Standard Scores

Fundamentals of Social Statistics by Adam J. McKee

One method of helping consumers of research understand the score of a particular individual is to convert the raw score into a standard score.  A standard score, which is also called a z-score, indicates how many standard deviation units a person is from the mean and whether that score is above or below the mean.  A z-score of 1.00 means that the person scored exactly 1 standard deviation unit above the mean, a z-score of 2.00 means that the person scored exactly 2 standard deviation units above the mean, and a z-score of 3.00 means that the person scored exactly 3 standard deviation units above the mean.

We know that each of these scores is above the mean because the value of the z-score is positive.  A negative sign indicates that the z-score is that many standard deviation units below the mean.  Drawing on what we already know about the normal distribution, we can conclude that a z-score of 1.00 puts that person above about 84% of the other subjects.  We know this because 50% of all cases will fall below the mean, and the area between the mean and the first standard deviation to the right of the mean contains 34% of cases, which we know from the 68% Rule.

We can also infer the practical range of z-scores from the 99.7% Rule.  Since almost every case will fall within three standard deviation units of the mean, we can conclude that almost everybody in a given sample will have a z-score of less than 3.0 but not less than -3.0.

Computing an Individual’s z-Score

To compute an individual’s z-score, use the following formula:

z-score formula

X is the person’s raw score

X-baris the mean of the raw scores

S is the standard deviation of the raw scores

Note that the numerator in the z-score formula is a difference score.  When the difference score is divided by the standard deviation, the result is how many standard deviation units that a particular person is away from the mean.

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Last Modified:  06/03/2021


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