A weighted mean lets some values contribute more than others. Each value is multiplied by its weight, the products are added, and that total is divided by the sum of the weights. The method covers repeated observations, course grades with different percentages, price indices, and group means based on different sample sizes.
Weights describe influence, not decoration
The weights must match the values one by one and must represent the same kind of influence. They may be counts, percentages, probabilities, durations, areas, or reliability measures. Multiplying every weight by the same positive constant leaves the weighted mean unchanged because both numerator and denominator scale together.
Change the three values and their weights below. The result stays between the smallest and largest value when all weights are nonnegative and their total is positive.
Build a weighted mean
Increase a weight to pull the result toward its matching value.
Worked example: a final course grade
Suppose assignments count for 25%, a project for 35%, and an exam for 40%. A student scores 80, 92, and 75 in those components. Convert the percentages to weights or use 25, 35, and 40 directly, multiply each score by its matching weight, and divide by the total weight.
The result is 81.2. A quick reasonableness check places it between 75 and 92 and somewhat closer to the heavily weighted exam and project scores. The ordinary mean, about 82.33, would incorrectly give all three components equal influence.
Percentages need a consistent denominator
If weights are 25%, 35%, and 40%, they already total 100%. If they total something else because one component is missing or the listed items cover only part of a whole, decide whether to normalize the available weights or retain the missing contribution. Those are different questions.
The formula always divides by the actual weight sum. Dividing by the number of values is correct only when all weights are equal. Likewise, entering percentages as 25, 35, and 40 gives the same result as 0.25, 0.35, and 0.40, but mixing the two scales in one calculation does not.
Combining group means without losing group size
Imagine one group of 10 people has mean 68 and another group of 30 has mean 76. The combined mean uses group sizes as weights: the reconstructed totals are 680 and 2280, producing 74 across 40 people. Simply averaging 68 and 76 gives 72 and overrepresents the smaller group.
This same reconstruction works for frequency tables because a frequency is a weight. It does not recover a combined variance from group means alone; that requires additional information about within-group spread.
Checks, limits, and interpretation
Negative weights occur in specialized mathematical models, but then the result may lie outside the range of the values and should not be interpreted as an ordinary average. A zero total weight makes the expression undefined. Missing values should not automatically receive zero scores unless the scoring rule explicitly says so.
State what the weights mean and whether they were normalized. Keep unrounded products until the final division. For equal weights, the result should match the arithmetic mean; to compare it with other measures of center, see mean, median, and mode.
