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Wikipedia / 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.

Background

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

  1. is the expected value operator.

For a random variable,

(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

This effectively proves the use of the divisor n1 in the calculation of an unbiased sample estimate of2.

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

.

will be zero.

It is now possible to calculate three sums of squares:

  1. Individual

    Treatments

are zero, the expectation simplifies to

  1. Combination

Sums of squared deviations

Under the null hypothesis, the difference of any pair of I, T, and C.

  1. 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 (n1), (k1), and (nk) 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

  1. Total squared deviations = 66 51.2 = 14.8 with 4 degrees of freedom.

  2. Treatment squared deviations = 62 51.2 = 10.8 with 1 degree of freedom.

  3. Residual squared deviations = 66 62 = 4 with 3 degrees of freedom.

Two-way analysis of variance

See also

References