores.

(2)

Distribution of scores using an average computed from the scores of 5 randomly selected students per university.will be unimodal, and symmetric (i.e., normal). by Central Limit Theorem.

So,

they would look different: distribution of individual scores positively skewed and distribution of averages normally distributed.