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In statistics, Cochran's theorem is used in the analysis of variance.

Suppose U₁, ..., U_n are independent standard normally distributed random variables, and an identity of the form

can be written where each Q_i is a sum of squares of linear combinations of the Us. Further suppose that

where r_i is the rank of Q_i. Cochran's theorem states that the Q_i are independent, and Q_i has a chi-square distribution with r_i degrees of freedom.

Cochran's theorem is the converse of Fisher's theorem.

[edit] Example

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

is standard normal for each i.

It is possible to write

(here, summation is from 1 to n, that is over the observations). To see this identity, multiply throughout by and note that

and expand to give

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

and the second term is just n identical terms added together.

Combining the above results (and dividing by σ²), we have:

Now the rank of Q₂ is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q₁ can be shown to be n − 1, and thus the conditions for Cochran's theorem are met.

Cochran's theorem then states that Q₁ and Q₂ are independent, with Chi-squared distribution with n − 1 and 1 degree of freedom respectively.

This shows that the sample mean and sample variance are independent; also

To estimate the variance σ², one estimator that is often used is

.

Cochran's theorem shows that

which shows that the expected value of is σ²(n − 1)/n.

Both these distributions are proportional to the true but unknown variance σ²; thus their ratio is independent of σ² and because they are independent we have

where F_1,n is the F-distribution with 1 and n degrees of freedom (see also Student's t-distribution).

This page uses content from the English-language version of Wikipedia. The original article was at Cochran's_theorem. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License.