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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[edit]

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

\operatorname {E} (X^{2})

, where

\operatorname {E}

is the expected value operator.

For a random variable $X$ with mean $\mu$ and variance $\sigma ^{2}$ ,

\sigma ^{2}=\operatorname {E} (X^{2})-\mu ^{2}.

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(Its derivation is shown here.) Therefore,

\operatorname {E} (X^{2})=\sigma ^{2}+\mu ^{2}.

From the above, the following can be derived:

\operatorname {E} \left(\sum \left(X^{2}\right)\right)=n\sigma ^{2}+n\mu ^{2},

\operatorname {E} \left(\left(\sum X\right)^{2}\right)=n\sigma ^{2}+n^{2}\mu ^{2}.

Sample variance[edit]

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

S=\sum x^{2}-{\frac {\left(\sum x\right)^{2}}{n}}

From the two derived expectations above the expected value of this sum is

\operatorname {E} (S)=n\sigma ^{2}+n\mu ^{2}-{\frac {n\sigma ^{2}+n^{2}\mu ^{2}}{n}}

which implies

\operatorname {E} (S)=(n-1)\sigma ^{2}.

This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ².

Partition — analysis of variance[edit]

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

\operatorname {E} (\mu _{i})=\mu +T_{i}

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:

Individual

I=\sum x^{2}

\operatorname {E} (I)=n\sigma ^{2}+n\mu ^{2}

Treatments

T=\sum _{i=1}^{k}\left(\left(\sum x\right)^{2}/n_{i}\right)

\operatorname {E} (T)=k\sigma ^{2}+\sum _{i=1}^{k}n_{i}(\mu +T_{i})^{2}

\operatorname {E} (T)=k\sigma ^{2}+n\mu ^{2}+2\mu \sum _{i=1}^{k}(n_{i}T_{i})+\sum _{i=1}^{k}n_{i}(T_{i})^{2}

Under the null hypothesis that the treatments cause no differences and all the $T_{i}$ are zero, the expectation simplifies to

\operatorname {E} (T)=k\sigma ^{2}+n\mu ^{2}.

Combination

C=\left(\sum x\right)^{2}/n

\operatorname {E} (C)=\sigma ^{2}+n\mu ^{2}

Sums of squared deviations[edit]

Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on $\mu$ , only $\sigma ^{2}$ .

\operatorname {E} (I-C)=(n-1)\sigma ^{2}

total squared deviations aka total sum of squares

\operatorname {E} (T-C)=(k-1)\sigma ^{2}

treatment squared deviations aka explained sum of squares

\operatorname {E} (I-T)=(n-k)\sigma ^{2}

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[edit]

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.

I={\frac {1^{2}}{1}}+{\frac {2^{2}}{1}}+{\frac {3^{2}}{1}}+{\frac {4^{2}}{1}}+{\frac {6^{2}}{1}}=66

T={\frac {(1+2+3)^{2}}{3}}+{\frac {(4+6)^{2}}{2}}=12+50=62

C={\frac {(1+2+3+4+6)^{2}}{5}}=256/5=51.2

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[edit]

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.

References[edit]

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

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

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