Some Distribution Theory

Here we outline some of the theory behind the chi-square, t- and F-distributions.

Recall from Inferential Statistics that the chi-squared test uses the chi-squared statistic to determine whether or not two population characteristics are related in some way. While a large value for the chi-squared statistic suggests that the two characteristics are not independent, a small value for the chi-squared statistic suggests that the two characteristics are independent.

The latter situation is the assumption underlying the chi-square distribution. In particular, the chi-square distribution is the probability function one would expect the chi-squared statistic to follow if the two characteristics being studied are indeed independent.

More theoretically, if Y1, Y2, …, Yn are normally and independently distributed random variables, each with mean 0 and variance 1, then

chi2 = sigma(Yi2)

follows a chi-square distribution with n degrees of freedom.

The assumption that the means of two samples represent two different normally distributed populations is the basis for constructing the Student’s t-distribution. In particular, the Student’s t-distribution is the probability function one would expect the t-statistic to follow if a sample standard deviation is used to approximate the population standard deviation.

More theoretically, if X and Y are independently distributed random variables then t = Xsqrt(n)/Y is a Student’s t-distribution with n degrees of freedom, assuming X is normally distributed with mean 0 and variance sigma2 and Y2/sigma2 follows a chi-square distribution with n degrees of freedom. Thus the Student’s t-distribution is constructed from normal and chi-square distributions. Since the chi-square distribution is constructed from normal distributions, this means the basic building block of the t-distribution is the normal distribution.

The theory of the F-distribution is based on two independently distributed random variables, X and Y. If X follows a chi-square distribution with m degrees of freedom and Y follows a chi-square distribution with n degrees of freedom, then F = X/m/(Y/n) is an F-distribution with m and n degrees of freedom. Thus the F-distribution is constructed from normal and chi-square distributions. Since the chi-square distribution is constructed from normal distributions, this means the basic building block of the F-distribution is the normal distribution…

(abstract)…