Calculating the Sum of Squares Within (SSW)

Summary statistics from the problem

\( \bar{x} \)	=	3.9333
\( \bar{x}_1 \)	=	1.8
\( \bar{x}_2 \)	=	5
\( \bar{x}_3 \)	=	5
\( n_1 \)	=	5
\( n_2 \)	=	6
\( n_3 \)	=	4

In the box below, please enter the sum of squares within (SSW) for the data, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

You got the correct answer of SSW = 484.8. Congratulations!

Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.

Show Formula

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Here is the formula you can use to calculate the within sum of squares.

In this formula, there are a few symbols to know:

Show Solution

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From these calculations, the within sum of squares is SSW = 484.8.

Show the R Code

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There are two ways of performing these calculations in R. The method you select will depend on how your data are stored.

Method 1: Wide Format

Copy and paste the following code into your R script window, then run it from there.

In the R output, the value of the sum of squares within is the number in the table under Sum Sq and to the right of Residuals. If you would like better precision for that value, or if you would like to have only that value, run the following code in addition to that above:

Here, the number outputted is the sum of squares between. How did you get the number? The summary table (also known as an ANOVA table) is just a table. Thus, the first line saves the table as the variable modSummary the last line looks inside that variable, selects the ANOVA table ([[1]]), and then selects the row 2, column 2 value.

Method 2: Long Format

Copy and paste the following code into your R script window, then run it from there.

As discussed above, in the R output, the value of the sum of squares within is the number in the table under Sum Sq and to the right of Residuals. If you would like better precision for that value, or if you would like to have only that value, run the following code in addition to that above:

Here, the number outputted is the sum of squares between. How did you get the number? The summary table (also known as an ANOVA table) is just a table. Thus, the first line saves the table as the variable modSummary the last line looks inside that variable, selects the ANOVA table ([[1]]), and then selects the row 2, column 2 value.

Note: The difference between wide and long formats is this: In wide formatted data, each group has its own variable. In long formatted data, the group number is a variable.