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

: <math>\operatorname{E}( X ^ 2 )</math>, where <math>\operatorname{E}</math> is the expected value operator.

For a random variable <math>X</math> with mean <math>\mu</math> and variance <math>\sigma^2</math>,

: <math>\sigma^2 = \operatorname{E}( X ^ 2 ) - \mu^2.</math><ref>Mood & Graybill: ''An introduction to the Theory of Statistics'' (McGraw Hill)</ref>

(Its derivation is shown here.) Therefore,

: <math>\operatorname{E}( X ^ 2 ) = \sigma^2 + \mu^2.</math>

From the above, the following can be derived:

: <math>\operatorname{E}\left( \sum\left( X ^ 2\right) \right) = n\sigma^2 + n\mu^2,</math>

: <math>\operatorname{E}\left( \left(\sum X \right)^ 2 \right) = n\sigma^2 + n^2\mu^2.</math>

== 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 <math display="block"> S = \sum x ^ 2 - \frac{\left(\sum x\right)^2}{n}. </math>

From the two derived expectations above the expected value of this sum is <math display="block"> \operatorname{E}(S) = n\sigma^2 + n\mu^2 - \frac{n\sigma^2 + n^2\mu^2}{n}, </math> which implies <math display="block"> \operatorname{E}(S) = (n - 1)\sigma^2. </math>

This effectively proves the use of the divisor ''n'' − 1 in the calculation of an '''unbiased''' sample estimate of ''σ''<sup>2</sup>.

== Partition — analysis of variance == {{main|Partition of sums of squares}}

In the situation where data is available for ''k'' different treatment groups having size ''n''<sub>''i''</sub> where ''i'' varies from 1 to ''k'', then it is assumed that the expected mean of each group is

: <math>\operatorname{E}(\mu_i) = \mu + T_i</math>

and the variance of each treatment group is unchanged from the population variance <math>\sigma^2</math>.

Under the Null Hypothesis that the treatments have no effect, then each of the <math>T_i</math> will be zero.

It is now possible to calculate three sums of squares:

;Individual

:<math>I = \sum x^2 </math>

:<math>\operatorname{E}(I) = n\sigma^2 + n\mu^2</math>

;Treatments

:<math>T = \sum_{i=1}^k \left(\left(\sum x\right)^2/n_i\right)</math>

:<math>\operatorname{E}(T) = k\sigma^2 + \sum_{i=1}^k n_i(\mu + T_i)^2</math>

:<math>\operatorname{E}(T) = k\sigma^2 + n\mu^2 + 2\mu \sum_{i=1}^k (n_iT_i) + \sum_{i=1}^k n_i(T_i)^2</math>

Under the null hypothesis that the treatments cause no differences and all the <math>T_i</math> are zero, the expectation simplifies to

:<math>\operatorname{E}(T) = k\sigma^2 + n\mu^2.</math>

;Combination

:<math>C = \left(\sum x\right)^2/n</math>

:<math>\operatorname{E}(C) = \sigma^2 + n\mu^2</math>

===Sums of squared deviations===

Under the null hypothesis, the difference of any pair of ''I'', ''T'', and ''C'' does not contain any dependency on <math>\mu</math>, only <math>\sigma^2</math>.

:<math>\operatorname{E}(I - C) = (n - 1)\sigma^2</math> total squared deviations aka ''total sum of squares''

:<math>\operatorname{E}(T - C) = (k - 1)\sigma^2</math> treatment squared deviations aka ''explained sum of squares''

:<math>\operatorname{E}(I - T) = (n - k)\sigma^2</math> 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.

:<math>I = \frac{1^2}{1} + \frac{2^2}{1} + \frac{3^2}{1} + \frac{4^2}{1} + \frac{6^2}{1} = 66</math>

:<math>T = \frac{(1 + 2 + 3)^2}{3} + \frac{(4 + 6)^2}{2} = 12 + 50 = 62</math>

:<math>C = \frac{(1 + 2 + 3 + 4 + 6)^2}{5} = 256/5 = 51.2</math>

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}}

==References== <References/>

Category:Statistical deviation and dispersion Category:Analysis of variance