General Question
Can you do the algebraic manipulation to solve this problem?
A while back I gave an intuition problem regarding weighted averages. I said that if as the weights increased the associated values also increased, what conclusion could be drawn? A number of people correctly answered that the weighted average would be greater than the simple average.
Intuition is one thing but an actual proof is something else. I wanted to see if there was a relatively easy way to confirm the intuition. Arranging the weights so that 0 <W1 < W2 < .... Wn we are given that the associated values satisfy 0 <V1 < V2 < ... Vn. What we want to show is that
(W1V1 + ... + WnVn)/(W1+...Wn) > (V1+...Vn)/n.
Multiplying by the terms in the denominator and bringing all the terms together, this is the same as showing that
n (W1V1+...+WnVn) – (V1+...Vn)(W1+...Wn) > 0
I was hoping that the formula could be reworked so as to make the inequality obvious. It turns out that it can and that the reworked formula for the difference can be expressed rather compactly using appropriate algebraic notation. If you try some simple cases, the formula should be fairly clear.
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