t-tests only test for a significant difference between two samples. If we have three samples (A, B, and C), we would have to do 3 different t-tests (AB, AC, BC); for four samples (A, B, C, and D) we would have to do 6 t-tests (AB, AC, AD, BC, BC, CD); for five samples 10 t-tests; etc. ANOVA, which stands for “analysis of variance”, allows us to do one test for more than two samples, and tells us if at least two samples are significantly different. One-way ANOVA is used when we only have one variable, or factor.

Henceforth, we’ll denote the number of samples as k. So we have k samples, with n1 values in the first sample, n2 values in the second sample, and so on until we have nk values in the kth sample. The null and alternative hypotheses for ANOVA are:

H0: μ1 = μ2 = μ3

Ha: At least two populations are significantly different

Recall that the t-statistic tells us whether or not two populations are most likely significantly different (based on the collected samples), and is a function of how far apart samples are from each other (the numerator, which is the difference between means), and the standard error (the denominator). Remember, the standard error is the estimated standard deviation of our expected population distribution (where this population is either based on a sample or the difference between dependent samples, or the result of subtracting two estimated populations based on two independent samples).

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