When, on the next page, we delve into the theory behind the analysis of variance method, we'll see that the F-statistic follows an F-distribution with m−1 numerator degrees of freedom and n− m denominator degrees of freedom. That is, the F-statistic is calculated as F = MSB/MSE. Because we want to compare the 'average' variability between the groups to the 'average' variability within the groups, we take the ratio of the Between Mean Sum of Squares to the Error Mean Sum of Squares. The F column, not surprisingly, contains the F-statistic.

The Error Mean Sum of Squares, denoted MSE, is calculated by dividing the Sum of Squares within the groups by the error degrees of freedom.

The Mean Sum of Squares between the groups, denoted MSB, is calculated by dividing the Sum of Squares between the groups by the between group degrees of freedom.

The mean squares ( MS) column, as the name suggests, contains the 'average' sum of squares for the Factor and the Error: We'll soon see that the total sum of squares, SS(Total), can be obtained by adding the between sum of squares, SS(Between), to the error sum of squares, SS(Error). As the name suggests, it quantifies the total variability in the observed data.

SS(Total) is the sum of squares between the n data points and the grand mean.

It quantifies the variability within the groups of interest.

Again, as we'll formalize below, SS(Error) is the sum of squares between the data and the group means.

As the name suggests, it quantifies the variability between the groups of interest.

As we'll soon formalize below, SS(Between) is the sum of squares between the group means and the grand mean.

If there are n total data points collected and m groups being compared, then there are n− m error degrees of freedom.

If there are m groups being compared, then there are m−1 degrees of freedom associated with the factor of interest.

If there are n total data points collected, then there are n−1 total degrees of freedom.

Let's start with the degrees of freedom ( DF) column: Yikes, that looks overwhelming! Let's work our way through it entry by entry to see if we can make it all clear. Hover over the lightbulb for further explanation. With the column headings and row headings now defined, let's take a look at the individual entries inside a general one-factor ANOVA table:

Total means 'the total variation in the data from the grand mean' (that is, ignoring the factor of interest).

Error means 'the variability within the groups' or 'unexplained random error.' Sometimes, the row heading is labeled as Within to make it clear that the row concerns the variation within the groups.

And, sometimes the row heading is labeled as Between to make it clear that the row concerns the variation between the groups. Sometimes, the factor is a treatment, and therefore the row heading is instead labeled as Treatment. In the learning example on the previous page, the factor was the method of learning.

Factor means 'the variability due to the factor of interest.' In the tire example on the previous page, the factor was the brand of the tire.

MS means 'the mean sum of squares due to the source.'.

SS means 'the sum of squares due to the source.'.

DF means 'the degrees of freedom in the source.'.In the learning study, the factor is the learning method. In the tire study, the factor is the brand of tire. The factor is the characteristic that defines the populations being compared. Source means 'the source of the variation in the data.' As we'll soon see, the possible choices for a one-factor study, such as the learning study, are Factor, Error, and Total.

In working to digest what is all contained in an ANOVA table, let's start with the column headings:

Calculate degrees of freedom in r