CK-12 Probability and Statistics - Advanced

5.1. The Standard Normal Probability Distribution http://www.ck12.org

Z-Scores

Az−scoreis a measure of the number of standard deviations a particular data point is away from the mean. For
example, let’s say the mean score on a test for your statistics class were an 82 with a standard deviation of 7 points.
If your score was an 89, it is exactly one standard deviation to the right of the mean, therefore yourz−score would
be 1. If, on the other hand you scored a 75, your score is exactly one standard deviation below the mean, and your
z−score would be−1. To show that it is below the mean, we will assign it az−score of negative one. All values that
are below the mean will have negativez−scores. Az−score of negative two would represent a value that is exactly
2 standard deviations below the mean, or 82− 14 =68 in this example.

To calculate az−score in which the numbers are not so obvious, you take the deviation and divide it by the standard
deviation.

z=

Deviation

Standard Deviation

You may recall that deviation is the observed value of the variable, subtracted by the mean value, so in symbolic
terms, thez−score would be:

z=

x−x ̄

sd

Ex. What is thez−score for anAon this test? (assume that anAis a 93).

z=

x−x ̄

sd

z=

93 − 82

7

z=

11

7

≈ 1. 57

It is not necessary to have a normal distribution to calculate az−score, but thez−score has much more significance
when it relates to a normal distribution. For example, if we know that the test scores from the last example are