﻿ Did a Series of Samples Come from the Same Normal Distribution? - ASQ

# Did a Series of Samples Come from the Same Normal Distribution?

By Lloyd S. Nelson

DIFFERENCES among several means from normally distributed populations can be examined using the analysis of variance or the analysis of means. Differences among several variances from normally distributed populations can be examined using Bartlett's test (Dixon and Massey (1983) or an analysis of means type test (Wludyka and Nelson (1997)). In what follows, the problem of testing normally distributed data simultaneously for differences among either means or variances or both is illustrated. This is referred to as a general test of homogeneity. It has the advantage of involving a single significance level.

Problem Illustration: Batches of items are purchased from a company that manufactures them. Prior experience indicates that the characteristic of interest meets the requirement of normality. You wish to test whether the mean and/or the variance have changed over time. Time constraints and economics dictate the batch size, which must be constant, and the frequency of sampling from the supplier s process.

Solution: Imagine that it is feasible to obtain a constant number n = 4 observations taken each week for k = 6 weeks. Table 1 shows an example.

The null hypothesis to be tested is H0: Neither the mean nor the variance has changed over the time of sampling. The alternative hypothesis is H1: Either the mean or the variance or both have changed over the time of sampling. The required statistic is

where is the variance based on all the nk values and the are the variances of the k samples each of size n. All variances are calculated using n - 1 in the denominator. The statistical significance of L0 is evaluated using the critical values in Table 2.

Key Words: Monte Carlo Simulation, Significance Test.