8.5: Chi-square Distribution

How does one determine if bingo numbers are evenly distributed or if some numbers occurred with a greater frequency? Or if the types of movies people preferred were different across different age groups or if a coffee machine dispensed approximately the same amount of coffee each time. These questions can be addressed by conducting a hypothesis test. One distribution that can be used to find answers to such questions is known as the chi-square distribution. The chi-square distribution has applications in tests for independence, goodness-of-fit tests, and test of a single variance.

The properties of the chi-square distribution are as follows:

1. The curve is nonsymmetrical and skewed to the right.
2. There is a different chi-square curve for each degree of freedom (df).
3. The test statistic for any test is always greater than or equal to zero.
4. When df > 90, the chi-square curve approximates the normal distribution.
5. The mean, μ, is located just to the right of the peak.

This text is adapted from 11.1 Facts About the Chi-Square Distribution - Introductory Statistics OpenStax

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.