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Squared deviations from the mean

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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

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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 ,

[1]

(Its derivation is shown here.) Therefore,

From the above, the following can be derived:

Sample variance

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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

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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

Sums of squared deviations

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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.

Example

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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 degrees of freedom.

Two-way analysis of variance

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In statistics, the two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable. The two-way ANOVA not only aims at assessing the main effect of each independent variable but also if there is any interaction between them.

See also

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References

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  1. ^ Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)