Two-way ANOVA Test: Concepts, Formula & Examples

two-way ANOVA test formula

The two-way analysis of variance (ANOVA) test is a powerful tool for analyzing data and uncovering relationships between a dependent variable and two different independent variables. It’s used in fields like psychology, medicine, engineering, business, and other areas that require a deep understanding of how two separate variables interact and impact dependent variable. With the right knowledge, you can use this test to gain valuable insights into your data. Through a two-way ANOVA, data scientists are able to assess complex relationships between multiple variables and draw meaningful conclusions from the data. This helps them make informed decisions and identify patterns in the data that may have gone unnoticed otherwise. Let’s dive into what the two-way ANOVA test is and how it works. You may as well want to check my related blog on one-way ANOVA test.

What is the Two-Way ANOVA Test?

The two-way ANOVA test is a statistical tool used to examine the influence of two different independent variables on one dependent variable. It is a statistical method that tests the group means of two or more factors. This type of analysis examines groups that have been divided into multiple categories based on the values of both independent variables. A two-way ANOVA test can be performed using either an unbalanced or balanced design. In an unbalanced design, there are unequal amounts of observations in each group; in a balanced design, each group contains equal amounts of observations.

In an experiment determining the effects of age and food type on weight, a two-way ANOVA can be used to determine whether the average weight gain of participants differs based on their age group and type of food consumed. This hypothesis is generated from a specific formula which calculates the overall variation in a dataset, allowing you to compare expected results with experimental results. Two-way ANOVA is considered more comprehensive than traditional tests such as chi-square or Student’s t-test because it is capable of testing multiple hypotheses at once, making it useful for a variety of applications.

Difference between Two-way & One-way ANOVA test

One-way Analysis of Variance (ANOVA) and two-way ANOVA are both statistical tests that compare the means of three or more groups. The main difference between these two tests is the number of independent factors analyzed. One-way ANOVA looks at the differences between three or more levels with respect to one factors, while two-way ANOVA analyzes differences between multiple levels with respect to two factors.

A real-life example to better explain the differences between one-way and two-way ANOVA is a survey about job satisfaction among IT professionals. With one-way ANOVA, you could survey IT professionals in different cities to determine if there are any significant differences in their level of job satisfaction. On the other hand, with two-way ANOVA you can investigate how two factors such as gender and age affect job satisfaction among IT professionals in different cities.

For example, let’s say that we surveyed 100 IT professionals across four cities (25 in each city). We could use a one-way ANOVA to find out if there were significant differences in job satisfaction based on city alone, but with a two way ANOVA we could look at additional factors like gender and age to see if they have an impact on job satisfaction as well. In this case, our sample size would be divided into eight groups of seven people each – Male/Female in each City – and then compared using two way ANOVA.

It’s important to note that while one way ANOVA allows us to analyze only one independent variable (city), two way analysis allows us to analyze multiple independent variables (gender and age). This means that it can provide more detail about how different factors interact with each other in order to produce different outcomes regarding job satisfaction.

Here is another example sighting the difference between one-way ANOVA and two-way ANOVA:

A real life example of the difference between one-way and two-way ANOVA can be seen when comparing the mean grades of students for a certain class. For one-way analysis, you can assess if there is any significant difference in grades from different sections of the same course, such as from those enrolled in the morning versus afternoon time slots. On the other hand, for two-way analysis, you can look at how students fared based on both their attendance records and their section in addition to just their section. This would allow a more detailed level of comparison since it takes into account an additional variable (attendance).

Furthermore, with an extra variable added into a two-way test, not only do you get more insight into what causes differences in mean values but also which factors interact with each other to affect them. This interaction allows us to understand whether the effect of one variable on another is equal across all levels or if it changes depending on the level chosen by researchers. In relation back to our example, this would mean determining whether attending classes regularly has an impact on grades regardless of what section they’re enrolled in or if it differs depending on section timings.

Two-Way ANOVA Test: Formula

The formula for calculating a two-way ANOVA involves several components: SS (sum of squares) total, SS within subjects, SS between subjects A and B, MS (mean square) for subjects A and B, df (degrees of freedom) for subjects A and B, and F ratio for subjects 1 and 2. These components are all important factors in determining whether or not there is a statistically significant difference between the groups being examined. The following represents the formula for two-way ANOVA test:

two-way ANOVA test formula

Two-Way ANOVA Test: Examples

Two-way ANOVA tests can be used in many situations where you need to compare differences between groups divided by two independent variables. For example, if you wanted to analyze how years of experience and age affect job performance at your company, you could use a two-way ANOVA test to examine those factors together. 

Here is an example of business world where the ask is to measure the impact of marketing campaign and gender on the sales performance.

Let’s say, a grocery store wants to examine how effective their marketing campaigns were in driving sales for two different genders: male and female. They decided to conduct a two-way ANOVA study to find out what is the effect of both the marketing campaigns and gender on sales performance.

The grocery store randomly selected 200 customers from each gender group, making a total of 400 customers. These customers were divided into four different groups, each receiving a different kind of marketing campaign. Group 1 was exposed to an email campaign, Group 2 was exposed to a television commercial, Group 3 was exposed to newspaper advertising, while Group 4 received no advertisement at all (control group). After collecting data on the sales performance of each customer following exposure to the advertisements, the grocery store then conducted the two-way ANOVA test between gender and the four different advertisement types – email, television commercial, newspaper advertisement, control group.

By conducting this study using two-way ANOVA test, it will help us understand if there is any significant difference in sales performance between males and females with respect to each type of advertisement or no advertisement at all (control group). The results from this study will give us insight into which type of advertisement works better for either male or female customers, as well as provide evidence that whether running all types of advertisements works better than running only one type of advertisement for either male or female customers. Knowing this information will enable the grocery store to optimize their marketing strategies and allocate resources more efficiently in order to drive higher revenues from their target audiences.


The two-way ANOVA test is an extremely useful tool for examining data with multiple independent variables. By understanding how this type of analysis works—including its components such as sum of squares total and mean square—you can gain valuable insights into your data that may have gone undetected before. Whether you’re analyzing job performance or salary across different locations or any other scenario involving multiple independent variables, the two-way ANOVA test will help you uncover relationships between them that may have previously been hidden from view.

Ajitesh Kumar
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Ajitesh Kumar

I have been recently working in the area of Data analytics including Data Science and Machine Learning / Deep Learning. I am also passionate about different technologies including programming languages such as Java/JEE, Javascript, Python, R, Julia, etc, and technologies such as Blockchain, mobile computing, cloud-native technologies, application security, cloud computing platforms, big data, etc. For latest updates and blogs, follow us on Twitter. I would love to connect with you on Linkedin. Check out my latest book titled as First Principles Thinking: Building winning products using first principles thinking. Check out my other blog,
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