Weighted Average Calculator

Weighted average = Σ wᵢxᵢ ÷ Σ wᵢ

xᵢ

One value in the list — a score, a price, a measurement. Every value still has to be given, including the ones whose weight turns out to be zero; a value left out is not the same as a value weighted at nothing

wᵢ

The weight of that value, written in the same position in the second box. It may be a percentage, a count of credits, a number of people or any other share — only the ratio between the weights matters, and the page does not require them to be whole numbers or to add up to 100

Σ wᵢxᵢ

The weighted total: every value multiplied by its own weight, then added. This is the numerator, and it is what makes the answer move towards the entries with the larger weights

Σ wᵢ

The total weight, printed on the results panel. Dividing by it is what turns the weighted total back into an average — and it is also the check: if this number is not what you expected, a weight was mistyped, and the answer is wrong in a way no other line will reveal

Use a weighted average whenever the entries differ in how much they should count, and use a plain average when they do not. Coursework is the everyday case: a final exam worth 30% and a quiz worth 10% are not two marks to be averaged as equals, and the same arithmetic covers a grade point average, a portfolio's average purchase price, and a customer satisfaction score where each branch has a different number of replies. It is also the right tool for a number that is itself an average: if branch A reports an average of 40 from 10 sales and branch B reports 90 from 2, averaging 40 and 90 gives 65, which no customer paid anything like, while weighting by the counts gives the figure the whole company actually earned. Do not use it to give a value more influence because it is more reliable or more important in some other sense — the weight is a share of the total, and the answer is exactly what a plain average would give if each value were repeated as many times as its weight.

1. Five test scores with percentage weights

   1. Multiply each score by its weight: 85 × 20 = 1700, 90 × 20 = 1800, 78 × 10 = 780, 92 × 20 = 1840, 88 × 30 = 2640
   2. Add the products: 1700 + 1800 + 780 + 1840 + 2640 = 8760
   3. Add the weights: 20 + 20 + 10 + 20 + 30 = 100
   4. Divide: 8760 ÷ 100 = 87.6

   The plain average of these five scores is 86.6, so the weighting is worth 1.0 mark here. It comes from where the weight sits: 78 is the lowest score and carries only 10, while 88 carries 30, the largest share. The weighted average is therefore dragged up — not because the weighting is unfair, but because this is what the weighting was asked for. Reading the same numbers the other way rounds out the picture: had the 30 sat on 78 instead of 88, the answer would be 85.6, one mark below the plain average. A weighted average is only as meaningful as the weights, which is why the total weight is printed next to it.

2. A grade point average across five courses

   1. Multiply each grade point by that course's credits: 4 × 3 = 12, 3.7 × 4 = 14.8, 3.3 × 3 = 9.9, 3 × 2 = 6, 4 × 3 = 12
   2. Add the products: 12 + 14.8 + 9.9 + 6 + 12 = 54.7
   3. Add the credits: 3 + 4 + 3 + 2 + 3 = 15
   4. Divide: 54.7 ÷ 15 = 3.6467

   This is the calculation behind a grade point average, and it is the reason the page prints the total weight: 15 credits is the denominator of the fraction, and a GPA that came from 15 credits is not the same claim as one that came from 60. The plain average of the five grade points is 3.6, so the credits move the answer by 0.0467 — a smaller shift than the previous example, because the credits range only from 2 to 4 while the weights there ranged from 10 to 30. The unweighted figure is worth knowing too: if every course were worth the same, this student's record would read 3.6 rather than 3.65.

3. Weights that do not add up to 100

   1. Multiply each value by its weight: 10 × 1 = 10, 20 × 2 = 40, 30 × 3 = 90
   2. Add the products: 10 + 40 + 90 = 140
   3. Add the weights: 1 + 2 + 3 = 6
   4. Divide: 140 ÷ 6 = 23.3333

   These weights sum to 6, and that is perfectly acceptable — the answer would be identical if they were written 10, 20 and 30 or 0.1, 0.2 and 0.3. Only the ratio matters, so a set of weights is best read as a recipe rather than as percentages: whatever the weights add up to, the values are being mixed in those proportions. The one thing that does change is the total weight printed beside the answer, which follows the weights you typed. Note also that the answer sits much closer to the largest value, 30, than the plain average of 20 would: the heaviest weight is on the largest entry, and the weighted average is a balance point, so it slides towards whatever carries the most weight.

The weights have to be zero or positive, and at least one of them has to be greater than zero: a list whose weights are all zero has no weighted average, and the page refuses it rather than dividing by nothing. A single weight of zero is allowed and means that entry is ignored — it still counts in the count and still appears among the values, but it contributes nothing to the answer, which is worth knowing before reading a zero as merely a small weight. Nothing is normalised on your behalf, so weights that add up to 150 are fine and so are weights that add up to 0.4; only the ratio between them affects the result. The two boxes are positional, so the weights must be written in the same order as the values and there must be the same number of them — the page refuses a mismatch rather than guessing which weight belongs to which value, and that refusal is the guard against the worst failure mode here, which is a correct-looking answer built from misaligned pairs. It is not a weighted average of a frequency table, where a value stands for many readings; for that, enter the value once with the count as its weight and the answer will be the same, but a table copied in raw will not work. The lists are capped at 200 pairs, and a token like 1,500 is refused rather than guessed at, because a comma between digits is a decimal comma in much of the world; write 1500 or 1.5.

How do I calculate a weighted average?

Multiply each value by its weight, add those products up, then divide by the sum of the weights. For the five test scores on this page the products are 1700, 1800, 780, 1840 and 2640, which add to 8760, and the weights add to 100, so the weighted average is 8760 ÷ 100 = 87.6. Both operands of that division are printed on the results panel, so the calculation can be checked without redoing the multiplication by hand.

What is the difference between a weighted average and a weighted mean?

They are the same number and the two names are used interchangeably. Weighted mean is the more formal of the two and turns up in statistics texts, while weighted average is what most people search for. The idea behind both is the same: each value is given a weight, and the answer is a balance point that sits closer to the values carrying more weight. A plain average is what this turns into when every weight is equal.

Do the weights have to add up to 100?

No, and nothing on this page divides by 100. The weights are a share, so only the ratio between them matters: 1, 2 and 3 give the same answer as 10, 20 and 30, and the same again as percentages. Groups that do not add up to 100 are common once the weights are not percentages at all — course credits, numbers of responses, shares of a portfolio. The one number that does follow what you typed is the total weight, which the page prints beside the answer.

What happens if one weight is zero?

That value is left out of the average entirely, as if it were never in the list — but it still counts in the count and it still has to be entered in both boxes, because the two boxes are matched position by position. A zero is not a small weight; it is no weight at all, and the two are different claims. What the page will not accept is every weight being zero: then the total weight is zero and the division has no answer, so the entry is refused instead of producing an infinity.

Can I use it for test scores or a grade point average?

Yes, and both are the standard uses. Weighted test scores are the same arithmetic as the first example: put the marks in the first box and the weight each assessment carries in the second. A grade point average works the same way with the grade points as the values and the credits as the weights, which is the second example — the answer is the number a registrar would compute, and the total weight printed beside it is the credit total.

Why is the total weight shown if the answer does not depend on it?

Because it is the denominator you would use to check the division, and because it is where a typing mistake shows up first. If the total weight is 97 rather than 100 when you meant percentages, a weight was mistyped — and the weighted average that comes out of it is wrong in a way that no other line on the panel reveals, since the values and the count both look right. If the weights are a count of something real, credits or responses or people, the total is a number worth having on its own.

• Weighted Mean — MathWorld — Wolfram MathWorld
• Measures of Scale — e-Handbook of Statistical Methods, section 1.3.5.6 (why the mean is a balance point, and how weights move it) — National Institute of Standards and Technology (NIST)