Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of variance is either the expected value of the SDM (when considering a theoretical distribution) or its average value (for actual experimental data). Computations for analysis of variance involve the partitioning of a sum of SDM.

An understanding of the computations involved is greatly enhanced by a study of the statistical value

For a random variable ${\displaystyle X}$ with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$,

(Its derivation is shown here.) Therefore,

From the above, the following can be derived:

The sum of squared deviations needed to calculate sample variance (before deciding whether to divide by n or n − 1) is most easily calculated as $${\displaystyle S=\sum x^{2}-{\frac {\left(\sum x\right)^{2}}{n}}.}$$

From the two derived expectations above the expected value of this sum is $${\displaystyle \operatorname {E} (S)=n\sigma ^{2}+n\mu ^{2}-{\frac {n\sigma ^{2}+n^{2}\mu ^{2}}{n}},}$$ which implies $${\displaystyle \operatorname {E} (S)=(n-1)\sigma ^{2}.}$$

This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ².