Squared deviations from the mean

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 where is the expected value operator. For a random variable with mean and variance , Therefore, From the above, the following can be derived: Sample variance 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 From the two derived expectations above the expected value of this sum is which implies This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ2. Partition of sums of squares In the situation where data is available for k different treatment groups having size ni where i varies from 1 to k, then it is assumed that the expected mean of each group is and the variance of each treatment group is unchanged from the population variance . Under the Null Hypothesis that the treatments have no effect, then each of the will be zero. It is now possible to calculate three sums of squares: Individual Treatments Under the null hypothesis that the treatments cause no differences and all the are zero, the expectation simplifies to Combination Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on , only . total squared deviations aka total sum of squares treatment squared deviations aka explained sum of squares residual squared deviations aka residual sum of squares The constants (n − 1), (k − 1), and (n − k) are normally referred to as the number of degrees of freedom. In a very simple example, 5 observations arise from two treatments.