What a weighted average is for
A weighted average is what you use when the things you are averaging do not count equally. A final exam worth 35% of a grade should not be averaged on the same footing as a quiz worth 5%. A branch with 400 customers should not count the same as a branch with 12 when you calculate a company-wide satisfaction score. A five-credit course should move a grade point average more than a one-credit course.
The mechanism is simple: multiply each value by its weight, add those products, then divide by the total weight. Dividing by the total is the step that makes the scale of the weights irrelevant. Weights of 20, 30, 15 and 35 produce exactly the same answer as 0.20, 0.30, 0.15 and 0.35, or as 4, 6, 3 and 7. Only the ratios between weights matter, which is why you never need to rescale your weights to sum to 1 before entering them here.
The ordinary arithmetic mean is the special case where every weight is equal. That is worth stating explicitly, because it is the source of the most common error in this area: taking a plain average of numbers that are themselves averages of groups of different sizes.
The formula, and what the weights actually mean
Write the weighted mean as sum(wixi) divided by sum(wi). There are two equivalent ways to read it.
As a total divided by a total. If the values are prices and the weights are quantities, then sum(wixi) is the total amount spent and sum(wi) is the total quantity, so the weighted mean is the average price actually paid per unit. Read this way the formula is not a statistical construction at all; it is just a total cost divided by a total count.
As a sum of shares. Divide each weight by the total first, giving normalised weights that add to 1. Then the weighted mean is the sum of each value multiplied by its share. This is the form the table on this page uses, and it makes the contribution of each item visible: a value of 95 with a 35% share contributes 33.25 to the answer.
The two readings answer different questions and it pays to know which you are in. In the first, the weight is a physical quantity - hours, units, dollars, respondents. In the second, the weight is a policy decision - the syllabus says the final is worth 35%. Both go into the same formula, but only the second is something you can argue about.
One structural property is worth remembering: as long as every weight is non-negative and not all are zero, the weighted mean always lies between the smallest and largest value. It cannot escape the range of the data. Introduce a negative weight and that guarantee disappears, which is why this calculator flags negative weights rather than silently accepting them.
Worked example: a course grade with four components
A syllabus assigns homework 20%, a midterm 30%, a project 15% and a final exam 35%. A student scores 88, 92, 79 and 95.
- Multiply each score by its weight. 88 x 20 = 1760; 92 x 30 = 2760; 79 x 15 = 1185; 95 x 35 = 3325.
- Add the products. 1760 + 2760 + 1185 + 3325 = 9030.
- Add the weights. 20 + 30 + 15 + 35 = 100.
- Divide. 9030 / 100 = 90.3.
Compare that with the simple average: (88 + 92 + 79 + 95) / 4 = 354 / 4 = 88.5. The weighted result is 1.8 points higher, and you can see exactly why from the contributions. The two components the student did best on, the midterm and the final, carry 30% and 35% of the weight between them - 65% of the total - while the weakest score of 79 carries only 15%.
Now check that the scale of the weights really does not matter. Divide every weight by 100 to get 0.20, 0.30, 0.15 and 0.35. The products become 17.6, 27.6, 11.85 and 33.25, summing to 90.3, and the weights sum to 1.00, so the answer is 90.3 / 1.00 = 90.3. Identical, as it must be.
Those four products in the second version are the contributions shown in this page's table: 17.6 + 27.6 + 11.85 + 33.25 = 90.3. Each one is the number of grade points that component put on the board.
Reading the gap between weighted and simple averages
The calculator reports the weighted mean, the simple mean, and the difference between them. That difference is the entire effect of the weighting, and it is guaranteed to be zero in two cases: when all the weights are equal, and when all the values are equal. It can also come out at zero by coincidence - values 1, 2 and 3 with weights 3, 0 and 3 give (3 + 0 + 9) / 6 = 2, exactly the simple mean of 2 - so a zero gap is not by itself evidence that the weights were uniform.
The sign of the gap tells you where the weight sits. If the weighted mean exceeds the simple mean, the larger values are carrying more than their equal share of the weight. If it falls below, the weight is concentrated on the smaller values. Neither is good or bad in itself - it is a description of the data, not a verdict on it.
The size of the gap is worth attention when it is large relative to the spread of the values. If your values range over ten points and the weighting moves the average by four, then the choice of weights is doing more work than the measurements are, and anyone reading the number deserves to see the weights. The largest-single-weight figure on this page is a quick check on that: when one item holds more than half the total weight, the average is largely reporting that one item.
Weighted averages are also the correct way to combine group means. If three stores average 4.2, 3.8 and 4.6 on a satisfaction survey with 500, 90 and 40 responses, the company average is not (4.2 + 3.8 + 4.6)/3 = 4.20. It is (4.2 x 500 + 3.8 x 90 + 4.6 x 40) / 630 = (2100 + 342 + 184) / 630 = 2626 / 630 = 4.168. The simple average overstates the result because it gives the two small stores a third of the influence each despite holding a fifth of the responses between them.
Common weighting schemes and what the weight is
| Setting | Value | Weight | Result |
|---|---|---|---|
| Course grade | Component score | Percentage of the final grade | Final percentage |
| Grade point average | Grade points for each course | Credit hours | GPA |
| Combining group means | Each group's mean | Group sample size | Overall mean |
| Inventory costing | Unit cost of each purchase lot | Units in the lot | Weighted average unit cost |
| Portfolio return | Return of each holding | Market value of the holding | Portfolio return |
| Cost of capital | Cost of each funding source | Market value of that source | Weighted average cost of capital |
| Survey with unequal sampling | Respondent answer | Inverse of the selection probability | Population estimate |
The last row is the one people forget. When some groups are deliberately over-sampled, the raw average describes the sample rather than the population until the design weights are applied.
Mistakes that produce the wrong weighted average
- Averaging averages without weighting. The plain mean of several group means equals the overall mean only when the groups are the same size. This is the single most frequent error in reporting.
- Mismatched list lengths. A missing weight silently shifts every later pairing, so the third value gets the fourth weight. The calculator counts both lists and tells you when they disagree.
- Weights on inconsistent scales. Mixing percentages with counts - 35 meaning 35% next to 400 meaning 400 respondents - gives the count-scaled item almost all the influence. Check that every weight measures the same thing.
- Weighting a value that is itself already weighted. Applying credit hours to a course grade that was already computed as a weighted average of its own components is correct; applying them twice is not. Be clear about which level you are at.
- Averaging rates that share a denominator. To combine speeds over equal distances, or costs per unit across lots, the natural weight is often the denominator quantity, not the count of items. Averaging 30 mph and 60 mph over equal distances gives 40 mph, not 45.
- Treating a zero weight as an exclusion that changes n. A zero weight removes an item's influence, which is right, but it still appears in the item count. It cannot affect the answer, so do not go looking for its contribution.
Related averages and when to use them
The weighted mean is one member of a family. The plain arithmetic mean is the equal-weight case. The geometric mean, the n-th root of the product, is the right average for multiplicative quantities such as growth rates and index numbers; a weighted geometric mean raises each value to the power of its normalised weight and multiplies. The harmonic mean is the right average for rates when the fixed quantity sits in the denominator.
Weighting also carries into measures of spread. A weighted variance uses the same normalised weights against the squared deviations from the weighted mean, and the choice of denominator for a weighted sample variance depends on whether the weights are frequencies or reliabilities - a genuine subtlety that catches out spreadsheets as often as students. For unweighted data the variance calculator and standard deviation calculator show the standard forms.
When weights are integer counts, a weighted mean is exactly the mean of the expanded data set, and the mean, median and mode calculator accepts frequencies directly for that case. That equivalence gives you a useful check: if your weights are whole-number counts, writing the data out in full and taking a simple average must reproduce the weighted mean to the last decimal.
Finally, remember that weighting changes the question rather than improving the answer. A weighted average describes a specific population under a specific set of weights. Change the weights and you have described a different population, so publish the weights alongside the result.