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

Let U₁, ..., U_N be i.i.d. standard normally distributed random variables, and ${\displaystyle U=[U_{1},...,U_{N}]^{T}}$. Let ${\displaystyle B^{(1)},B^{(2)},\ldots ,B^{(k)}}$ be symmetric matrices. Define r_i to be the rank of ${\displaystyle B^{(i)}}$. Define ${\displaystyle Q_{i}=U^{T}B^{(i)}U}$, so that the Q_i are quadratic forms. Further assume ${\displaystyle \sum _{i}Q_{i}=U^{T}U}$.

Cochran's theorem states that the following are equivalent:

Often it's stated as ${\displaystyle \sum _{i}A_{i}=A}$, where ${\displaystyle A}$ is idempotent, and ${\displaystyle \sum _{i}r_{i}=N}$ is replaced by ${\displaystyle \sum _{i}r_{i}=rank(A)}$. But after an orthogonal transform, ${\displaystyle A=diag(I_{M},0)}$, and so we reduce to the above theorem.

The following version is often seen when considering linear regression. Suppose that ${\displaystyle Y\sim N_{n}(0,\sigma ^{2}I_{n})}$ is a standard multivariate normal random vector (here ${\displaystyle I_{n}}$ denotes the n-by-n identity matrix), and if ${\displaystyle A_{1},\ldots ,A_{k}}$ are all n-by-n symmetric matrices with ${\displaystyle \sum _{i=1}^{k}A_{i}=I_{n}}$. Then, on defining ${\displaystyle r_{i}=\operatorname {Rank} (A_{i})}$, any one of the following conditions implies the other two:

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

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

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