Working with Z Scores

Example Problem

Imagine that a nationwide mathematics aptitude test is normally distributed with a mean of 80 and a standard deviation of 12. Here’s a look at the graph of the distribution of mathematics aptitude test scores.

On a separate sheet of paper, draw a normal curve like on the one above and label the mean and the standard deviation.

Percentage of Scores Below a Given Score

Now imagine our score on the aptitude test was 90 and we were curious what percentage of scores were below our score. So we were interested in the area below a score of 90 as shown in the shaded area below.

Draw the histogram above, noting the mean and standard deviation, and then shade in the area the problem is analyzing, below a score of 90.

To perform this calculation, we first find the z score associated with the score of 90:

(90 - 80) / 12

[1] 0.8333333

Then we use pnorm to find the percentage associated with that z score:

pnorm(.83)

[1] 0.7967306

So a score of 90 is at the 80th percentile — approximately 80% of the scores on the aptitude test are less than a score of 90.

Percentage of Scores Above a Given Score

Now let’s say we want to know the percentage of scores above a score of 65, or the area described below.

Draw the histogram above, noting the mean and standard deviation, and then shade in the area the problem is analyzing, above a score of 65.

First, find the z score associated with a score of 65:

(65 - 80) / 12

[1] -1.25

Then subtract the pnorm of that z score from 1.00, because we are looking for the scores above it:

1.00 - pnorm(-1.25)

[1] 0.8943502

So approximately 89% of the scores are above a score of 65.

Rule of thumb: To find the percentage of scores below a value, use pnorm(z). To find the percentage above a value, use 1 - pnorm(z).

Percentage of Scores Between Two Values

You can also find the percentage of scores between two scores by finding the z scores for each and subtracting their pnorm values.

For example, what percentage of scores are between a score of 72 and 106?

Draw the histogram above, noting the mean and standard deviation, and then shade in the area the problem is analyzing, between a score of 72 and 106.

Find the z score for each value, then subtract the smaller pnorm from the larger:

z1 <- (72 - 80) / 12
z2 <- (106 - 80) / 12
pnorm(z2) - pnorm(z1)

[1] 0.7323773

Approximately 74% of scores fall between 72 and 106.