Média Ponderada

Weighted mean: not all values count equally

The weighted mean gives each value xᵢ a weight pᵢ and computes WM = Σ(xᵢ · pᵢ) / Σ pᵢ. When every weight is the same, you get the ordinary arithmetic mean back. Take a grade example: test 1 = 7 (weight 2) and test 2 = 9 (weight 3) give WM = (7·2 + 9·3) / (2 + 3) = (14 + 27) / 5 = 8.2, which is not the plain mean of 8. A university GPA works the same way, averaging course grades by their credit hours. Reach for it any time your observations carry different importance, frequency or sample size.

Applications

School and university grades (GPA by credits) lean on it, as does IPCA / CPI, where consumption baskets are weighted by each item's share of the household budget. It also shows up in cap-weighted indexes such as the S&P 500 and Ibovespa as opposed to equal-weighted ones, in portfolio expected return Σ wᵢ·Rᵢ, in a weighted Sharpe ratio, and in the survey weights used to correct for non-response.

FAQ

Do weights need to sum to 1? No. Since the formula divides by Σ pᵢ, any positive numbers will do. Normalizing the weights so they add up to 1 only saves you that division.

Can weights be zero or negative? A weight of zero drops the observation entirely. Negative weights are odd and seldom mean anything sensible, so check your model before reaching for them.

Weighted mean or arithmetic mean? Go weighted when the values don't count equally, whether that's more credits, higher frequency, a larger sample or bigger market cap. The moment all the weights match, the weighted mean is just the arithmetic mean.