Multiply each value by its weight, add those up, and divide by the total of the weights. Marks of 90, 80 and 70 weighted 50, 30 and 20 give (4500 + 2400 + 1400) ÷ 100 = 83 — not the 80 you get by averaging them plainly. The weights do not have to add to 100: 5, 3 and 2 gives the same 83, because the total is what the formula divides by.
Not the average of the averages
This is the mistake worth naming. Given marks of 90, 80 and 70, the obvious move is to add them and divide by three, which gives 80. If the components are worth 50%, 30% and 20%, the answer is 83.
Three marks is a small enough example that the gap looks minor. It is not always: a 95 on something worth 5% and a 60 on something worth 95% averages plainly to 77.5 and weights to 61.75 — a difference between a good grade and a poor one.
The page shows both numbers every time for exactly that reason. If they differ, the weights are doing something, and the difference is the size of what they are doing.
Weights do not have to add to 100
A weighted average divides by the total of the weights. So the SCALE of the weights cancels entirely: 2, 3 and 5 gives the same answer as 20%, 30% and 50%, or as 200, 300 and 500.
Tools that demand percentages have added a constraint the mathematics does not have, and the usual consequence is people normalising by hand — dividing by the total, rounding each to two places, and introducing an error that was never necessary.
Use whatever the weights naturally are. Credits, quantities purchased, headcounts, hours worked — all of them work as they stand.
Where weighted averages come up
Course marks. Components worth different percentages. The commonest reason anyone visits a page like this.
GPA. Grade points weighted by credit hours, so a four-credit course moves the average twice as far as a two-credit one. TheGPA Calculator does that specific job end to end.
Average price paid. Buying 100 at 12.50 and 250 at 9.75 is not an average of 11.13 — it is 10.54, because more units came at the lower price.
Index numbers. A consumer price index is a weighted average whose weights are how much people actually spend on each thing. Choosing those weights is most of the work in building one.
A zero weight is not a missing value
Giving a row a weight of zero removes it from the answer completely while leaving it visible in the list. That is often exactly what you want.
A dropped assignment, an excluded outlier, a product that was not stocked this month — all of them are better recorded as present with no weight than quietly deleted, because the row still shows what was excluded.
A NEGATIVE weight is a different matter and is refused here. It would subtract a value from the average rather than counting it less, which is not what anyone means by weighting.
Weighting is a judgement, not a calculation
The arithmetic on this page is exact. The weights are not: they are a claim about how much each value should matter, and that claim comes from outside the mathematics.
It is also where weighted averages get abused. Any desired answer can be produced by choosing weights to suit it, and a weighted figure quoted without its weights is not checkable at all.
So publish the weights alongside the answer. This page shows every share as a percentage for that reason — the reader can see what was decided before they see what it produced.
Checking a weighted answer
It must lie between the smallest and largest value. Always, whatever the weights. An answer outside that range means a sign error or a mispaired weight.
It leans towards the heavily weighted values. If the largest weight sits on the smallest number, the answer should be below the plain average. The page shows each row's distance from the answer so that lean is visible.
Equal weights must reproduce the plain average. A quick sanity check on any implementation, and one the suite runs.
Sources and methodology
The weighted mean is elementary; these are where its uses are defined.
Method. Values and weights are paired in order and the mean is Σ(value × weight) ÷ Σ(weight). No normalisation is applied first, because none is needed and doing it would introduce a rounding step. The unweighted mean is computed separately and shown alongside, and the suite asserts scale invariance directly: multiplying every weight by a constant must not move the answer. That engine is verified on every change against 61 hand-written assertions, including that scaling every weight by any constant leaves the answer unchanged across fifteen hundred generated cases. The count and the per-case breakdown are published on the formula verification page.
Read the guide
If your quantities multiply rather than add — growth rates, ratios — the Geometric Mean Calculator is the right average. If they are rates over a fixed distance or quantity, the Harmonic Mean Calculator is.