Question

In a study with three groups and 13 participants in each group, the sum of squares for the within-groups source of variation is 18. What is the value for the mean square within groups in this study

Answer 1

For this case we know this
`SS_(within)= 18` and we also know that we have 3 groups each one of 13 so in total we have 13*3 = 39 individuals

The degrees of freedom for the numerator on this case is given by
`df_(num)=df_(between)=k-1=3-1=2` where k =2 represent the number of groups.

The degrees of freedom for the denominator on this case is given by
`df_(error)=df_(within)=N-K=3*13-3=36`.

And the total degrees of freedom would be
`df=N-1=3*13 -1 =38`

And the mean square within groups would be given by:

`MSE_(within)= (SSE_(within))/(df_(within))= (18)/(36)= 0.5`

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`

For this case we know this
`SS_(within)= 18` and we also know that we have 3 groups each one of 13 so in total we have 13*3 = 39 individuals

The degrees of freedom for the numerator on this case is given by
`df_(num)=df_(between)=k-1=3-1=2` where k =2 represent the number of groups.

The degrees of freedom for the denominator on this case is given by
`df_(error)=df_(within)=N-K=3*13-3=36`.

And the total degrees of freedom would be
`df=N-1=3*13 -1 =38`

And the mean square within groups would be given by:

`MSE_(within)= (SSE_(within))/(df_(within))= (18)/(36)= 0.5`