Question

Where can I find a proof for:

SST = SSM + SSE

Answer 1

SST = ∑X^2 - (∑X)^2/N
...where ∑X^2 = sum of squared scores and (∑X)^2 = square of the summed scores. N = total sample size.

SSM = ∑A^2/n - (∑X)^2/N
...where A^2 = square of the sum of scores in each group and n = sample size per group.

SSE = ∑X^2 - ∑A^2/n

SST = ∑A^2/n - (∑X)^2/N + ∑X^2 - ∑A^2/n = ∑X^2 - (∑X)^2/N

I hope this will help.

Answer 2

Wow. That's right, but how do you get from

SST = ∑((x - MEAN)^2)
to this
SST = ∑(x^2) - (∑(x))^2/N
?

The former is the definition I'm used to. The latter is what you used in your simple proof. I tested it out and they are equal, but can you prove that?

Answer 3

Got it...

SST = ∑((x - MEAN)^2)
= ∑(x^2 - 2 * x1 * MEAN + MEAN^2)
= ∑(x^2) - 2 * MEAN * ∑(x) + N * MEAN^2
MEAN = ∑(x)/N
= ∑(x^2) - 2 * ∑(x)^2/N + ∑(x)^2/N
= ∑(x^2) - ∑(x)^2/N

Awesome! Thanks!

Answer 4

OK, SST makes sense, but I can't see how to derive your SSM or SSE formulas:

I get this:

SSM = ∑((MODEL - MEAN)^2)
= ∑(MODEL^2 - 2 * MODEL * MEAN + MEAN^2)
= ∑(MODEL^2) - 2 * MEAN * SIGMA(MODEL) + N * MEAN^2

SSE = ∑((X - MODEL)^2)
= ∑(X^2 - 2 * X * MODEL + MODEL^2)
= ∑(X^2) - 2 * ∑(X * MODEL) + ∑(MODEL^2)

When I add those two, terms don't cancel out and I don't get SST. What am I missing?