Weighted Average
$$\bar{x}_w = \frac{x_1 w_1 + x_2 w_2 + \cdots + x_n w_n}{w_1 + w_2 + \cdots + w_n}$$
What is Weighted Average?
A weighted average assigns each value a weight that reflects how much data it represents before the average is computed. The formula multiplies each value by its weight, sums the products, and divides by the total weight. The result is identical to what you would get by combining all individual data points and computing a single overall rate.
The concept appears whenever groups of different sizes are combined. A simple arithmetic mean treats every value equally, so smaller groups get the same influence as larger ones - producing a misleading result. A weighted average corrects for this by scaling each value to the size of the group it came from.
Weights can be any unit that represents the denominator of each value: sample size, total marks, hours, impression counts, or investment amounts. The only requirement is that all weights use the same unit.
Weighted Average explained to a beginner
Think of two test scores: you scored 90% on a 10-question quiz and 70% on a 100-question exam. Simple average: (90 + 70) / 2 = 80%. But the exam had ten times as many questions, so your overall performance is much closer to 70%.
Weighted average: (90 x 10 + 70 x 100) / (10 + 100) = (900 + 7000) / 110 = 72.7%. That number matches what you would get if you counted all 110 questions directly. The weighted average always matches the direct count.
When to use Weighted Average
Use a weighted average any time the values you are combining came from groups or items of different sizes. Common cases: combining pass rates from cohorts with different enrollment counts, calculating course grades from assignments worth different marks, blending ad campaign click-through rates from runs with different impression counts, or computing portfolio returns weighted by position size. When all groups have equal size, simple average and weighted average give the same result.
Worked examples for Weighted Average
This table quickly gives you the overview you need to understand Weighted Average and its most important comparisons.
| Scenario | Values | Weights | Simple avg | Weighted avg |
|---|---|---|---|---|
| Pass rates, unequal groups | 60%, 80% | 200, 50 people | 70% | 64% |
| Course grades | 80%, 90%, 70% | 30, 10, 60 marks | 80% | 75% |
| Campaign CTR | 3%, 8% | 10,000, 1,000 impressions | 5.5% | 3.45% |
| Survey satisfaction | 72%, 88% | 500, 100 respondents | 80% | 74.7% |
| Equal groups (same size) | 70%, 80% | 100, 100 people | 75% | 75% |
Common pitfalls
The most common error is treating percentages as their own weights - for example, using "40% of the course" as the weight for a grade rather than the underlying mark total. This works only when those percentage weights already sum to 100. A second pitfall is mixing units across weights: combining people with hours, or marks with credit hours, produces a meaningless result. Always confirm all weights share the same unit.
Frequently asked questions about Weighted Average
Is weighted average always more accurate than simple average?
When groups differ in size, yes - weighted average is the only mathematically correct method. When all groups are equal in size, both methods give identical results and either is valid. "More accurate" only applies when there is a size difference to account for.
What should I use as the weight for each percentage?
Use whatever was the denominator when the percentage was originally calculated: the number of people in a group, the total marks available for an assignment, the number of impressions in a campaign, or the value of an investment position. If you know the raw counts, they are always the correct weight.
Can the weights be decimals or fractions?
Yes. Weights do not need to be whole numbers or sum to any particular value. What matters is that they are proportional to the actual sizes of the groups. If one group is 2.5 times the size of another, its weight should be 2.5 times larger.
Quiz: how well do you know weighted average?
1. A class of 200 students has a 60% pass rate. A second class of 50 students has an 80% pass rate. What is the combined pass rate?
2. When does weighted average give the same result as simple average?
3. What should you use as the weight for a test score percentage?
4. Which of the following is a pitfall when calculating a weighted average?