WeightedAverage

Weighted average is a measure that allows different values to contribute to an overall average according to their importance or frequency. Each value x_i is paired with a weight w_i that represents its relative significance. The weighted average of values x1, x2, ..., xn with weights w1, w2, ..., wn is defined as the sum over i of w_i times x_i, divided by the sum over i of w_i. If the weights are normalized so that their sum equals 1, the weighted average reduces to the simple sum of w_i x_i.

Example: with values 80, 90, and 70 and weights 0.5, 0.3, and 0.2, the weighted average is (0.5*80 + 0.3*90 + 0.2*70) / (0.5+0.3+0.2) = 81.

Weights should generally be nonnegative, reflecting relative importance, frequency, or confidence. They can also be any positive constants; some formulations use weights that sum to 1 for convenience, but this is not required. The weighted average differs from the ordinary arithmetic mean by incorporating these weights to emphasize certain observations more than others.

Common considerations include how to determine weights, how to handle missing values (e.g., excluding a value and its weight or imputing), and how the choice of weights affects the result. In statistics and data analysis, weighted averages appear in contexts such as grade calculations, survey analysis, index construction, and portfolio performance, where different components contribute unevenly to the overall measure. In some optimization settings, weights may be chosen to optimize estimator properties, such as giving more weight to measurements with lower variance.