In statistics, Cochran's theorem, devised by William G. Cochran,^[1] is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.^[2]

Examples

Sample mean and sample variance

If X₁, ..., X_n are independent normally distributed random variables with mean μ and standard deviation σ then

${\displaystyle U_{i}={\frac {X_{i}-\mu }{\sigma }}}$

is standard normal for each i. Note that the total Q is equal to sum of squared Us as shown here:

${\displaystyle \sum _{i}Q_{i}=\sum _{jik}U_{j}B_{jk}^{(i)}U_{k}=\sum _{jk}U_{j}U_{k}\sum _{i}B_{jk}^{(i)}=\sum _{jk}U_{j}U_{k}\delta _{jk}=\sum _{j}U_{j}^{2}}$

which stems from the original assumption that ${\displaystyle B_{1}+B_{2}\ldots =I}$. So instead we will calculate this quantity and later separate it into Q_i's. It is possible to write

${\displaystyle \sum _{i=1}^{n}U_{i}^{2}=\sum _{i=1}^{n}\left({\frac {X_{i}-{\overline {X}}}{\sigma }}\right)^{2}+n\left({\frac {{\overline {X}}-\mu }{\sigma }}\right)^{2}}$

(here ${\displaystyle {\overline {X}}}$ is the sample mean). To see this identity, multiply throughout by ${\displaystyle \sigma ^{2}}$ and note that

${\displaystyle \sum (X_{i}-\mu )^{2}=\sum (X_{i}-{\overline {X}}+{\overline {X}}-\mu )^{2}}$

and expand to give

${\displaystyle \sum (X_{i}-\mu )^{2}=\sum (X_{i}-{\overline {X}})^{2}+\sum ({\overline {X}}-\mu )^{2}+2\sum (X_{i}-{\overline {X}})({\overline {X}}-\mu ).}$

The third term is zero because it is equal to a constant times

${\displaystyle \sum ({\overline {X}}-X_{i})=0,}$

and the second term has just n identical terms added together. Thus

${\displaystyle \sum (X_{i}-\mu )^{2}=\sum (X_{i}-{\overline {X}})^{2}+n({\overline {X}}-\mu )^{2},}$

and hence

${\displaystyle \sum <!--\; \sum \; -->{\left(\frac{X_i-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}\right)}^2=\sum <!--\; \sum \; -->{\left(\frac{X_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{X}}{\mathrm{\sigma <!--\; \sigma \; -->}}\right)}^2+n{\left(\frac{\stackrel{¯<!--\; ¯\; -->}{X}-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}\right)}^2=\stackrel{Q_1}{\stackrel{⏞<!--\; ⏞\; -->}{\underset{i}{\sum <!--\; \sum \; -->}{\left(U_i-<!--\; -\; -->\frac{1}{n}\underset{j}{\sum <!--\; \sum \; -->}{U}_j\right)}^2}}+\stackrel{Q_2}{\stackrel{⏞<!--\; ⏞\; -->}{\frac{1}{n}{\left(\underset{j}{\sum <!--\; \sum \; -->}{U}_j\right)}^2}}=Q_1+Q_2.}$Zdroj:https://en.wikipedia.org?pojem=Cochran's_theorem
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