Weighted Average

A weighted average is a type of mean in which the values being averaged do not all count equally. Each value is multiplied by a weight, the products are added, and the total is divided by the sum of the weights: $\bar{x}_w = \sum w_i x_i / \sum w_i$, where $x_i$ are the values and $w_i$ their weights. If all weights are equal, the result is the ordinary arithmetic average. An investor who buys 100 shares at $10 and 300 shares at $12 has an average cost of $(100 \times 10 + 300 \times 12) / 400 = 4{,}600 / 400 = 11.50$ dollars, not the simple average of $11.

Weighted averages are used throughout finance. The volume-weighted average price weights each trade price by the quantity traded. A stock market index is commonly weighted by market capitalization. The return of a portfolio is the average of its holdings' returns weighted by their share of its value, and an expected value is an average of outcomes weighted by probability.