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Weighted average
Definition
A weighted average combines values while giving each a specified relative influence. It is calculated by multiplying each value by its weight, adding those products and dividing by the sum of the weights. With nonnegative weights and a positive total weight, the result lies between the smallest and largest values. The meaning of the average depends on what the weights represent, such as group sizes, population shares, importance or measurement precision.
What it means
An ordinary arithmetic mean gives every listed value equal weight. That can be inappropriate when the values summarize groups of different sizes or when a target population has a different composition from the observed sample. Weights make the intended contribution explicit. They are not automatically correct merely because they are numerical. Different weighting schemes answer different questions, and a weighted average should be interpreted together with its target, units and construction.
How it works
For values xi and weights wi, calculate sum(wi times xi) divided by sum(wi). If weights already sum to one, the denominator is one. Multiplying all weights by the same positive constant leaves the average unchanged. For group means, weighting by group sample sizes reproduces the overall mean of the underlying observations when the groups partition those observations. Survey weights, frequency weights and inverse-variance weights serve different purposes and require different uncertainty calculations. A weighted average formula alone does not specify a valid standard error.
Three practical examples
Example 1: A company combines branch success rates using the number of cases in each branch. Example 2: An AI evaluation combines subgroup performance using the intended deployment population shares. Example 3: A measurement analysis combines independent unbiased estimates using precision-related weights under suitable assumptions. These applications share the arithmetic but differ in interpretation. Importance weights chosen by preference do not have the same statistical meaning as weights derived from sampling probabilities.
Think of it this way
Think of a balance where some values carry more mass than others. The balancing point moves toward the values with greater total weight. What the mass represents matters: it may count how many observations a value summarizes or express a deliberate priority. The arithmetic can be correct even when the chosen priorities do not answer the intended question.
Worked example
Suppose branch A processes thirty cases with an 80% success rate and branch B processes seventy with a 50% success rate. The weighted rate is (30 times 80 + 70 times 50)/(30 + 70) = 59%. Equivalently, the branches have twenty-four and thirty-five successes, giving 59 successes out of 100 cases. The unweighted average of the two branch rates is 65%, which gives each branch equal influence rather than each case equal influence. Both calculations are arithmetically valid but answer different questions. Multiplying the weights by ten leaves the weighted rate at 59%. The example assumes that the branch counts and rates refer to the same case definition and period.
Illustrative case study
In this fictional case study, Alder Services compares an AI-assisted process across offices. A dashboard averages office percentages equally, even though office workloads differ sharply. Analysts add a case-weighted overall rate and retain the equal-office summary only when that is the explicit reporting objective. They also show office-level results so that an overall improvement does not hide a decline in a smaller office. The team documents the weighting rule rather than treating one unlabeled average as universally representative.
Common misconceptions and mistakes
Common mistakes include averaging percentages without their denominators, failing to normalize weights, and mixing weights that represent incompatible concepts. Another mistake is assuming a weighted average describes every subgroup. A changing population mix can change the overall result even when subgroup values are unchanged. Negative weights can place the result outside the observed range and need a specialized justification; the usual intuitive averaging interpretation assumes nonnegative weights. Missing values also require a documented rule for which weights remain in the denominator.
Business use
Businesses should choose weights according to the decision and target population, then label the result clearly. For operational totals, case counts may be appropriate; for a standardized comparison, a fixed reference mix may be preferable. Show subgroup values when they matter, and distinguish changes in performance from changes in composition. If weights are estimated or highly unequal, uncertainty can be affected. Do not use weighted arithmetic to hide inconsistent definitions, unreliable measurements or a mismatch between evaluation and deployment populations.
People also ask
Do weights have to sum to one initially?
No; the formula divides by their total.
Does scaling every weight change the result?
No, if the same positive multiplier is used.
Is an equal-weight average a special case?
Yes.
Can averaging group percentages equally differ from the overall case rate?
Yes.
Does the weighted mean alone provide its standard error?
No.
Check your understanding
What is the case-weighted success rate in the example?
Answer: 59%.
What does the 65% figure represent?
Answer: The equally weighted average of the two branch rates.
Why does branch B influence the case-weighted result more?
Answer: It contributes seventy of the 100 cases.
What must be consistent before combining the rates?
Answer: Their outcome definition, denominator meaning and reporting period.
Related terms
- Weighted sum
- Related terms include arithmetic mean
- population weight
- frequency weight
- inverse-variance weighting and standardization
Learn next
Learn next how changing subgroup shares affects aggregate rates and how a fixed weighting standard supports comparisons. Compare weighted means with weighted medians, which solve a different summary problem. Then study uncertainty formulas for the particular weight type rather than applying an unweighted standard-error formula automatically.
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Sources
Source: University of Vermont, Statistical Methods for Psychology glossary, Weighted average and Weighted means, https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html. This primary teaching source provides the weighted-average definition and group-size context. The two-branch calculation and weight-rescaling check are original and independently verified; the service-dashboard case is fictional.
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