- Squared deviations
probability theoryand statistics, the definition of varianceis either the expected value(when considering a theoretical distribution), or average (for actual experimental data) of squared deviations from the mean. Computations for analysis of varianceinvolve the partitioning of a sum of squared deviations. An understanding of the complex computations involved is greatly enhanced by a detailed study of the statistical value:
It is well-known that for a
random variablewith mean and variance :
: [Mood & Graybill: "An introduction to the Theory of Statistics" (McGraw Hill)]
From the above, the following are readly derived:
The sum of squared deviations needed to calculate 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
This effectively proves the use of the divisor in the calculation of an unbiased sample estimate of
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 Hyporthesis that the treatments have no effect, then each of the will be zero.
It is now possible to calculate three sums of squares:
Under the null hypothesis that the treatments cause no differences and all the are zero, the expectation simplifies to
ums 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
: treatment squared deviations
: residual squared deviations
The constants ("n" − 1), ("k" − 1), and ("n" − "k") are normally referred to as the number of degrees of freedom.
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.
: 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
The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
Errors and residuals in statistics
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