Analysis of Variance, or ANOVA, is a statistical method used to compare the means of three or more groups to determine if there are any statistically significant differences among them. ANOVA assesses the variability within and between groups to help researchers understand if the observed differences are due to chance or indicate true effects.
This method is widely used across various fields, such as biology, psychology, business, and social sciences. In this guide, we’ll cover the basics of ANOVA, including its formulas, types, and practical examples.
History of ANOVA
The t- and z-test methods developed in the 20th century were used for statistical analysis. In 1918, when Ronald Fisher created the analysis of variance method.1 For this reason, ANOVA is also called the Fisher analysis of variance, and it’s an extension of the t- and z-tests. The term became well-known in 1925 after appearing in Fisher’s book, “Statistical Methods for Research Workers.”2 It was first employed in experimental psychology and later expanded to other subjects.
The ANOVA test is the first step in analyzing factors that affect a given data set. Once the test is finished, an analyst performs further testing on the factors that measurably might be contributing to the data’s inconsistency. The analyst utilizes the ANOVA test results in an F-test to generate further data that aligns with the proposed regression models.
Analysis of Variance (ANOVA)
ANOVA is a statistical test used to examine differences among the means of three or more groups. Unlike a t-test, which only compares two groups, ANOVA can handle multiple groups in a single analysis, making it an essential tool for experiments with more than two categories.
For example, if a researcher wants to test the effects of three different study methods on student performance, ANOVA can help determine if there are significant performance differences among the groups.
When should I use ANOVA?
If your response variable is numeric, and you’re looking for how that number differs across several categorical groups, then ANOVA is an ideal place to start. After running an experiment, ANOVA is used to analyze whether there are differences between the mean response of one or more of these grouping factors.
ANOVA can handle a large variety of experimental factors such as repeated measures on the same experimental unit (e.g., before/during/after).
If instead of evaluating treatment differences, you want to develop a model using a set of numeric variables to predict that numeric response variable, see linear regression and t tests.
ANOVA Formula
ANOVA formula is made up of numerous parts. The best way to tackle an ANOVA test problem is to organize the formulae inside an ANOVA table. Below are the ANOVA formulae.
| Source of Variation | Sum of Squares | Degree of Freedom | Mean Squares | F Value |
|---|---|---|---|---|
| Between Groups | SSB = Σnj(X̄j– X̄)2 | df1 = k – 1 | MSB = SSB / (k – 1) | f = MSB / MSEor, F = MST/MSE |
| Error | SSE = Σnj(X̄- X̄j)2 | df2 = N – k | MSE = SSE / (N – k) | |
| Total | SST = SSB + SSE | df3 = N – 1 |
where,
- F = ANOVA Coefficient
- MSB = Mean of the total of squares between groupings
- MSW = Mean total of squares within groupings
- MSE = Mean sum of squares due to error
- SST = total Sum of squares
- p = Total number of populations
- n = The total number of samples in a population
- SSW = Sum of squares within the groups
- SSB = Sum of squares between the groups
- SSE = Sum of squares due to error
- s = Standard deviation of the samples
- N = Total number of observations
How do I read and interpret an ANOVA table?
Interpreting any kind of ANOVA should start with the ANOVA table in the output. These tables are what give ANOVA its name, since they partition out the variance in the response into the various factors and interaction terms. This is done by calculating the sum of squares (SS) and mean squares (MS), which can be used to determine the variance in the response that is explained by each factor.
If you have predetermined your level of significance, interpretation mostly comes down to the p-values that come from the F-tests. The null hypothesis for each factor is that there is no significant difference between groups of that factor. All of the following factors are statistically significant with a very small p-value.
Why not calculate multiple t-tests?
ANOVA is used when there are more than two groups. Of course, it would also be a possibility to calculate a t-test for each combination of the groups. The problem here, however, is that every hypothesis test has some degree of error. This probability of error is usually set at 5%, so that, from a purely statistical point of view, every 20th test gives a wrong result
If, for example, 20 groups are compared in which there is actually no difference, one of the tests will show a significant difference purely due to the sampling.
Types of ANOVA
There are several types of ANOVA, each with its own specific use case. The most commonly used types of ANOVA include:
1.One-Way ANOVA
One-way ANOVA is used to compare the means of three or more groups based on a single independent variable. It shows if there is a significant difference among the group means.
- Example: A researcher tests three different fertilizers on plant growth. The independent variable is the fertilizer type, and the dependent variable is the growth rate of plants.
2.Two-Way ANOVA
Two-way ANOVA is used when there are two independent variables, allowing researchers to explore individual and interactive effects.
- Formula:
- The formula involves calculating three F-ratios: one for each main effect and one for their interaction.
- Example: A researcher examines how different teaching methods (lecture vs. discussion) and class times (morning vs. afternoon) impact student performance.
3.Repeated Measures ANOVA
This type is used when the same subjects are tested under different conditions over time, controlling individual variability.
- Example: A psychologist tests stress levels in patients before, during, and after a treatment. Repeated measures ANOVA identifies changes in stress over time.
4.MANOVA (Multivariate Analysis of Variance)
MANOVA is an extension of ANOVA that handles multiple dependent variables, analyzing group differences across several outcomes.
- Example: A company evaluates the effect of training programs on both productivity and job satisfaction.
Difference between one-way and two-way ANOVA
The one-way analysis of variance only checks whether an independent variable has an influence on a metric dependent variable. This is the case, for example, if it is to be examined whether the place of residence (independent variable) has an influence on the salary (dependent variable). However, if two factors, i.e. two independent variables, are considered, a two-way analysis of variance must be used.
| One-way ANOVA | Two-way ANOVA |
|---|---|
| Does a person’s place of residence (independent variable) influence his or her salary? | Does a person’s place of residence (1st independent variable) and gender (2nd independent variable) affect his or her salary? |
Two-way analysis of variance tests whether there is a difference between more than two independent samples that are split between two variables or factors.
Steps in Conducting ANOVA
- State the Hypotheses
- Null Hypothesis (H₀): All group means are equal, indicating no significant difference.
- Alternative Hypothesis (H₁): At least one group mean differs significantly from the others.
- Calculate Sums of Squares
Compute SS Total, SS Between, and SS Within. - Calculate Mean Squares
Divide each sum of squares by its respective degrees of freedom to obtain MSB and MSW. - Compute the F-Ratio
Divide MSB by MSW to calculate the F-ratio. - Interpret the Results
Compare the F-ratio to a critical value from the F-distribution or use the p-value to determine significance.
How to Use ANOVA in Excel?
Here’s a how to use ANOVA in Excel:
1. Enable Data Analysis Toolpak (if necessary):
- Go to File > Options (or Excel > Preferences on Mac).
- Click on Add-Ins.
- In the Manage dropdown menu, select Excel Add-Ins and click GO.
- Check the box for Analysis ToolPak and click OK.
2. Perform ANOVA:
- Now that the Data Analysis Toolpak is enabled, switch to the Data tab.
- Click on Data Analysis in the Analysis section.
- Choose the appropriate ANOVA test based on your data structure:
- Anova: Single Factor for one categorical independent variable and one continuous dependent variable.
- Anova: Two Factor With Replication for two categorical independent variables and one continuous dependent variable (with equal replications in each group).
3. Setting up the ANOVA Dialog Box:
- Input Range: Select the range of your data, including column headers if you have them.
- Grouped By: Choose “Columns” if you have organized each group in a separate column, or choose “Rows” if you have placed each group in a separate row.
- Labels: Check this box if the first row of your data range contains column headers.
- Alpha: This is the significance level (usually 0.05 by default).
4. Run the Analysis:
- Click OK to run the ANOVA test. Excel will generate an output table with various statistics like Sum of Squares, Degrees of Freedom, and F-statistic to help you assess the significance of differences between your groups.pen_spark
Example of a One-Way ANOVA Calculation
Consider a simple one-way ANOVA example:
Scenario: A researcher wants to test the effect of three diets (A, B, and C) on weight loss. Data is as follows:
- Diet A: 5, 6, 7
- Diet B: 4, 5, 6
- Diet C: 3, 5, 7
Step 1: Calculate the group means and overall mean:
- Mean of Diet A: 6
- Mean of Diet B: 5
- Mean of Diet C: 5
- Overall Mean: 5.44
Step 2: Calculate SS Between and SS Within.
Step 3: Compute MS Between and MS Within.
Step 4: Compute F-ratio and interpret it against a critical value or use the p-value to check significance.
Key Takeaways
- ANOVA is a statistical formula used to compare variances across the means (or average) of different groups.
- There are two types of commonly used ANOVA; one-way ANOVA and two-way ANOVA.
- To analyze variance (ANOVA), statisticians or analysts use the F-test to compute the feasibility of variability amongst two groups more than the variations observed within the said groups under study.
Advantages and Limitations of ANOVA
Advantages:
- Efficiently Tests Multiple Groups: ANOVA allows comparing more than two groups in a single test.
- Reduces Type I Error: By testing all groups together, it minimizes the likelihood of finding false significance.
Limitations:
- Assumptions: Assumes normal distribution, homogeneity of variances, and independent observations.
- No Specific Group Identification: ANOVA shows if a difference exists but doesn’t specify which groups differ significantly without additional post-hoc tests.
REFERENCES
- DATAtab Team (2025). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net
- Dr. Shaun V. Ault, Dr. Soohyun Nam Liao, Larry Musolino, OpenStax Principles of Data Science, Jan 24, 2025, https://openstax.org/books/principles-data-science/pages/4-4-analysis-of-variance-anova
- Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
- McDonald, J. H. (2014). Handbook of Biological Statistics (3rd ed.). Sparky House Publishing.
- Mertler, C. A., & Reinhart, R. V. (2016). Advanced and Multivariate Statistical Methods: Practical Application and Interpretation. Routledge.
- https://www.analyticsvidhya.com/blog/2018/01/anova-analysis-of-variance/
- https://datatab.net/tutorial/anova
- https://www.graphpad.com/guides/the-ultimate-guide-to-anova
- [1] Sapkota, Anupama. 2023. “ANOVA: Definition, One-Way, Two-Way, Table, Examples, Uses.” Microbe Notes. August 3. https://microbenotes.com/anova/.
- [2] Hassan, Muhammad. 2024. “ANOVA (Analysis of Variance) – Formulas, Types, and Examples.” Research Method. November 12. https://researchmethod.net/anova/.
- https://www.investopedia.com/terms/a/anova.asp
Stories are the threads that bind us; through them, we understand each other, grow, and heal.
JOHN NOORD
Connect with “Nurses Lab Editorial Team”
I hope you found this information helpful. Do you have any questions or comments? Kindly write in comments section. Subscribe the Blog with your email so you can stay updated on upcoming events and the latest articles.
