How To Calculate A Sum Of Squared Deviations From The Mean (Sum Of Squares)

Concepts such as ​mean​ and ​deviation​ are to statistics what dough, tomato sauce and mozzarella cheese are to pizza: Simple in principle, but having such a variety of interrelated applications that it is easy to lose track of basic terminology and the order in which you must perform certain operations.

Calculating the sum of the squared deviations from the mean of a sample is a step along the way to computing two vital descriptive statistics: the variance and the standard deviation.

To calculate a mean (often referred to as an average), add the individual values of your sample together and divide by ​n​, the total items in your sample. For example, if your sample includes five quiz scores and the individual values are 63, 89, 78, 95 and 90, the sum of these five values is 415, and the mean is therefore

\(415 ÷ 5 = 83\)

In the present example, the mean is 83, so this subtraction exercise yields values of

\((63-83) = -20\)
\((89-83) = 6\)
\((78-83) = -5\)
\((95-83) = 12\)
\((90-83) = 7\)

These values are called the deviations, because they describe the extent to which each value deviates from the sample mean.

In this case:

\((-20)^2 = 400\)
\(6^2 = 36\)
\((-5)^2 = 25\)
\(12^2 =144\)
\(7^2 = 49\)

These values are, as you would expect, the squares of the deviations determined in the previous step.

To get the sum of the squares of the deviations from the mean, and thereby complete the exercise, add the values you calculated in step 3. In this example, this value is

\(400 + 36 + 25 + 144 + 49 = 654\)

The sum of the squares of the deviations is often abbreviated SSD in stats parlance.

This exercise does the bulk of the work involved in calculating the variance of a sample, which is the SSD divided by n − 1, and the standard deviation of the sample, which is the square root of the variance.