Question

Bill Board is “lording” his SAT score over his friend, Rhoda Dendron, who took the ACT. “You only got a 25 in math,” he chortled, “while I got a 300 in math.” Given that the SAT has a μ of 500 and a σ of 100, and the ACT has a μ of 20 and a σ of 5, what is wrong with Bill’s logic (give the answer in both z scores and percentile ranks).

Answer 1

It is all about relative scores that can be calculated by Z scores and percentile ranks. So for SAT score of bill

Z score is calculated by

$Z=\frac{X-\mu}{\sigma}=\frac{300-500}{100}=-2$

And his percentile will be the p value according to the Z score from the table shown below as 0.02275 hence he has the percentile rank of 2.28

And if we look at Z score of SAT score of Rhoda then by zZscore formula

$Z=\frac{X-\mu}{\sigma}=\frac{25-20}{5}=1$

And percentile or P value according to Z score is 0.841. So the percentile rank of Rhode is 84.1

Hence however bill got more marks but he still way behind the Rhode score and percentile rank.