weightedavg

A weighted average, also known as a weighted mean, is a measure of central tendency in which each data value x_i is multiplied by a corresponding weight w_i reflecting its importance, frequency, or reliability. The products are summed and then divided by the total of the weights to produce the final value.

Mathematically, the weighted average is defined as W = (Σ w_i x_i) / (Σ w_i), where i ranges over all data items, the weights w_i are nonnegative, and the sum of weights is greater than zero. If the weights sum to 1, the weighted average simplifies to W = Σ w_i x_i.

Example: for values 3, 5, 7 with weights 2, 1, 3, the weighted average is (3*2 + 5*1 + 7*3) / (2+1+3) = (6 + 5 + 21) / 6 = 32/6 ≈ 5.33.

Common applications include combining disparate data sources, computing grade point averages with course credits as weights, constructing index values, and evaluating portfolio performance where weights reflect allocation or confidence.

Notes and caveats: the choice of weights determines the result and can be subjective, so weights should be chosen to reflect true importance or frequency. Weights are typically nonnegative; allowing negative weights changes the interpretation and is less common. When dealing with missing data, weights may be adjusted or items omitted. Related concepts include the unweighted mean, weighted median, and weighted moving averages.