Percentages do not average unless the groups are the same size.
Weighted by the group sizes
Which is the only correct way.
of
of
101 out of 102
99.019607…%
Averaging the percentages directly would give 75%, which is wrong here — the groups are not the same size, so they do not carry equal weight.
Weighted average
99.019607…%
the correct answer
Plain mean
75%
what most people reach for
Total counted
101
the numerators added
Total group size
102
the denominators added
What each group contributes
Each group with its percentage, its size, the count that implies, and its share of the total weight
Group
Percentage
Size
Count
Weight
1
50%
2
1
1.960784…%
2
100%
100
100
98.039215…%
The weight column is each group’s share of the total size, and it is what decides how much its percentage moves the answer. A group holding 2% of the total can be at 100% and shift the average by almost nothing.
The correct average puts the counts back together: 101 out of 102, which is 99.019607…%.
Averaging the percentages directly gives 75%, which is wrong here by 24.019607… percentage points. The plain mean is only right when every group is the same size.
This is the same reason a batting average across two seasons is not the average of the two averages, and why a company-wide conversion rate is not the mean of its regional ones.
Carried as exact fractions, so a set of shares still sums to exactly 100% rather than to 99.99999%.
What this tool shows
One success out of two is 50%. A hundred out of a hundred is 100%. Together that is 101 out of 102, which is 99.02% — not the 75% you get by averaging the two percentages. The group sizes are the weights, and leaving them out is what makes the answer wrong.
The correct average of several percentages
Each group’s share of the total weight
The count each percentage implies
The plain mean, for contrast
When the two agree, and why
Any number of groups
Weighted by group size The naive answer shown too The gap named Any number of groups
Updated 7 September 2026 · Works in any browser, no installation
Put the counts back together, then divide once. Multiply each percentage by its group size to recover the count it stands for, add the counts, add the sizes, and divide. For 50% of 2 and 100% of 100 that is 101 ÷ 102 = 99.02%. Averaging the percentages instead gives 75%, and it is simply the wrong calculation.
At a glance
Formula shown
The weighted average is \u03a3(p\u1d62 \u00d7 n\u1d62) \u00f7 \u03a3n\u1d62, which is the pooled count over the pooled base. The plain mean \u03a3p\u1d62 \u00f7 k equals it only when every n\u1d62 is the same.
Scenario support
Combining conversion rates across regions of different sizes; a pass rate across classes of different sizes; a company-wide figure from departmental ones.
Educational estimate
Planning support from the values you enter — not professional advice.
Why the plain mean fails
Averaging percentages treats every percentage as equally important. They almost never are.
Say a shop converts 1 visitor out of 2 on Monday and 100 out of 100 on Tuesday. Monday’s 50% describes two people. Tuesday’s 100% describes a hundred. Giving them equal say means letting two people outvote a hundred.
The correct figure comes from the underlying counts: 101 conversions from 102 visitors, which is 99.02%. Every percentage has to be weighted by how many things it is a percentage OF.
This is not a subtlety. The naive answer here is wrong by 24 percentage points, and nothing about 75% looks suspicious.
When the plain mean is right
There is exactly one case: every group is the same size.
80% of 50 and 90% of 50 average to 85%, and the plain mean gives 85% too. With equal weights, the weighting does nothing.
That is why averaging percentages so often seems to work — in a classroom exercise the groups usually are equal. Outside one they usually are not, and the habit survives into places where it produces a wrong number quietly.
The page reports when the groups happen to be equal, so you can see that the agreement is a property of your data rather than of the method.
Simpson’s paradox
Weighting can do something stranger than shifting the answer: it can reverse it entirely.
A treatment can have a higher success rate than a rival in every subgroup and a lower rate overall. This happens when the subgroups are unequal in a way that correlates with the outcome — if the better treatment was tried mostly on harder cases, its pooled rate suffers.
The famous case is the 1973 Berkeley admissions data: women were admitted at a lower rate overall while being admitted at an equal or higher rate in most individual departments. Women applied in larger numbers to the departments with lower admission rates for everyone.
Neither the pooled figure nor the subgroup figures is “the” answer. Which one is right depends on the question, and that is a modelling decision this page cannot make for you — but seeing the weights makes it visible.
Choosing the weights
The right weight is whatever each percentage is a percentage OF. Usually that is obvious; sometimes it needs a moment.
Rates. A conversion rate weights by visitors, a pass rate by candidates, a defect rate by units produced.
Growth rates. These do not weight, they COMPOUND. Averaging annual growth rates needs a geometric mean rather than a weighted arithmetic one, and this page is the wrong tool.
Percentages of a total. If they already sum to 100%, averaging them is not a meaningful operation at all. Ask what question the average is supposed to answer first.
Percentages of different things
Sometimes the percentages are not comparable at all, and no weighting fixes it.
“Sales rose 20% and costs rose 5%, so on average things rose 12.5%” is not a statement about anything. The two percentages measure different quantities, and averaging them produces a number with no referent.
The test is whether the underlying counts can be pooled. Conversions from two days pool into conversions. Sales and costs do not pool into anything.
If the counts cannot be added, the average cannot be taken — and that is a modelling answer rather than an arithmetic one.
Where it goes wrong in practice
Dashboards. A tool averaging a metric across regions without weighting will overstate small regions badly, and nothing on the dashboard says so.
Batting and shooting averages. A season average is not the average of the two half-season averages unless the same number of attempts fell in each.
Survey results. Combining a 200-response survey with a 2,000-response one by averaging the percentages gives the small survey ten times the influence it earned.
Grades. A module average across courses of different credit weights is a weighted average, which is exactly why credit-weighted GPA exists as a separate calculation.
Sources and methodology
Weighting by base is standard statistical practice, and the failure mode has a name; these are the references.
Method. Each percentage is converted back to the count it represents, the counts and the bases are pooled, and the answer is that single fraction — which is what averaging percentages is trying to approximate. Everything is carried as exact rationals, so a set of shares still sums to exactly 100%. The plain mean is computed alongside so the difference is visible rather than argued. The suite checks the result against pooling raw counts by hand on fifteen hundred generated sets, and asserts across three thousand more that the weighted answer always lies between the smallest and largest input percentage. That engine is verified on every change against 81 hand-written assertions, including that the weighted average always lies between the smallest and the largest input across three thousand generated sets, which a weighting error would break immediately. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Weighted AverageEach value carries the weight you give it, with every item's share of the total shown as a percentage so you can see what is actually driving the answer.
PercentageSolve X% of Y, what percent X is of Y, reverse percentage, increase/decrease, discounts, and tax, tip, or commission.
Percentage ChangeWork out percentage increase or decrease, reverse change, loss recovery, percentage points, multi-period change, and CAGR.
RatioSimplify a ratio and share a total by it, with each term's fraction of the whole shown — because 3 : 2 means three fifths, not three halves.
Geometric MeanThe average for things that compound. Growth of +50% then -50% averages to zero arithmetically and to a real 13.4% loss geometrically, which is what actually happened.
Percentage PointA move from 4% to 6% is two percentage points and a 50% increase. Both are computed and named, so the two can never be quietly swapped.
For values that are not percentages — course grades, portfolio returns — the Weighted Average Calculator applies the same weighting to any quantity.
Educational use disclaimer
This is an educational tool. The arithmetic is exact; whether the group sizes you supply are the right weights for your question is a judgement the page cannot make.
Published the average percentage page asking for the group sizes, because the plain mean of two percentages is only correct when the groups behind them are the same size.
The naive answer is shown beside the correct one with the gap named, since 75% against a true 99.02% looks entirely plausible on its own.
Covers Simpson's paradox, where weighting does not merely shift the answer but reverses it.
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