squared deviations from the mean
{{Short description|Calculations in probability theory}}
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
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 ,
: Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)
(Its derivation is shown here.) Therefore,
:
From the above, the following can be derived:
:
:
Sample variance
{{main|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
:
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
{{main|Partition of sums of squares}}
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 .
: 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=
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=
{{excerpt|Two-way analysis of variance}}