Weighted Mean

A weighted mean is an average in which each value is multiplied by a weight reflecting its importance or frequency before the sum is taken, and the total is then divided by the sum of the weights. Where the ordinary arithmetic mean treats every observation as equally important, the weighted mean lets some count more than others. Its formula in words: multiply each value by its weight, add up those products, and divide by the total of the weights. With all weights equal it reduces exactly to the ordinary mean, which is why it is best understood as the general form of which the plain mean is the special case.

The weighted mean is the right tool whenever values are not interchangeable, and the most familiar example is a course grade: if coursework is worth 40 percent and the exam 60 percent, a student scoring 90 on coursework and 70 on the exam finishes on 0.4 × 90 plus 0.6 × 70, which is 78, not the plain average of 80. The weights need not sum to one, since dividing by their total normalises them, so weights can be written as credit hours, counts, or percentages interchangeably. The grade point average is the same idea, with credit hours as weights, so an A in a four-credit course moves the result more than an A in a one-credit course. The effect is that the answer is pulled toward the heavily weighted values, and so a weighted mean can lie anywhere between the smallest and largest value depending on where the weight is concentrated.

A second, very common use of the weighted mean is to combine groups: the overall mean of several groups is the mean of the group means weighted by group sizes, and averaging the group means without weights is a classic error. If 10 students average 80 and 90 students average 60, the combined mean is 62, since weighting by size gives (10 × 80 + 90 × 60) divided by 100, not 70, which the unweighted average of 80 and 60 would suggest. The same principle computes the mean from a frequency table, where each class value or midpoint is weighted by its frequency, and it explains why a mean of percentages across groups of different sizes is misleading and why pooled rates must be computed from the underlying counts.

Weighted means underlie a great deal of applied statistics, often hidden inside other names. Stock market indices weight constituents by market capitalisation, and consumer price indices weight price changes by the share of household spending on each item, with the weights revised periodically. In survey statistics, design weights, in the tradition of the Horvitz-Thompson estimator published in 1952, correct for unequal selection probabilities and nonresponse so that a sample's weighted mean estimates the population mean. In meta-analysis, study results are combined by weighting each by the inverse of its variance, giving precise studies more say. In machine learning, class weights, sample weights, and the weighted averaging of F1 scores across classes use the same arithmetic, and exponentially weighted moving averages weight recent observations more heavily, which underlies smoothing in forecasting and the momentum terms in optimisers.

The pitfalls concern where the weights come from. Weights are a modelling choice: reweighting changes the answer, and a weighting scheme chosen after seeing the results can steer a conclusion, so weights should be justified independently of the outcome. Extreme weights make a result hinge on one or two observations, inflating the variance of the estimate, which is why weights are often capped or trimmed in survey work. Negative weights are generally meaningless. Weighted means also inherit the sensitivity of the arithmetic mean to outliers, so a heavy weight on an outlier is doubly harmful. And a weighted mean should not be confused with the median or with the geometric mean, which handles multiplicative rather than unequal-importance structure.

Read the weighted mean as the arithmetic mean with a vote proportional to importance: the plain mean is just the case where every value gets the same vote, and the discipline lies in choosing weights for reasons stated in advance, and in remembering that combining groups with unweighted averages is almost always a mistake.