The formula
weighted average = Σ(value × weight) ÷ Σ(weight)
Multiply each value by its weight, add those products, then divide by the total of the weights. A plain average is the same formula with every weight set to 1.
A worked example
A course scored out of 100, with four pieces of work:
- Homework 92, weight 10%
- Quizzes 85, weight 20%
- Midterm 78, weight 30%
- Final 90, weight 40%
Products: 9.2 + 17 + 23.4 + 36 = 85.6. Weights add to 100, so the average is 85.6.
The plain average of those four scores is 86.25. The weighted figure is lower because the weakest score carries the most weight.
The weights do not have to add to 100
Dividing by the total of the weights handles any scale. Credits of 3, 4, 3 and 5 work exactly the same way as percentages of 20, 27, 20 and 33: only the ratios matter. If your weights happen to add to 100 the division is by 100, which is why the step is easy to forget.
GPA is a weighted average
Grade points weighted by credit hours:
- Biology, 4.0, 4 credits → 16
- Calculus, 3.0, 3 credits → 9
- History, 3.7, 3 credits → 11.1
- Writing, 3.3, 2 credits → 6.6
Total 42.7 over 12 credits = 3.56 GPA. The plain average of the four grades is 3.5, and the gap grows as the credit values spread out.
Two mistakes to watch for
Averaging percentages of different-sized groups. If one store converts 10% of 1,000 visitors and another converts 30% of 100, the combined rate is 130 ÷ 1,100 = 11.8%, not the 20% you get by averaging the two rates. Weight by the group size.
Averaging rates that should be harmonic. Driving 60 km at 30 km/h and 60 km at 60 km/h does not average 45 km/h. It takes 2 h + 1 h for 120 km, so the average speed is 40 km/h. Anything "per unit" needs the totals, not the mean of the rates.
Where else it turns up
A portfolio's return is its holdings weighted by value. A survey result is weighted by how far each group is over- or under-represented. A blended interest rate is each loan's rate weighted by its balance. In every case the weight is the size of the thing behind the number.
Tools
The Weighted Average Calculator takes any list of values and weights, in percentages or credits. The GPA Calculator does the credit-hour version above, and the Standard Deviation Calculator shows how spread out the values behind the average are.