Squared deviations from the mean


Squared deviations from the mean are involved in various calculations. In probability theory and statistics, the definition of variance is either the expected value of the SDM or its average value. Computations for analysis of variance involve the partitioning of a sum of SDM.

Introduction

An understanding of the computations involved is greatly enhanced by a study of the statistical value
For a random variable with mean and variance,
Therefore,
From the above, the following can be derived:

Sample variance

The sum of squared deviations needed to calculate sample variance 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

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

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

Two-way analysis of variance

The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
Extra CO2Extra humidity
No fertiliser7, 2, 17, 6
Nitrate11, 610, 7, 3
Phosphate5, 3, 411, 4

Five sums of squares are calculated:
FactorCalculationSum
Individual64115
Fertilizer × Environment556.16676
Fertilizer525.43
Environment519.26792
Composite504.61

Finally, the sums of squared deviations required for the analysis of variance can be calculated.
FactorSumTotalEnvironmentFertiliserFertiliser × EnvironmentResidual
Individual6411511
Fertiliser × Environment556.166761−1
Fertiliser525.431−1
Environment519.267921−1
Composite504.61−1−1−11
Squared deviations136.414.66820.816.09984.833
Degrees of freedom141229