Question

When utilizing ANOVA, what does the between group sum of squares measure?

a. The sum of the squared deviations between each group mean and every other group mean

b. The sum of the squared deviations between each group mean and the mean across all groups

c. The sum of the squared deviations between each individual observation and its group mean

d. The sum of the squared deviations between each individual observation and the mean across all groups

Answer 1

b. The sum of the squared deviations between each group mean and the mean across all groups

Explanation:

Previous concepts

Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"

Solution to the problem

If we assume that we have
`p` groups and on each group from
`j=1,\dots,p` we have
`n_j` individuals on each group we can define the following formulas of variation:

`SS_(total)=\sum_(j=1)^p \sum_(i=1)^(n_j) (x_(ij)-\bar x)^2`

`SS_(between)=SS_(model)=\sum_(j=1)^p n_j (\bar x_(j)-\bar x)^2`

`SS_(within)=SS_(error)=\sum_(j=1)^p \sum_(i=1)^(n_j) (x_(ij)-\bar x_j)^2`

And we have this property

`SST=SS_(between)+SS_(within)`

As we can see the sum of squares between represent the sum of squared deviations between each group mean and the mean across all groups.

`SS_(between)=SS_(model)=\sum_(j=1)^p n_j (\bar x_(j)-\bar x)^2`

So then the best option is:

b. The sum of the squared deviations between each group mean and the mean across all groups