Weighted Mean
Statistics & ProbabilityA 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.
Example
A grade is $70\%$ tests and $30\%$ homework. Test average: $85$, Homework average: $90$. Weighted mean $= (0.70)(85) + (0.30)(90) = 59.5 + 27 = 86.5$.
Key Insight
Use a weighted mean whenever not all values are equally important or not all groups are the same size.
Definition
The weighted mean 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. When all weights are equal ($w_i = 1$), it reduces to the arithmetic mean. Weighted means arise in grading, survey analysis, and portfolio returns.
Example
Portfolio return: $40\%$ in Stock A (return $8\%$), $35\%$ in Stock B (return $5\%$), $25\%$ in Stock C (return $12\%$). Weighted return $= 0.40(8) + 0.35(5) + 0.25(12) = 3.2 + 1.75 + 3.0 = 7.95\%$.
Key Insight
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.
Definition
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
In meta-analysis, studies are combined using inverse-variance weighting: $w_i = 1/s_i^2$. This assigns 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
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.