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.

Background

An understanding of the computations involved is greatly enhanced by a study of the statistical value For a random variable X with mean \mu and variance \sigma^2, (Its derivation is shown here.) 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 — analysis of variance

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 \sigma^2. Under the Null Hypothesis that the treatments have no effect, then each of the T_i will be zero. It is now possible to calculate three sums of squares: Under the null hypothesis that the treatments cause no differences and all the T_i are zero, the expectation simplifies to

Sums of squared deviations

Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on \mu, only \sigma^2. The constants (n − 1), (k − 1), and (n − k) are normally referred to as the number of degrees of freedom.

Example

In a very simple example, 5 observations arise from two treatments. The first treatment gives three values 1, 2, and 3, and the second treatment gives two values 4, and 6. Giving

Two-way analysis of variance