ANOVA STEP-BY-STEP
The Sums of Squares:
1) Computing SStotal: The SStotal is the SS based on the entire set of scores in the
study. So computing this SS is the same as if you just stacked your different
treatment samples together to form a single sample and then computed the Sum of
2
SS total =  X 
2
G
N
Squares on that one larger sample. In terms of our new notation:
So for the Alcohol data, you first need to make X2 columns for each of the
treatment levels, then add up the columns and then add the three ΣX2 's together.
Then you subtract G2/N:
1oz
0
1
0
2
1
4
X2
0
1
0
4
1
6
3oz
2
3
0
3
1
9
X2
4
9
0
9
1
23
5oz
4
6
3
2
3
18
X2
16
36
9
4
9
74
k = 3, n = 5, N = 15, T1 = 4, T2 = 9, and T3 = 18, and G = 31
ΣX2 = sum of the ΣX2 for each level of the factor = 6 + 23 + 74 = 103
So SStotal = 103 - 312 = 103 - 961/15 = 38.933
15
2) Computing SSwithin:
To compute SSwithin you first need to compute the SS
within each level of the IV using the regular SS formula:
(  X)
SS =  X 
n
2
2
Then to get SSwithin you simply add up the SS from within each level of the IV:
SSwithin = ΣSS
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ANOVA STEP-BY-STEP
So for the Alcohol data there are three SS you need to compute first, one for each
level:
SS1 = 6 - 42/5 = 2.800
SS2 = 23 - 92/5 = 6.800
SS3 = 74 - 182/5 = 9.200
Then you add these three SS up to get SSwithin:
SSwithin = 2.800 + 6.800 + 9.200 = 18.800
3) Computing SSbetween:
Recall that the variance between treatments measures the differences or variance
between the treatment means. This implies one way we could find the
SSbetween would be to compute a SS using the X
- 's as the scores. That
is, we could consider our deviations (that we will square and sum) as
the deviation of each individual mean from the grand mean (the grand
mean is the over all mean of the entire set of data or G/N). Of course,
there is a computational formula that looks different from that, but is
much easier to use:
2
2
G
T
SS betw = [  ] 
n
N
So for the Alcohol data,
SSbetween = (42/5 + 92/5 + 182/5) - 312/15 = 84.2 - 64.067 = 20.133
Note: You should Always check to see if:
SStotal = SSbetween + SSwithin
If this check does not come out right - you have made a mistake in your
calculations.
So for the Alcohol data:
Does 20.133 + 18.8 = SStotal? Yes 38.933 = 38.933.
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ANOVA STEP-BY-STEP
In computing the degrees of freedom, you should keep in mind that:
1) Each df is associated with a specific SS
2) The df are approximately equal to the number of items that went into
computing the corresponding SS minus 1. So if n things went in, then df = n
- 1.
Computing the df:
1) dftotal = N - 1 Because SStotal was computed using the entire set of N scores.
So for the Alcohol study, dftotal = 15 - 1 = 14
2) dfwithin = N - k To get the SSwithin we first computed the SS for each level and
then added them up. This is the same for dfwithin in a sense. For each level we have
"n - 1" degrees of freedom. Then we sum those n - 1 degrees of freedom across the
levels: (n - 1) + (n - 1) + (n - 1) + ... If you simplify this, you get N - k which is
the right number for the dfwithin.
So for the Alcohol study, dfwithin = 15 - 3 = 12.
3) dfbetween = k - 1 Because SSbetween is really based on the deviations of each
treatment mean from the grand mean, the number of items in this SS is the number
of treatment means = k. So the dfbetween = k - 1.
So for the Alcohol study, dfbetween = 3 - 1 = 2.
Note: You should Always check to see if:
dftotal = dfbetween + dfwithin
If this check does not come out right - you have made a mistake in your
calculations.
For the Alcohol study: does 2 + 12 = dftotal? Yes 14 = 14!
So now we have the 3 SS and the corresponding 3 df. What we need now is to
compute the variances.
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ANOVA STEP-BY-STEP
Computing the between and within variances:
Recall that a variance is SS/df. In Anova the variances we compute are called
Mean Squares, symbolized "MS" (Because they are essentially mean squared
deviations)
So we can compute:
MSbetween = SSbetween
dfbetween
= the variance between treatments
and
MSwithin = SSwithin
dfwithin
= the variance within treatments
So for the Alcohol study,
MSbetween = 20.133/2 = 10.067
MSwithin = 18.800/12 = 1.567
Note: In general we do not compute MStotal. Also, it is NOT TRUE that MStotal =
MSbetween + MSwithin.
Finally, because the F test is the variance between divided by the variance within,
we get our F-ratio:
F = MSbetween
MSwithin
So for the Alcohol study, F = 10.067/1.567 = 6.426.
Finally, you should always present what is called an Anova Summary Table that
contains the results of all of these calculations. It should look like:
Source
SS
df
MS
F
_____________________________________________________________
Between
20.133
2
10.067
6.426
Within
18.800
12
1.567
Total
38.933
14
_____________________________________________________________
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