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$$ \begin{align}
\text{SSW} &= \sum_{i=1}^g \sum_{j=1}^{n_i}\ (x_{i,j} - \bar{x}_i)^2 \\[3em]
&= \sum_{i=1}^{4} \sum_{j=1}^{n_i}\ (x_{i,j} - \bar{x}_i)^2 \\[1em]
&= \sum_{j=1}^{6}\ (x_{1,j} - 4.5)^2 + \sum_{j=1}^{8}\ (x_{2,j} - 3)^2 + \sum_{j=1}^{9}\ (x_{3,j} - 6.6667)^2 + \sum_{j=1}^{9}\ (x_{4,j} - 2)^2 \\[1em]
&= (x_{1,1} - 4.5)^2\ + (x_{1,2} - 4.5)^2\ + (x_{1,3} - 4.5)^2\ + (x_{1,4} - 4.5)^2\ + (x_{1,5} - 4.5)^2\ + (x_{1,6} - 4.5)^2\ + \\
& \qquad
(x_{2,1} - 3)^2\ + (x_{2,2} - 3)^2\ + (x_{2,3} - 3)^2\ + (x_{2,4} - 3)^2\ + (x_{2,5} - 3)^2\ + (x_{2,6} - 3)^2\ + (x_{2,7} - 3)^2\ + (x_{2,8} - 3)^2\ + \\
& \qquad
(x_{3,1} - 6.6667)^2\ + (x_{3,2} - 6.6667)^2\ + (x_{3,3} - 6.6667)^2\ + (x_{3,4} - 6.6667)^2\ + (x_{3,5} - 6.6667)^2\ + (x_{3,6} - 6.6667)^2\ + (x_{3,7} - 6.6667)^2\ + (x_{3,8} - 6.6667)^2\ + (x_{3,9} - 6.6667)^2\ + \\
& \qquad
(x_{4,1} - 2)^2\ +\ (x_{4,2} - 2)^2\ +\ (x_{4,3} - 2)^2\ +\ (x_{4,4} - 2)^2\ +\ (x_{4,5} - 2)^2\ +\ (x_{4,6} - 2)^2\ +\ (x_{4,7} - 2)^2\ +\ (x_{4,8} - 2)^2\ +\ (x_{4,9} - 2)^2 \\[1em]
&= (2 - 4.5)^2\ + (0 - 4.5)^2\ + (9 - 4.5)^2\ + (3 - 4.5)^2\ + (2 - 4.5)^2\ + (11 - 4.5)^2\ + \\
& \qquad
(-6 - 3)^2\ + (11 - 3)^2\ + (3 - 3)^2\ + (6 - 3)^2\ + (0 - 3)^2\ + (-2 - 3)^2\ + (9 - 3)^2\ + (3 - 3)^2\ + \\
& \qquad
(10 - 6.6667)^2\ + (4 - 6.6667)^2\ + (8 - 6.6667)^2\ + (14 - 6.6667)^2\ + (-1 - 6.6667)^2\ + (4 - 6.6667)^2\ + (8 - 6.6667)^2\ + (3 - 6.6667)^2\ + (10 - 6.6667)^2\ + \\
& \qquad
(-4 - 2)^2\ +\ (4 - 2)^2\ +\ (-1 - 2)^2\ +\ (10 - 2)^2\ +\ (-7 - 2)^2\ +\ (-1 - 2)^2\ +\ (11 - 2)^2\ +\ (-1 - 2)^2\ +\ (7 - 2)^2 \\[1em]
&= (-2.5)^2\ + (-4.5)^2\ + (4.5)^2\ + (-1.5)^2\ + (-2.5)^2\ + (6.5)^2\ + \\
& \qquad
(-9)^2\ + (8)^2\ + (0)^2\ + (3)^2\ + (-3)^2\ + (-5)^2\ + (6)^2\ + (0)^2\ + \\
& \qquad
(3.3333)^2\ + (-2.6667)^2\ + (1.3333)^2\ + (7.3333)^2\ + (-7.6667)^2\ + (-2.6667)^2\ + (1.3333)^2\ + (-3.6667)^2\ + (3.3333)^2\ + \\
& \qquad
(-6)^2\ +\ (2)^2\ +\ (-3)^2\ +\ (8)^2\ +\ (-9)^2\ +\ (-3)^2\ +\ (9)^2\ +\ (-3)^2\ +\ (5)^2 \\[1em]
&= (6.25)\ + (20.25)\ + (20.25)\ + (2.25)\ + (6.25)\ + (42.25)\ + \\
& \qquad
(81)\ + (64)\ + (0)\ + (9)\ + (9)\ + (25)\ + (36)\ + (0)\ + \\
& \qquad
(11.1111)\ + (7.1111)\ + (1.7778)\ + (53.7778)\ + (58.7778)\ + (7.1111)\ + (1.7778)\ + (13.4444)\ + (11.1111)\ + \\
& \qquad
(36)\ +\ (4)\ +\ (9)\ +\ (64)\ +\ (81)\ +\ (9)\ +\ (81)\ +\ (9)\ +\ (25) \\[1em]
&= 97.5\ + 224\ + 166\ + 293 \\[1em]
&= 805.5 \\[1em]
\end{align}
$$
From these calculations, the within sum of squares is SSW = 805.5.
Hide the R Code
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.
## Import data
treatment1 = c(2, 0, 9, 3, 2, 11)
treatment2 = c(-6, 11, 3, 6, 0, -2, 9, 3)
treatment3 = c(10, 4, 8, 14, -1, 4, 8, 3, 10)
treatment4 = c(-4, 4, -1, 10, -7, -1, 11, -1, 7)
## Change to Long Format
mmt = c( treatment1, treatment2, treatment3, treatment4 )
grp = c( rep("trt1",6), rep("trt2",8), rep("trt3",9), rep("trt4",9) )
## Model the data
mod = aov(mmt~grp)
summary(mod)
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:
modSummary = summary(mod)
modSummary[[1]][2,2]
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.
## Import data
yields = c(2, 0, 9, 3, 2, 11, -6, 11, 3, 6, 0, -2, 9, 3, 10, 4, 8, 14, -1, 4, 8, 3, 10, -4, 4, -1, 10, -7, -1, 11, -1, 7)
grp = c('trt1', 'trt1', 'trt1', 'trt1', 'trt1', 'trt1', 'trt2', 'trt2', 'trt2', 'trt2', 'trt2', 'trt2', 'trt2', 'trt2', 'trt3', 'trt3', 'trt3', 'trt3', 'trt3', 'trt3', 'trt3', 'trt3', 'trt3', 'trt4', 'trt4', 'trt4', 'trt4', 'trt4', 'trt4', 'trt4', 'trt4', 'trt4')
## Model the data
mod = aov(yields~grp)
summary(mod)
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:
modSummary = summary(mod)
modSummary[[1]][2,2]
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.
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