Question

A new version of the Medical College Admissions Test (MCAT) was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded was modified, with the total score of the four sections on the test ranging from 472 to 528. In spring 2015, the mean score was 500.0 with a standard deviation of 10.6.

Required:
What are the median and the first and third quartiles of the MCAT scores?

Answer 1

The median of the MCAT scores was of 500.

The first quartile of MCAT scores was of 492.845.

The third quartile of MCAT scores was of 507.155.

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

The mean score was 500.0 with a standard deviation of 10.6.

This means that
`\mu = 500, \sigma = 10.6`

Median:

In a normal distribution, the median is the same as the mean, so the median of the MCAT scores was of 500.

First quartile:

This is the 100*(1/4) = 25th percentile, which is X when Z has a pvalue of 0.25. So X when Z = -0.675.

`Z = (X - \mu)/(\sigma)`

`-0.675 = (X - 500)/(10.6)`

`X - 500 = -0.675*10.6`

`X = 492.845`

The first quartile of MCAT scores was of 492.845.

Third quartile:

This is the 100*(3/4) = 75th percentile, which is X when Z has a pvalue of 0.75. So X when Z = 0.675.

`Z = (X - \mu)/(\sigma)`

`0.675 = (X - 500)/(10.6)`

`X - 500 = 0.675*10.6`

`X = 507.155`

The third quartile of MCAT scores was of 507.155.