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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

In the general case, suppose that ${\displaystyle \mathbf {X} =[x_{1},\dots ,x_{n}]^{T}}$, ${\displaystyle \mathbf {C} }$ is the covariance matrix relating the quantities ${\displaystyle x_{i}}$, ${\displaystyle {\bar {x}}}$ is the common mean to be estimated, and ${\displaystyle \mathbf {J} }$ is a design matrix equal to a vector of ones ${\displaystyle [1,...,1]^{T}}$ (of length ${\displaystyle n}$). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:

, ${\displaystyle \mathbf {C} }$ is the covariance matrix relating the quantities ${\displaystyle x_{i}}$, ${\displaystyle {\bar {x}}}$ is the common mean to be estimated, and ${\displaystyle \mathbf {J} }$ is a design matrix equal to a vector of ones ${\displaystyle [1,...,1]^{T}}$ (of length ${\displaystyle n}$). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by: