Squared deviations from the mean
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Squared deviations from the mean (SDM) are involved in various calculations. 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.
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 ,
From the above, the following can be derived:
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
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:
Under the null hypothesis that the treatments cause no differences and all the are zero, the expectation simplifies to
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.
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.
|Extra CO2||Extra humidity|
|No fertiliser||7, 2, 1||7, 6|
|Nitrate||11, 6||10, 7, 3|
|Phosphate||5, 3, 4||11, 4|
Five sums of squares are calculated:
|Fertilizer × Environment||556.1667||6|
Finally, the sums of squared deviations required for the analysis of variance can be calculated.
|Factor||Sum||Total||Environment||Fertiliser||Fertiliser × Environment||Residual|
|Fertiliser × Environment||556.1667||6||1||−1|
|Degrees of freedom||14||1||2||2||9|
- Absolute deviation
- Algorithms for calculating variance
- Errors and residuals
- Least squares
- Mean squared error
- Residual sum of squares
- Variance decomposition
- Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)