Introduction to One-Way ANOVA – Fresh Take

  • Write a null and alternative hypothesis for a one-way ANOVA hypothesis test
  • Discuss the error sum of squares and group sum of squares

The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test actually uses variances to help determine if the means are equal or not.

The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different.

hypotheses

The null hypothesis for a one-way ANOVA states that all the group/population means are the same. This can be written as:

[latex]H_0: \mu_1 = \mu_2 = ... = \mu_k[/latex]

where [latex]k[/latex] is the number of independent groups or samples.

 

The alternative hypothesis for a one-way ANOVA should be written as:

[latex]H_{A}:[/latex] At least two of the group means are different.

If the null hypothesis is false, then the variance of the combined data is larger, which is caused by the different means as shown in the second graph (green boxplots).3 boxplots that are orange labeled (a), and 3 boxplots that are green labeled (b). The boxplots correspond to the description below.
(a) [latex]H_0[/latex] is true. All means are the same; the differences are due to random variation. (b) [latex]H_0[/latex] is not true. All means are not the same; the differences are too large to be due to random variation.

But what does it mean when we reject the null hypothesis? Remember that an ANOVA only tells us that there is a difference, not which group(s) are different. Let’s use colors to understand it better.