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Mean1 = <math>\frac{(4+6+8+4+5+3)}{6}=\frac{30}{6}= 5.0 </math>

Mean2 = <math>\frac{(7+5+8+9+7+9)}{6}= \frac{45}{6}=7.5 </math>

Then a grand mean over all observations:

Mean = <math>\frac{30+45}{6+6} = 6.25</math>

Variance is always a sum of square deviations divided by degree of freedom: SS/df. This is also called a mean squared deviation MS.

ANOVA begins by expressing the deviation of each observation from the grand mean as a sum of two terms: the difference of the observation from its group mean, plus the difference of the group mean from the grand mean. Writing this out explicitly for the example, we have, for the placebo group:

(4 - 6.25) = (4 - 5.0) + (5.0 - 6.25)
(6 - 6.25) = (6 - 5.0) + (5.0 - 6.25)
...
(3 - 6.25) = (3 - 5.0) + (5.0 - 6.25)

and for the vitamin ME group:

(7 - 6.25) = (7 - 7.5) + (7.5 - 6.25)

(5 - 6.25) = (5 - 7.5) + (7.5 - 6.25)
...
(9 - 6.25) = (9 - 7.5) + (7.5 - 6.25)

The magic (actually the Pythagorean Theorem in an appropriate dimensional space) is that the sums of squares decompose in this way.

<math>(4-6.25)^2 +...+(9-6.25)^2 =</math> [(4-5.0)^2+...+(9 - 7.5)^2] + [(5.0 - 6.25)^2+...+(7.5 - 6.25)^2]
Check: 46.25 = 27.5 + 18.75

In the usual abbreviations:

SST = SSE + SSG