- Complete a one-way ANOVA hypothesis test
- Write the conclusion of a one-way ANOVA hypothesis test in context of the problem
Suppose a researcher wants to investigate the effect of the amount of fertilizer on the height of a common houseplant. More specifically, the researcher is interested in determining if there is a difference in the mean height of plants between those receiving one of the following three different fertilizer levels: high, medium, and low.
The following data are the simulated results of this controlled experiment.
(Note that this small data set is used to introduce the concept and make calculations easier. When conducting an ANOVA, larger sample sizes are usually needed to meet assumptions.)
| Fertilizer Level | Height of Plant (inches) |
| Low | 23.2, 20.9, 21.5, 25.3 |
| Medium | 24.6, 27.7, 22.5, 30.1 |
| High | 29.2, 30.2, 31.1, 33.6 |
As we saw previously, conducting a one-way ANOVA involves comparing the variation within each of the groups to the variation between each of the groups. When the variation between each of the groups is significantly larger than the variation within each of the groups, we might conclude that there is a statistically significant difference among the means.
ANOVA table
In an ANOVA table, the calculation illustrating the total variation within the groups of interest is known as the error sum of squares (SSError). The calculation illustrating the total variation between the groups is known as the group sum of squares (SSGroup).
Two other essential columns found in an ANOVA table are the degrees of freedom (df) and the mean square.
The following table illustrates how these values are calculated for each of the given sources: Group or Error (i.e., between and within).
When calculating these values, it is important to know that represents the number of groups being considered and represents the total number of data values among all groups.
ANOVA table
| Source | Degrees of Freedom (df) | Sum of Squares | Mean Square | F-Statistic |
| Group |
[latex]k-1[/latex]
(The number of groups minus 1) |
SSGroup | [latex]\dfrac{\text{SSGroup}}{k-1}[/latex] | [latex]\dfrac{\text{MSGroup}}{\text{MSError}}[/latex] |
| Error |
[latex]N-k[/latex]
(The total number of data points minus the number of groups) |
SSError | [latex]\dfrac{\text{SSError}}{N-k}[/latex] | |
| Total |
[latex]N-1[/latex]
(The total number of data points minus 1) |
SSGroup + SSError |