# Using ANOVA to Analyze Within-Group Variance

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

ANOVA is used to determine if there are differences between multiple groups of data. In this lesson, learn how to calculate within-group variance, a critical component of ANOVA.

## What is ANOVA?

Imagine that you are a teacher, and you want to know if the place where a student sits in the classroom has an effect on his or her exam grades. You decide to record each student's grade in a table, organized by the row where the student's desk is located.

Now that you have all this data, what do you do with it? How can you use it to tell if there are actually any differences in exam grades between students in different rows?

Analysis of variance, more commonly known as ANOVA, is a statistical test that allows you to compare more than two groups of data and determine if there are differences between groups. ANOVA would be a great tool to help you determine if there are any statistically significant differences between the rows in your class.

## What is Variance?

Variance measures how much spread there is a data set. Two groups of numbers with similar means could have very different variances, depending on how far apart all the values are from each other. If the values in the data set are all very close to each other and to the mean, then the variance will be small. However, if there is a lot of variability in the data, then the variance will be large. In order to tell if two groups of data are actually different from each other, it is critical to know both the mean of each group AND the variance. Even if the means appear different, they may not be if the data is really spread out. Variance is one way to quantify this, and so it is an important tool in statistical analysis.

One important part of performing an ANOVA is calculating the variance of the data both within groups and between groups. Comparing these two variances allows you to determine how likely it is that there are significant differences between the groups. If the variance between groups is bigger than the variance within the groups, this indicates that it is likely that there are differences between the groups. In this lesson, we will just focus on calculating the variance within the groups, known as within-group variance.

## How to Calculate Within-Group Variance

To calculate the within-group variance in a data set with several groups of data, you need to take into account how far each measurement is from the group mean and also the total number of measurements and the number of groups.

1. First, find the mean of each group by adding up all the values in that group and then dividing by the total number of values. This was already done in the data table shown above.

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