Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
Home Univariate Data Distributions Sampling Distributions ChiSquare Distribution  
See also: sampling distributions, tdistribution, Fdistribution  
ChiSquare DistributionIn order to make inferences about the population variance on the basis of the sample variance, we have to consider a special distribution, called chisquare (χ^{2}) distribution: if a random variable Y is normally distributed with mean µ and variance σ^{2}, then the quantity shows a χ^{2} distribution with n1 degrees of freedom for a random sample of size n. Several examples of χ^{2} distributions for different degrees of freedom are shown in the figure below. As you can see, the χ^{2} distribution is skewed and is always positive. The mean of the χ^{2} distribution is equal to the number of degrees of freedom n1, the variance is twice the degrees of freedom. The χ^{2} distribution is tabulated in statistical tables, or can be calculated online by means of the distribution calculator. The χ^{2} distribution is used to test differences between population and sample variances, and between theoretical and observed distributions.
An important property of the χ^{2} distribution is its additivity: if two independent variables follow a χ^{2} distribution (exihibiting the degrees of freedom f_{1} and f_{2}), then the sum of the two variables is also χ^{2}distributed with a degree of freedom of f_{1}+f_{2}.


Home Univariate Data Distributions Sampling Distributions ChiSquare Distribution 