Squared difference from population mean

A squared standard normal variable has a chi-squared distribution with 1 degree of freedom,

We will find it more useful to deal with the squared difference of Y from µ without dividing by σ. This has a scaled version of the chi-squared distribution,

Sum of n squared normals

A random sample consists of n such independent normal random variables, Y1, Y2, ..., Yn. The sum of their squared differences from µ has a more general form of chi-squared distribution — a chi-squared distribution with n degrees of freedom.


Shape of the chi-squared distribution (n d.f.)

The diagram below shows the chi-squared distribution for different values of n, its degrees of freedom.

Note the following properties of the distribution.

  • The mean of the distribution equals n, its degrees of freedom.
  • As n increases, the distribution approaches the normal distribution, as should be expected from the Central Limit Theorem.