Let’s start by defining when a one-way Analysis of Variance is adequate. One of the assumptions in the one-way ANOVA is that the dependent variable is of a continuous nature and the independent variable is categorical. A categorical variable takes values that represent characteristics such as gender (male, female), location (rural, urban, metro).

Let’s look at the simplest case of a researcher who wants to compare 3 groups or conditions on an outcome of interest. These groups are also called levels and the independent variable is known as factor. For example let’s imagine that our researcher is interested in comparing the production of apples across 3 types of soil (volcanic, clay and rocky). If we had only two groups this comparison could be done using a two sample Student’s t-test. Because we have 3 groups we need to perform a one-way Analysis of Variance. Therefore a one-way ANOVA can be understood as an extension of a t-test for more than 2 groups. Note, that if you have a two level factor the one-way ANOVA will give you the exact same p-value than the two-sample t-test.

The main output that you will obtain with any statistical package is the ANOVA table. You want to look at the Between Subjects row and the p-value corresponding to the F statistic. The F test assesses the null hypothesis that the three means are statistically equal versus the alternative that there are significant differences between the three means. A p-value less than or equal to 0.05 means that the null hypothesis is rejected, that is, there are significant differences in the mean production of apples between the three types of soil.

However, realize that this is an omnibus test statistic, in the sense that having a significant p-value for the F test does not tell you which are the soil means that are different. For instance, it could be that volcanic is better than both clay and rocky or that volcanic is only better than rocky. The way to solve these questions is with the so-called post hoc tests. Tukey is the most common one, it will produce all the pairwise comparisons, in our case this is a total number of 6. Each comparison will be accompanied by the mean difference, standard error and the p-value. If a pair shows a significant p-value it will indicate that there is a statistically significant difference between the two levels that are being compared. You will need to look at all the individual p-values to know where the differences are.

And remember that you don’t need to use a post hoc test if the omnibus test is not significant (p-value greater than .05)or if there are only two categories in your factor since you already know the differences are between the two groups that are being compared.