Weighted Mean
A weighted mean is an average that gives more importance to certain values by multiplying each value by its assigned weight before averaging.
Formula
\text{Weighted Mean} = \dfrac{\sum w_i x_i}{\sum w_i}
Definition
A weighted mean is an average where some values count more than others: you multiply each value by how much it should count (its weight), add those up, and divide by the total weight. Formally, it is a generalization of the arithmetic mean: $\bar{x}_w = \sum w_i x_i / \sum w_i$, where $w_i$ are non-negative weights, reducing to the arithmetic mean when all weights are equal ($w_i = 1$). The weighted mean is the expectation of $X$ under a probability measure defined by the normalized weights $w_i/\sum w_j$; in survey sampling, inverse probability weighting (IPW) sets $w_i = 1/\pi_i$ (where $\pi_i$ is the inclusion probability), ensuring unbiased estimation of the population mean under design-based inference.
Example
If a grade is $70\%$ tests and $30\%$ homework, with test average $85$ and homework average $90$, the weighted mean $= (0.70)(85) + (0.30)(90) = 59.5 + 27 = 86.5$. A portfolio with $40\%$ in Stock A (return $8\%$), $35\%$ in Stock B (return $5\%$), and $25\%$ in Stock C (return $12\%$) has weighted return $= 0.40(8) + 0.35(5) + 0.25(12) = 7.95\%$. In meta-analysis, studies are combined using inverse-variance weighting, $w_i = 1/s_i^2$, assigning higher weight to more precise studies, and the combined estimate is the minimum-variance linear unbiased estimator (BLUE) of the common effect size.
Key Insight
Use a weighted mean whenever not all values are equally important or not all groups are the same size; survey data often uses sampling weights to adjust for unequal probabilities of selection, making the weighted mean representative of the full population despite a non-proportional sample. The Horvitz-Thompson estimator extends IPW to complex survey designs, giving a design-unbiased estimator for any population total or mean regardless of the sampling scheme used.