Z Scores

Definition of z Scores

In order to compute a z score, we start with an original score (called Y) obtained from a sample (or a population) with a mean of MY and a standard deviation of SY. We eliminate the mean by subtracting it from the score, which transforms the original score into a deviation from its mean. We eliminate the original unit of measurement by dividing the score deviation by the standard deviation. Specifically, the formula for calculating a z score is

We say that subtracting the mean centers the distribution and that dividing by the standard deviation normalizes the distribution. The interesting properties of the z scores are that they have a zero mean (effect of centering) and a variance and standard deviation of 1 (effect of normalizing). It is because all distributions expressed in z scores have the same mean (0) and the same variance (1) that we can use z scores to compare observations coming from different distributions.

An Example

Applying the formula for a z score to a score of Y =85 coming from a sample of mean MY = 75 and standard deviation SY = 17 gives

Effect of z Scores

When a distribution of numbers is transformed into z scores, the shape of the distribution is unchanged, but this shape is translated in order to be centered on the value 0, and it is scaled such that its area is now equal to 1.

As a practical guide, when a distribution is normal, more than 99% of the z scores lie between the values −3 and +3. Also, because of the central limit theorem, a z score with a magnitude larger than 6 is extremely unlikely to occur regardless of the shape of the original distribution.

Demonstrating That z Scores Have a Mean of 0 and a Variance of 1

In order to show that the mean of the z scores is equal to 0, it suffices to show that the sum of the z scores is equal to 0. This is shown by developing the formula for the sum of the z scores: In order to show that the variance of the z scores is equal to 1, it suffices to show that the sum of the squared z scores is equal to (N − 1) (where N is the number of scores). This is shown by developing the formula for the sum of the squared z

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