Chi-square Distribution

Consider a normally distributed population from which several independent samples of size n are drawn, and the sample variance is calculated. The resulting distribution is called the chi-square distribution. The chi-square distribution is used to estimate the population variance and the standard deviation.

Unlike the normal and t distributions, the chi-square distribution is skewed to the right. 

However, the shape of the distribution curve varies for each degree of freedom, where the number of degrees of freedom is generally n minus one.

As the degrees of freedom increase, the symmetry of the curve approaches that of the normal distribution. At degrees of freedom greater than 90, the chi-square distribution approximately resembles a normal distribution.

As one can see, the chi-square test statistic can be greater than or equal to zero but never negative.

This distribution has wide applications in tests of independence, goodness-of-fit tests, and single variance tests.