Averaging percentages looks like ordinary averaging, but it hides a trap: a percentage is a ratio, and ratios from groups of different sizes cannot be averaged directly. A 90% score in a class of 10 and a 70% score in a class of 90 do not average to 80%. This calculator weights every percentage by its sample size, reports the correct overall figure, and shows how far a naive average would have been off.
How to use the average percentage calculator
- Enter one group per line: the percentage, then the sample size, separated by a comma — for example
85, 120. - Optionally start a line with a name and a colon:
Section B: 85, 120. - Read the weighted average on the tape, along with the plain average, the difference between them, the total sample and the combined count. A table and bar chart summarize the groups.
If you leave out every sample size, the calculator gives the ordinary (unweighted) mean of the percentages.
Weighted average percentage formula
Convert each percentage back into a count, add the counts, and divide by the total size:
Here pi is each group’s percentage and ni its size. This is a weighted mean with the sample sizes as weights. The plain average, (p1 + … + pk)/k, is correct only when all the ni are equal.
Worked example
Three sections of a course report their pass rates:
| Section | Pass rate | Students | Students who passed |
|---|---|---|---|
| A | 72% | 50 | 36 |
| B | 85% | 120 | 102 |
| C | 64% | 30 | 19.2 |
| Total | 200 | 157.2 |
Counts: 0.72 × 50 = 36, 0.85 × 120 = 102 and 0.64 × 30 = 19.2.
Weighted average: 157.2 ÷ 200 = 0.786, so the overall pass rate is 78.6%.
Plain average: (72 + 85 + 64) ÷ 3 = 73.67%, almost 5 percentage points too low, because the large, high-scoring Section B is under-represented.
The fractional count of 19.2 simply means the 64% figure was rounded; with real data, the counts are whole numbers.
When to use which average
| Question | Use |
|---|---|
| What share of all students passed? | weighted average |
| What is the typical section’s pass rate? | plain average |
| Combining survey results with different sample sizes | weighted average |
| Averaging monthly growth rates | neither — multiply growth factors |
The last row deserves a note. Percentage changes over time compound, so an average growth rate is a geometric mean, not an arithmetic one. The percent of a percent calculator shows how successive changes combine.
Simpson’s paradox in brief
Weighting matters most when group sizes and percentages are tied together. Suppose treatment A succeeds more often than B for both mild and severe cases, but A was mostly given to severe cases. Pooled together, A can look worse than B. The pooled weighted percentage is arithmetically correct; the lesson is that it answers a different question from the subgroup comparison, so report both.
Related calculators
For general weighted means, such as grades with category weights, use the weighted average calculator or the grade calculator. For an ordinary mean of numbers, use the mean calculator, and for a single percentage of a number, the percentage calculator.
Frequently asked questions
Can I just average the percentages?
Only if every group has the same size. Otherwise a small group counts as much as a large one and the result is wrong. Weight each percentage by its sample size, or equivalently add up the underlying counts and divide by the total.
How do I enter the data?
One group per line: the percentage, a comma, then the sample size, for example 72, 50. You can put a name before a colon, such as Section A: 72, 50. Leave out the sizes for a plain average.
Why is the weighted average closer to one group?
Larger groups pull the average toward their own percentage, because they contribute more people (or items) to the combined total.
Can the averages tell different stories?
Yes. In extreme cases, called Simpson's paradox, one option can have the higher percentage in every subgroup yet the lower percentage overall, because the subgroups have very different sizes. Always check the sizes behind percentages.