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An **ANOVA** (“analysis of variance”) is used to determine whether or not the means of three or more independent groups are equal.

An ANOVA uses the following null and alternative hypotheses:

**H**All group means are equal._{0}:**H**At least one group mean is different from the rest._{A}:

Whenever you perform an ANOVA, you will end up with a summary table that looks like the following:

Source | Sum of Squares (SS) | df | Mean Squares (MS) | F | P-value |
---|---|---|---|---|---|

Treatment | 192.2 | 2 | 96.1 | 2.358 | 0.1138 |

Error | 1100.6 | 27 | 40.8 | ||

Total | 1292.8 | 29 |

Two values that we immediately analyze in the table are the **F-statistic** and the corresponding **p-value**.

**Understanding the F-Statistic in ANOVA**

The **F-statistic** is the ratio of the mean squares treatment to the mean squares error:

- F-statistic: Mean Squares Treatment / Mean Squares Error

Another way to write this is:

- F-statistic: Variation between sample means / Variation within samples

The larger the F-statistic, the greater the variation between sample means relative to the variation within the samples.

Thus, the larger the F-statistic, the greater the evidence that there is a difference between the group means.

**Understanding the P-Value in ANOVA**

To determine if the difference between group means is statistically significant, we can look at the **p-value** that corresponds to the F-statistic.

To find the p-value that corresponds to this F-value, we can use an F Distribution Calculator with numerator degrees of freedom = df Treatment and denominator degrees of freedom = df Error.

For example, the p-value that corresponds to an F-value of 2.358, numerator df = 2, and denominator df = 27 is **0.1138**.

If this p-value is less than α = .05, we reject the null hypothesis of the ANOVA and conclude that there is a statistically significant difference between the means of the three groups.

Otherwise, if the p-value is not less than α = .05 then we fail to reject the null hypothesis and conclude that we do not have sufficient evidence to say that there is a statistically significant difference between the means of the three groups.

In this particular example, the p-value is 0.1138 so we would fail to reject the null hypothesis. This means we don’t have sufficient evidence to say that there is a statistically significant difference between the group means.

**On Using Post-Hoc Tests with an ANOVA**

If the p-value of an ANOVA is less than .05, then we reject the null hypothesis that each group mean is equal.

In this scenario, we can then perform post-hoc tests to determine exactly which groups differ from each other.

There are several potential post-hoc tests we can use following an ANOVA, but the most popular ones include:

- Tukey Test
- Bonferroni Test
- Scheffe Test

Refer to this guide to understand which post-hoc test you should use depending on your particular situation.

**Additional Resources**

The following resources offer additional information about ANOVA tests:

An Introduction to the One-Way ANOVA

An Introduction to the Two-Way ANOVA

The Complete Guide: How to Report ANOVA Results

ANOVA vs. Regression: What’s the Difference?