ANOVA (Analysis of Variance)

Analysis of Variance (ANOVA) is a statistical concept that features in many statistical problem-solving methodologies like Design of Experiments, Regression, and Measurement Systems Analysis.

ANOVA tests identify statistical differences between the means of three or more unrelated groups and determine how independent variables influence dependent variables. For example, you can use it to compare group means, run an exploratory data analysis, or measure one variable's influence over another.

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One-Way ANOVA vs. Two-Way ANOVA

ANOVA tests break into two main categories—One-Way and Two-Way.


One-Way ANOVAs are used to test hypotheses containing a single independent variable and are designed to determine whether the means of two or more groups are different.

You'd typically use this type of test to compare groups of individuals engaged in different tasks.

For example, you might use One-Way ANOVA to study the impact of different exercise routines on weight loss across three groups: swimming, jogging, and yoga.


Two-Way ANOVAs build on the One-Way test measuring two or more independent variables.

For example, a Two-Way ANOVA allows organizations to compare job applicants based on two independent variables like experience level and salary expectations. This test is used to identify the relationship between variables and test the impact of both factors simultaneously.

What Are the ANOVA Assumptions?

There are three primary ANOVA assumptions related to "residuals."

Residuals represent the difference between an actual data point and the fitted value. When sample sizes are equal, the “fits” should be equivalent to the mean of each group. Studying residuals allows practitioners to identify erratic or misleading ANOVA results before using their findings as the basis for real-world decision-making.

Here's a quick overview of the assumptions:

  • Residuals Must Be Random and Independent. We check this with the residuals versus order graph.
  • Residuals Must Be Distributed Normally. We'll use the normal probability plot and histogram to help determine this.
  • Residuals Must Have Constant Variance Across All Levels. If a pattern is present, it’s time to loop back.

If any of these three assumptions fail, you'll need to go back and check for issues like unstable conditions, poorly-defined measurement systems, or data entry errors before running the test again.

How to Practice ANOVA

Below, we’ve outlined some general steps for practicing One-Way and Two-Way ANOVA testing. Most practitioners use software to streamline ANOVA testing. You can use Excel, though platforms like Minitab or SigmaXL are more useful for running ANOVA tests on more complex data sets.

Here’s a brief look at the steps involved with both tests.

Running a One-Way ANOVA:

  • Randomly select at least 15 responses for each factor level.
  • Next, you'll want to determine the null and alternative hypotheses for the ANOVA and set the alpha value of the test.
  • The third step is running the ANOVA test and interpreting the results.
  • Then use Tukey's Test to run multiple comparisons. Here, you're testing multiple hypotheses at a time, checking for differences between pairs of levels in a study.
  • And finally, you'll want to look at the assumptions we went over in the last section to catch any issues before moving forward.

Running a Two-Way ANOVA:

  • Start by randomly selecting data for each factor level—data should be in a stable, controlled environment to ensure that you're analyzing each group under the same conditions.
  • The second step is determining the null and alternative hypothesis.
  • Next, run the Two-Way ANOVA test and interpret the results.
  • The final step is working through the three assumptions.

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