The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.
The standard deviation measures the spread in the same units as the data. The variance is defined as the square of the standard deviation. Thus, its units differ from that of the original data. The sample variance is represented by , while the population variance is represented by .
For variance, the calculation uses a division by n – 1 instead of n because the data is a sample. This change is due to the sample variance being an estimate of the population variance. Based on the theoretical mathematics behind these calculations, dividing by (n – 1) gives a better estimate of the population variance.
This text is adapted from Openstax, Introductory Statistics, Section 2.7 Measure of the Spread of the Data.