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Because m = 3, there are m−1 = 3−1 = 2 degrees of freedom associated with the factor.Because n = 15, there are n−1 = 15−1 = 14 total degrees of freedom.Now, having defined the individual entries of a general ANOVA table, let's revisit and, in the process, dissect the ANOVA table for the first learning study on the previous page, in which n = 15 students were subjected to one of m = 3 methods of learning: Therefore, we'll calculate the P-value, as it appears in the column labeled P, by comparing the F-statistic to an F-distribution with m−1 numerator degrees of freedom and n− m denominator degrees of freedom. 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.
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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).
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