One-way ANOVA test: Concepts, Formula & Examples

One way ANOVA table example

An ANalysis Of VAriance (ANOVA) test, also known as a one-way ANOVA test, is a hypothesis test used to determine whether there is a significant difference between the means of three or more groups. In other words, it can be used to answer the question of whether the averages of three or more populations are equal. If there is a need to compare the means of two populations (independent or pairwise), t-tests can be used. One-way ANOVA test or single-factor Anova test is often used in experiments with only one independent variable. As data scientists, it is of utmost importance to understand the ANOVA test as it is an important statistical tool used in regression models and hypothesis testing. In this blog post, we will discuss the concepts behind the one-way ANOVA test, as well as how to calculate and interpret the results. We will also provide some examples to help illustrate how this test works.

What is one-way ANOVA test?

One-way ANOVA test is defined as statistical hypothesis test to determine the equality of means from several groups. The reason why we need one-way ANOVA test is that when we have more than two groups, t-test cannot be used. Let’s say we want to compare the average height of an individual in different countries or regions (for example: UK, US and Japan). In this case, an one-way ANOVA test can be used as it allows us to determine if there is a significant difference between the average heights of  men above 20 years of age in different countries or regions. Once the difference between different groups is found to be significant, you can perform further analysis to explore the source of this difference. The hypothesis test is done as a measure of F-statistics. One-way ANOVA test is also termed as single-factor ANOVA test as the means are compared across different groups based on single common factor. For example, in the example relating to comparing heights of men accross different countries such as US, UK and India, the single factor is country. The following is how the sample data look like with single factor as country. The hypothesis that need to be tested is that there is no significant difference between the mean heights of men above 20 years of age across three different countries such as US, UK and India.

One-way Anova test sample data 

F-statistics is defined as a ratio of mean sum of squares between the groups (MSB) to the mean sum of squares within groups (MSW). The formula for F-statistics would look like the following:

F = MSB / MSW

Mean sum of squares between the group (MSB) can be calculated as the following:

MSB = Sum of squares between the group (SSB) / DFb

Where

DFb = degrees of freedom = K – 1 where K is the number of group, and,

Sum of squares between the group (SSB) can be calculated as the following:

SSB = Σ(Xi – Xt)² where Xi is mean of group i and Xt is mean of all the observations.

Mean sum of squares within the group (MSB) can be calculated as the following:

MSW = Sum of squares within the group (SSW) / DFw

Where

DFw = degrees of freedom = N – K where K is the number of group, and N is total number of observations in all the group

Sum of squares within the group (SSW) can be calculated as the following:

SSW = Σ(Xij – Xj)² where Xij is the observation of each group j

The above information can be put together in what can be called as ANOVA table that looks like the following:

One way ANOVA table example

Steps for performing one-way ANOVA test

The following represents the steps of performing one-way ANOVA test with two or more groups:

  • Make an assumption to test the equality of population means: The normality assumption and equal variance assumption
  • Formulate the null hypothesis that there is no difference between the means of different groups or population
  • Formulate the alternate hypothesis that there is a significant difference between the means of two or more groups
  • Calculate the sum of squares between the groups (SSB) for each group, and the degrees of freedom (dfb)
  • Based on the above, calculate the mean sum of squares between the groups (MSB) as MSB = SSB / dfb
  • Calculate the sum of squares within the groups (SSW) for each group, and the degrees of freedom (dfw)
  • Based on the above, calculate the mean sum of squares within the groups (MSW) as MSW = SSW / dfw
  • Calculate the F-statistics as MSB/MSW
  • Use F-table to find the critical value of F at a particular level of significance (such as 0.05) and degrees of freedom as dfb (numerator) and dfw (denominator)

Real-world examples of One-way ANOVA test

The following represents a few real-world examples where an one-way ANOVA test can be used:

  • Evaluation of academic performance of students from different schools
  • Assessment of customer satisfaction between two or more products
  • Determining difference in quality of service among different branches of a company
  • Comparing the average weight of individuals living in different countries or regions.

 

Summary

The one-way ANOVA test is a statistical hypothesis test that allows us to determine if there is a significant difference between the means of three or more different groups. By calculating the F-statistics, we can test the hypothesis. In order to perform an ANOVA test, we first formulate the null hypothesis to test the equality of population means and then calculate the sum of squares between the groups (SSB) and within the groups (SSW). We then use F-table to find the critical value of F at a particular level of significance. If the critical value of F is greater than the calculated F-statistics, we reject the null hypothesis and conclude that there is a significant difference between the means of different groups. Otherwise, we fail to reject the null hypothesis.

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
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