Introduction

The definition of variance is either the expected value (when considering a theoretical distribution), or average (for actual experimental data) of squared deviations from the mean. Computations for analysis of variance involve the partitioning of a sum of squared deviations. An understanding of the complex computations involved is greatly enhanced by a detailed study of the statistical value:
In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ...

It is well-known that for a random variable x with mean μ and variance σ²:

^[1]

Therefore:



From the above, the following are readly derived:

Sample Variance

The sum of squared deviations needed to calculate 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:



Giving:



This effectively proves the use of the divisor (n − 1) in the calculation of an unbiassed sample estimate of σ²

In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

Partition - Analysis of Variance

In the situation where data is available for k different treatment groups having size n_i 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 σ².
Under the Null Hyporthesis 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:

Individual

Treatments

Under the null hypothesis that the treatments cause no differences and all the T_i are zero, the expectation simplifies to:

Combination

Sums of Squared Deviations

Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on μ, only σ².

Total Squared Deviations

Treatment Squared Deviations

Residual Squared Deviations

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:
Total squared deviations = 66 - 51.2 = 14.8 with 4 degrees of freedom.
Treatment squared deviations = 62 - 51.2 = 10.8 with 1 degree of freedom.
Residual squared deviations = 66 = 62 = 4 with 3 dgrees of freedom.

Two-Way Analysis of Variance

The following hypothetical example gives the yields of 15 plants subject to two environmental variations, and three fertilisers.

Five sums of squares are calculated:

Factor	Calculation	Sum	σ2
Individual	72 + 22 + 12 + 72 + 62 + 112 + 62 + 102 + 72 + 32 + 52 + 32 + 42 + 112 + 42	641	15
Fertiliser x Environment		556.1667	6
Fertiliser		525.4	3
Environment		519.2679	2
Composite		504.6	1

Finally, the sums of squared deviations required for the Analysis of Variance can be calculated.
In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ...

References

1 Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)
2 Variance_decomposition
The well-known variance decomposition rule is given by: See also iterated expectations and law of total variance for proof. ...

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