F = msBetween ÷ msWithin
You need to complete a calculation table, a summary table and an F score that you compare pair to the critical value in an F table. And you need to do a lot of basic math steps. No one really calculates one-way ANOVAs by hand, except for illustrative purposes. This is a task computers are good at. But walk through the steps a couple of times and you’ll gain a real feel for how the statistic works.
One independent variable can have several levels, such as high, medium and low. In this example there are four levels, still only one independent variable. Eventually, we will complete a summary table and test for F. But first we’ve got some calculating to do.
Here is the general approach.
1. Summarize the subgroups
First, find the n (number of scores) for each group.
Second, find the sum for each group.
Third, square each number in the first group and sum them. Then square and sum the scores in each of the other groups.
Fourth, find the Sum of Squares (SS) for each group.
Fifth, find the totals for n, sum, squares, and SS. So, N = 20 (the sum of each group’s n’s). The SumX = 117 and so on.
Updating our example, it would look like this:
Group1 Group2 Group3 Group4
1 6 12 5
2 4 7 2
4 9 15 6
3 5 9 3
2 11 7 4
n 5 5 5 5 20
Sum 12 35 50 20 117
Squares 34 279 548 90 951
SS 5. 20 34 48 10 97.20
2. Find SSwithin
First, start by creating a summary table.
Second, write in the SSwithin. In the process of creating summarizing the groups, the SSwithin was calculated. It was 97.2. That is, SSwithin (within the experiment) is the sum of the SS that is in (within) each group.
Updating our summary table, it now looks like this:
SS df ms
Between ____ ___ ____
Within 97.20 ___ ____
Total ____ ___ ____
3. Find SSbetween
The formula for SSbetween looks more difficult than it is. Here’s the formula:
First, start with the sum of each group (12, 35, 50, 20). Square each of them and add them together:
122 + 352 + 502 + 202
So we get: 144 + 1225 + 2500 + 400 = 4269.
Second, divide Step1 by the n (the number of subjects in each group. NOTE: It is not the number of groups but the number of scores in each group. This gives us: 4269 divided by 5 = 853.8.
Third, take the Sum of X in the totals column (117) and square it, which equals 13689.
Fourth, divide Step3 by the N in the totals column (20). That is, 13689 divided by 20 = 684.5
Fifth, subtract Step4 from Step2. So, 853.8 – 684.5 = 169.35. This is the SSbetween.
Updating our summary table, it now looks like this:
SS df ms
Between 169.35 ___ ____
Within 97.20 ___ ____
Total ____ ___ ____
4. Find SStotal
The formula for SStotal is the same as any basic SS.
First, note that the sum of X-squares = 951.
Second, take the sum of X’s (117) and square it. This equals 13689.
Third, divide Step2 by 20 (big N), which equals 684.5
Fourth, subtract Step3 from Step1. That is, 951 – 684.5 = 266.55. This is the SStotal.
Updating our summary table, it now looks like this:
SS df ms
Between 169.35 ___ ____
Within 97.20 ___ ____
Total 266.55 ___ ____
To check the calculations, simply add SSbetween to SSwithin and see if they equal SStotal. It does, so we calculated everything correctly.
5. Find the degrees of freedom
First, enter the degrees of freedom (df) for Between, which is k-1 (columns minus one). Since our example has 4 columns, the df for Between = 3.
Second, enter the df for Within, which is N-k (number of people minus the number of columns). In our example, N = 20, so dfwithin = 16.
Third, enter the df for Total, which is N-1 (number of people minus one). So, dftotal = 19.
SS df ms
Between 169.35 3 ____
Within 97.20 16 ____
Total 266.55 19 ____
6. Find F
First, calculate the appropriate mean squares. Since mean squares is another name for variance (and SS divided by df equals variance), divide each SS by its respective df. Updating the table, we now have:
SS df ms
Between 169.35 3 56.45
Within 97.20 16 6.08
Total 266.55 19 14.03
Second, divide the mean squares of Between by the mean squares of Within. That is, 56.45 divided by 6.08 = 9.29. This ratio is called the F test, so F = 9.29.
7. Find the critical value
The Critical Values of the F Distribution table is actually a series of distributions. To enter the table, go across to the row whose number matches the degrees of freedom for Between (dfbetween). And go down the dfwithin.
In our example, go across to 3 and down to 16. The critical value (the value you have to beat) = 3.24 (at the .05 alpha level).
8. Decide what to do next
If F is not significant, there is nothing else to do. The differences between the groups is due to chance.
If F is significant, than t-tests are done: one between each pair of combinations (AB, AC, AD, BC, BD and CD).
To test for significance, the calculated value is compared to the F table. If the value you calculated is bigger than the value in the book, F is significant. In our example, we calculated F to be 9.29, which is bigger than the critical value of 3.24 we found at 3 and 16 degrees of freedom. So, F is significant and the t-tests are authorized.
