Yes, it’s been a few days since posting. I’ve been busy grading. Boooo.

Okay, so everyone should know that by adding a number to a set that is greater than the mean of said set will increase the mean. But I wasn’t sure I could prove it. Well, I did. And here it is. I did it by contradiction, but there’s probably a more straightforward approach too.

Meaning of variables:
xib = initial mean of i values
xi = ith value in x
xj = jth value in x (new value added)
N = number of values
xfb = final mean of i+1 values

Assume xj > xib means that xfb < xib

xib = sum(xi)/N
xfb = (sum(xi)+xj) / (N+1)

If xfb < xib then
(sum(xi)+xj) / (N+1) < sum(xi)/N
(sum(xi)+xj)*N < sum(xi)*(N+1)
N*sum(xi)+N*xj < N*sum(xi)+sum(xi)
N*xj < sum(xi)
xj < sum(xi) / N
xj < xib

Which is in contradiction to the initial assumption that xj > xib. Therefore, xfb > xib.

(Wouldn’t this look a lot prettier using LaTeX?)

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