Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Chi-Square Distribution

In order to make inferences about the population variance on the basis of the sample variance, we have to consider a special distribution, called chi-square (χ2) distribution: if a random variable Y is normally distributed with mean µ and variance σ2, then the quantity

shows a χ2 distribution with n-1 degrees of freedom for a random sample of size n. Several examples of χ2 distributions for different degrees of freedom are shown in the figure below.

As you can see, the χ2 distribution is skewed and is always positive. The mean of the χ2 distribution is equal to the number of degrees of freedom n-1, the variance is twice the degrees of freedom. The χ2 distribution is tabulated in statistical tables, or can be calculated online by means of the distribution calculator. The χ2 distribution is used to test differences between population and sample variances, and between theoretical and observed distributions.

An important property of the χ2 distribution is its additivity: if two independent variables follow a χ2 distribution (exihibiting the degrees of freedom f1 and f2), then the sum of the two variables is also χ2-distributed with a degree of freedom of f1+f2.