Weighted Arithmetic Mean

In an ordinary mean every item counts equally. Sometimes, though, items differ in importance, and it is right to give the more important ones greater influence. The weighted arithmetic mean does this by attaching a weight $w$ to each item.

Why weights matter. Suppose two commodities, mangoes and potatoes, have prices $P_1$ and $P_2$. A simple average of the two price changes is $\dfrac{P_1 + P_2}{2}$, which treats both alike. But if potatoes take a larger share of a consumer's budget, a rise in the potato price hurts more, so it deserves more weight. Using the budget shares $W_1$ and $W_2$ as weights, the average becomes:

General formula. For values $x_1, x_2, \ldots, x_n$ carrying weights $w_1, w_2, \ldots, w_n$:

When prices rise, you are usually more concerned with the goods that matter most to you — and the weighted mean captures exactly that emphasis. This idea is used again when constructing index numbers.