Squared deviations from the mean

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

From the two derived expectations above the expected value of this sum is $${\displaystyle \mathrm{E}(S)=n{\sigma }^2+n{\mu }^2-\frac{n{\sigma }^2+n^2{\mu }^2}{n},}$$ which implies $${\displaystyle \mathrm{E}(S)=(n-1){\sigma }^2.}$$

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

Partition — analysis of variance

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

${\displaystyle \mathrm{E}({\mu }_i)=\mu +T_i}$

and the variance of each treatment group is unchanged from the population variance ${\displaystyle {\sigma }^2}$.

Under the Null Hypothesis that the treatments have no effect, then each of the ${\displaystyle T_i}$ will be zero.

It is now possible to calculate three sums of squares:

Individual

${\displaystyle I=\sum x^2}$

${\displaystyle \mathrm{E}(I)=n{\sigma }^2+n{\mu }^2}$

Treatments

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

${\displaystyle \mathrm{E}(T)=k{\sigma }^2+\sum _{i=1}^k{n}_i(\mu +T_i{)}^2}$

${\displaystyle \mathrm{E}(T)=k{\sigma }^2+n{\mu }^2+2\mu \sum _{i=1}^k(n_i{T}_i)+\sum _{i=1}^k{n}_i(T_i{)}^2}$

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

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

Combination

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

${\displaystyle \mathrm{E}(C)={\sigma }^2+n{\mu }^2}$

Sums of squared deviations

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

${\displaystyle \mathrm{E}(I-C)=(n-1){\sigma }^2}$ total squared deviations aka total sum of squares

${\displaystyle \mathrm{E}(T-C)=(k-1){\sigma }^2}$ treatment squared deviations aka explained sum of squares

${\displaystyle \mathrm{E}(I-T)=(n-k){\sigma }^2}$ 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

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.

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

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

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

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

In statistics, the two-way analysis of variance (ANOVA) is used to study how two categorical independent variables affect one continuous dependent variable. It extends the One-way analysis of variance (one-way ANOVA) by allowing both factors to be analyzed at the same time. A two-way ANOVA evaluates the main effect of each independent variable and if there is any interaction between them.

Researchers use this test to see if two factors act independent or combined to influence a Dependent variable. It is used in the fields of Psychology, Agriculture, Education, and Biomedical research. For example, it can be used to study how fertilizer type and water level together affect plant growth. The analysis produces F-statistics that indicate whether observed differences between groups are statistically significant.