10.4 : One-Way ANOVA: Unequal Sample Sizes

Consider performing a one-way ANOVA test on a dataset with heights of students from three samples with unequal sample sizes.

The null hypothesis is that the mean heights of the three samples are equal, and the alternative hypothesis is that at least one of the mean heights is different.

Compute the F statistic using the ratio of the variance between samples and the variance within samples. Here, x̿ is the combined mean of all observations, ͞x_i is the mean of the i^th sample, n_i is the size of the i^th sample, k is the number of samples and s_i² is the variance of the i^th sample.

Observe that both variance estimates are weighted since they consider sample size to compute the F statistic.

From the P-value, we infer that at least one of the mean heights from the three samples is different. And hence, the null hypothesis is rejected.

Further, to determine which mean height is significantly different from the others, we may construct box plots, construct confidence intervals, or use multiple comparison tests.