Weighted Mean
A weighted mean is an average in which some values count for more than others. It is the honest average whenever the items are not equally important, which in school usually means coursework and exams.
The formula
x̄w = Σ(wx) ÷ Σw
Multiply each value by its weight, add the products, then divide by the total weight. With equal weights the formula collapses back into the ordinary mean, so it is a generalisation rather than a separate idea.
Worked example: grades
A student scores 85 on a piece of coursework worth 3 credits and 70 on a test worth 2 credits.
- Multiply: 85 × 3 = 255 and 70 × 2 = 140.
- Add the products: 255 + 140 = 395.
- Add the weights: 3 + 2 = 5.
- Divide: 395 ÷ 5 = 79.
The weighted mean is 79. The plain average of the two scores is 77.5, so weighting raised the result from 77.5 to 79, because the stronger score carried more weight. That difference is exactly the point of weighting.
Worked example: a survey
A survey asks how many hours a week students study, with 10 responses of 10 hours, 20 responses of 5 hours and 5 responses of 2 hours.
- Products: 10 × 10 = 100, 5 × 20 = 100 and 2 × 5 = 10, which total 210.
- Total weight: 10 + 20 + 5 = 35 responses.
- Weighted mean: 210 ÷ 35 = 6 hours.
When to use a weighted mean
- Grades, where components carry different credit.
- Survey results, where a response stands for a different number of people.
- Prices, where the same change matters more for a large purchase than a small one.
- Frequency tables, which are a weighted mean in disguise: the frequencies are the weights.
Common mistakes
- Adding the values without weighting them, which ignores why the weights were given.
- Dividing by the number of values instead of the total weight.
- Confusing weights with the values themselves, so 85 × 3 becomes 85 × 85.