Weighted Average
An average where some numbers count more than others. Enter a value and a weight for each row — the weights do not have to add to 100, or to anything in particular — and get the weighted mean, the plain unweighted mean of the same numbers, and the gap between them, which is usually the figure worth looking at.
Your values and their weights
3 values, weights totalling 100 — they do not have to add to 100, only to each other. The plain average of the same numbers is 85.00, so the weighting moves the answer by -0.70, which is small enough that the weights are barely doing anything here.
- Weighted average
- 84.30
- Plain average
- 85.00
- Difference
- -0.70
- Total weight
- 100
- Heaviest row
- Item 3 · 50%
| Label | Value | Weight | Share of total | Contribution |
|---|---|---|---|---|
| Item 1 | 92 | 20 | 20% | 18.40 |
| Item 2 | 78 | 30 | 30% | 23.40 |
| Item 3 | 85 | 50 | 50% | 42.50 |
- Only the ratios between your weights matter. Entering 2, 3 and 5 gives exactly the same average as 20, 30 and 50 — the total is divided out either way.
- A row weighted 0 contributes nothing to the weighted average but still counts in the plain one. That is often where a surprising gap between the two comes from.
- Neither average is the correct one on its own. Which to quote depends on what your weights mean, and that is a judgement this page cannot make for you.
How a weighted mean works, and why the total is free
Multiply each value by its weight, add those up, and divide by the total of the weights. The plain mean is the same operation with every weight equal.
x̄w = Σ(vᵢ × wᵢ) ÷ Σwᵢ • x̄ = Σvᵢ ÷ n • gap = x̄w − x̄- vᵢ
- Value i — a score, a price, a rating, a temperature, a rate
- wᵢ
- Weight i — how much value i counts, relative to the others
- Σwᵢ
- The total weight, whatever it is. This is the divisor, and the reason no particular total is required
- gap
- How far the weighting moved the answer away from a plain average of the same numbers
Dividing by Σwᵢ is what makes a weight relative rather than absolute. The lists 2 / 3 / 5, 20 / 30 / 50 and 200 / 300 / 500 all describe the same thing — the third row counts half — and all three produce an identical average, because the scaling factor appears in the numerator and the denominator and cancels. That is why this page does not ask your weights to total 100 and does not warn you when they do not. It prints your total instead, and gives each row’s share of it, so you can check the ratios are the ones you meant.
Two totals are refused. Zero, because there is nothing to divide by — a list where every row counts for nothing is a question with no answer, not an answer of none. And any total involving a negative weight, because negative importance is not a concept the formula supports: it lets Σwᵢ approach zero from either side, and an average that quietly runs off to an arbitrary magnitude is worse than an error message.
When the plain average passes and the weighted one fails
A course with two quizzes worth 10% each and a final worth 80%. A student scores 90 on each quiz and 55 on the final.
Plain average: (90 + 90 + 55) ÷ 3 = 78.33. Three numbers, three equal shares. On most scales that is a C+ and a comfortable pass.
Weighted average: (90 × 10 + 90 × 10 + 55 × 80) ÷ 100 = (900 + 900 + 4,400) ÷ 100 = 62.00. A D at most schools, and a fail at plenty of them.
Sixteen and a third points apart, on identical work. The reason is visible in the shares: the final holds 80% of the weight, while a plain average hands it 33% — one row out of three. When the heavily weighted item is also the outlier, the two averages pull in opposite directions, and the gap between them is a direct measure of how much the weighting is doing. Here it is 16.33 points out of a 35-point spread between the best and worst score, which is why the result panel raises it rather than leaving you to notice.
The same answer from three different weight totals
Now the defaults on this page: 92, 78 and 85, weighted 20, 30 and 50. The weighted mean is (1,840 + 2,340 + 4,250) ÷ 100 = 84.30. Change the weights to 2, 3 and 5 and the arithmetic becomes (184 + 234 + 425) ÷ 10, which is 84.30 again. Use 200, 300 and 500 and it is 84.30 a third time. Only the ratios ever mattered.
Notice, too, how small the gap is in that second example: the plain average is 85.00, so the weighting moves the answer by 0.70 — about a twentieth of the spread between the values. That is a case where arguing about the weighting scheme is not worth anyone’s afternoon. The first example is a case where it decides whether someone passed.
Which of these three tools you want
If every number counts the same, you do not need a weighted average at all — the mean, median, mode and range calculator gives you the centre and the spread of a plain list, which is more than an average and a better answer to “what is typical here”.
If you are working from a syllabus — scores out of points, categories with percentage weights, and a letter grade at the end — use the grade calculator. It understands what a category is, it treats a weight total that is not 100 as something worth telling you about, and it reports a letter.
This page is for everything else: costs weighted by quantity, ratings weighted by number of reviews, measurements weighted by sample size, regional rates weighted by population. It has no notion of a grade, no letter scale, and no opinion about what your weights should total.
What this calculator assumes
- Weights are relative. Only the ratios between them affect the answer, and the total is divided out — so no particular total is required and none is treated as a mistake.
- Every value counts exactly once in the plain average, no matter what its weight is. A row weighted 0 drops out of the weighted mean and stays in the plain one, which is a real difference and not a bug.
- A total weight of zero is refused, and so is any negative weight. Both are questions the formula cannot answer rather than edge cases with a sensible default.
- Values may be negative, and are not capped or normalised. Only the weights are divided by their total.
- Neither average is privileged. Which one to quote depends on what your weights represent, and nothing on this page can decide that for you.
- Figures are computed to four decimal places and displayed to two, so a third of the way between two values reads as 78.33 rather than 78.3333333333.
Weighted average FAQ
Do the weights have to add up to 100?
No. A weight says how much one value counts relative to the others and nothing more, so 2, 3 and 5 give exactly the same answer as 20, 30 and 50, and as 200, 300 and 500. The formula divides by the total of your weights, whatever that total happens to be, which is what makes all three lists equivalent. The only total this calculator refuses is zero, because a list where every row counts for nothing has no average to report.
Why is my weighted average lower than the plain average?
Because the values carrying the heaviest weights are the lower ones. That is the entire mechanism, and this page shows both figures so you can see it happen rather than guess. When the two averages are close, the weights are barely doing anything and the choice of method does not matter much. When they diverge sharply, one or two heavily weighted rows are deciding the answer, and it is worth checking those weights are what you meant before quoting the number.
What is the difference between a weighted average and a regular one?
A regular average is the special case of a weighted average where every weight is the same. Set all the weights here to 1 — or to 7, or to 0.5 — and the answer matches the plain average exactly, because equal weights divide out. The moment the weights differ, the values with the larger weights pull the answer toward themselves, and the difference between the two figures is a direct measure of how much that pulling matters.
Can a weight be zero? Can it be negative?
Zero is allowed and is useful: it takes a row out of the weighted average while leaving it visible in the table, and while still counting it in the plain average — which is often where a surprising gap between the two comes from. Negative weights are rejected. A negative importance is not a weight, and permitting them lets the total approach zero from both directions, which sends the average off to an arbitrary magnitude with no warning that anything went wrong.
Should I use this or the grade calculator?
Use the grade calculator when you have a syllabus: scores out of points, categories with percentage weights, and a letter grade at the end. It knows what a category is and reports a letter. Use this page for everything else — costs weighted by quantity, ratings weighted by number of reviews, measurements weighted by sample size, rates weighted by population. It has no notion of a grade and does not care whether your values are percentages at all.
My weights are already percentages that add to 100. Does anything change?
Nothing changes, and that is the point. Dividing by a total of 100 is the same operation as dividing by a total of 10 or of 3.7 — you just happen to be in the case where the division looks like it is not happening. Enter them as they are. The result panel prints your total weight so you can confirm it is what you expected, which is a useful check when a weight has been mistyped.
Sources and review notes
- NIST/SEMATECH e-Handbook of Statistical Methods — the standing federal reference for the weighted mean, including the convention that the estimator is normalised by the sum of the weights rather than by any fixed total
- NCI SEER Standard Populations — age-adjusted rates are weighted averages whose weights are population counts in the millions, a working demonstration that weights need no particular total to be valid
The arithmetic on this page is a weighted mean and cannot go stale. The claim that needs a source is the one this calculator is built around — that the weights are normalised by their own sum, so their scale is arbitrary and only their ratios carry meaning. Both references above rely on exactly that, one as a definition and one as a national statistic computed from it every year.