Weighted Average Calculator

Understanding the weighted average calculator — concept and principle

A weighted average is a mean in which some values count more than others. Each value is multiplied by a weight that expresses its importance, size or share, and the total is divided by the sum of the weights. It is used for course grades with different assessment shares, grade point averages with credit hours, portfolio returns, average prices of goods bought in different quantities and survey results adjusted by group size.

How it is calculated

Weighted average = Σ(xᵢ × wᵢ) ÷ Σwᵢ

• xᵢ are the values (scores, prices, rates). Their unit carries over to the result.
• wᵢ are the weights. They can be percentages, credits, quantities or any non-negative numbers; only their relative size matters.
• The calculator also shows the total weight Σwᵢ. Weights do not need to add up to 100 or 1, because the division by Σwᵢ rescales them automatically.

The principle

Think of the weights as the number of copies of each value. A value with weight 50 behaves as if it appeared 50 times in an ordinary list, so the weighted average is just the ordinary mean of that expanded list. Dividing by the total weight turns the weights into shares that add up to 1, so the result always lies between the smallest and largest value. If all weights are equal, the formula reduces to the ordinary arithmetic mean.

Worked example

1. Inputs: values 85, 90, 78 with weights 30, 50, 20 (for example, midterm 30%, final 50%, homework 20%).
2. Products: 85 × 30 = 2,550; 90 × 50 = 4,500; 78 × 20 = 1,560.
3. Sum of products = 2,550 + 4,500 + 1,560 = 8,610.
4. Total weight = 30 + 50 + 20 = 100.
5. Weighted average = 8,610 ÷ 100 = 86.1.

How to read the result

The result moves toward the values with the largest weights. The plain mean of 85, 90 and 78 is about 84.33, but because the 90 carries half of the weight, the weighted average rises to 86.1. To see how much a single item can change the outcome, multiply its weight share by the possible change: raising the homework score by 10 points adds 10 × 20 ÷ 100 = 2 points.

Common mistakes

• Entering weights in a different order from the values, so each weight attaches to the wrong item.
• Dividing by the number of items instead of the sum of the weights.
• Averaging group averages without weighting by group size.
• Using negative weights, which have no meaning as shares.

Frequently asked questions

Do the weights need to add up to 100%?

No. The formula divides by the total weight, so weights of 3, 5 and 2 give the same result as 30%, 50% and 20%.

How is a GPA a weighted average?

Each course grade point is weighted by its credit hours, then the total is divided by the total credits.

What score do I need on the last item to reach a target?

Solve target × Σw = Σ(known xᵢwᵢ) + x × w for x, where w is the weight of the remaining item.

Sources: G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press (weighted means) · D. Freedman, R. Pisani and R. Purves, Statistics, W. W. Norton