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variance and standard deviation

Variance is a measure of the dispersion of values in a distribution of values.

 In psychology and behavioral science statistics, the variance is typically a reference to the extent to which numerical values vary around the arithmetic mean of a data set. Theoretically, the values vary around a population mean but in most cases, researchers work with samples.

In statistics, write sigma squared for the population variance σ2

Write final form sigma squared for the sample variance ς2

In reports, write VAR for variance.

How it works

If we have a set of different numerical values such as scores on a test we can calculate a mean, which is the average of all the scores divided by the number of scores.

The difference of one score from the mean is a deviation score. X is a score and the Greek letter mu μ is the symbol for the population mean.

In a sample, which is what we normally have in psychology, we subtract a score X from the sample mean M. Thus, X - M = the deviation score. 

If a person earns a 7 on a test where the mean is 10 then their score is 3 points below the mean. The deviation score is -3 (minus three).

To find an average deviation for all the scores on the test, we must subtract each score from the mean. We end up with a set of positive and negative values.

We want to find an average but if we add the positive and negative values we will end up with zero. To solve the problem, 

square each deviation score

add them together to get the sum of the squares (SS)

 then divide by the number of scores (n) to get the variance.

The variance = the SS divided by n

Standard Deviation (SD)

The standard deviation is the average unit by which scores are distant from the mean.

Find the standard deviation by taking the square route of the variance.

In reports, write SD for standard deviation.

Examples in tests

A common mean for IQ tests is 100 and a common standard deviation is 15 points.

In personality tests, a common standard score is a T-score. The mean is T = 50 and SD = 10.

Key concepts in this post

deviation score


sum of squares

standard deviation


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