What is One Way ANOVA?
Use one way ANOVA to compare the means of three or more groups. This analysis is an inferential hypothesis test that uses samples to draw conclusions about populations. Specifically, it tells you whether your sample provides sufficient evidence to conclude that the groups’ population means are different. ANOVA stands for analysis of variance.
To perform one-way ANOVA, you’ll need a continuous dependent (outcome) variable and a categorical independent variable to form the groups.
For example, one-way ANOVA can determine whether parts made from four materials have different mean strengths.
In this post, learn about the hypotheses, assumptions, and interpreting the results for one-way ANOVA.
Related post: Descriptive vs. Inferential Statistics and Independent and Dependent Variables.
One Way ANOVA Hypotheses
One-way ANOVA has the following hypotheses:
- Null hypothesis: All population group means are equal.
- Alternative hypothesis: Not all population group means are equal.
Reject the null when your p-value is less than your significance level (e.g., 0.05). The differences between the means are statistically significant. Your sample provides sufficiently strong evidence to conclude that the population means are not all equal.
Note that one-way ANOVA is an omnibus test, providing overall results for your data. It tells you whether any group means are different—Yes or No. However, it doesn’t specify which pairs of means are different. To make that determination, follow up a statistically significant one-way ANOVA with a post hoc test that can identify specific group differences that are significant.
Related posts: Interpreting P Values and Null Hypothesis Definition and Inferential Statistics Definition & Examples.
One Way ANOVA Assumptions
For reliable one-way ANOVA results, your data should satisfy the following assumptions:
Use valid sampling methods
Use random sampling to help ensure your sample represents your target population. If your data do not reflect the population, your one-way ANOVA results will not be valid.
Additionally, the method assumes your sampling method obtains independent observations. Selecting one subject does not affect the chances of choosing any others.
Finally, the procedure uses independent samples. Each group contains a unique set of items.
Related posts: Representative Samples: Definition, Uses & Examples and Independent and Dependent Samples
Continuous data
One-way ANOVA requires continuous data. Typically, you quantity continuous variables using a scale that can be meaningfully divided into smaller fractions. For example, temperature, mass, length, and duration are continuous data.
Learn more about Hypothesis Tests for Continuous, Binary, and Count Data.
Data follows a normal distribution or each group has at least 15-20 observations
One-way ANOVA assumes your group data follow the normal distribution. However, your groups can be skewed if your sample size is large enough because of the central limit theorem.
Here are the sample size guidelines:
- 2 – 9 groups: At least 15 in each group.
- 10 – 12 groups: At least 20 per group.
For one-way ANOVA, unimodal data can be mildly skewed and the results will still be valid when all groups exceed the guidelines. Read here for more information about the simulation studies that support these sample size guidelines.
However, if your sample size is smaller, graph your data and determine whether the groups are skewed. If they are, you might need to use a nonparametric test. The Kruskal-Wallis test is the nonparametric test corresponding to one-way ANOVA.
Be sure to look for outliers because they can produce misleading results.
Related posts: Central Limit Theorem & Skewed Distributions
Groups can have equal or unequal variances but use the correct form of the test
One-way ANOVA has two methods for handling group variances. The traditional F-test ANOVA assumes that all groups have equal variances. On the other hand, Welch’s ANOVA does not assume they are equal. If in doubt, just use Welch’s ANOVA because it works well for either case.
Related posts: Variances and Standard Deviations
One Way ANOVA Example
Suppose we are a manufacturer testing four materials to make a part. We collect a random sample of parts made using the four materials and measure their strengths. Download the CSV dataset for this example: PostHocTests.
Use my free online ANOVA Calculator to perform one-way ANOVA!
First, I’ll graph the data to see what we’re working with.
The bar chart shows differences between the group means. However, a graph doesn’t indicate whether those differences are due to chance during random sampling or reflect underlying population differences. One-way ANOVA can help us out with that!
Let’s use one-way ANOVA to determine whether the mean differences between these groups are statistically significant. Below are the statistical results.
The p-value of 0.004 is less than our significance level of 0.05. We reject the null and conclude that all four population means are not all equal. While the Means table shows the group means at the bottom, we don’t know which differences between pairs of groups are statistically significant.
To perform pairwise comparisons between these four groups, we need to use a post hoc test, also known as multiple comparisons. To continue with this example and find the significant group differences, read my post Using Post Hoc Tests with ANOVA.
Related posts: How to do One-Way ANOVA in Excel
Comments and Questions