Your final grade: homework 20%, midterm 30%, final 50%. A plain average of your three scores is wrong — the final counts more. The weighted mean is how the real world actually averages.
A weighted mean lets some values count more than others. Multiply each value by its weight, sum those, then divide by the total weight.
Course grades, GPA, stock indices, opinion polls (adjusting for demographics), portfolio returns, physics centre-of-mass — whenever data points have unequal importance.
Scores 80, 90, 100 with weights 1, 2, 3. What's the weighted mean?
If all weights are equal, this collapses to the ordinary mean. Bigger weight = more pull on the result.
Homework 70 (weight 20%), midterm 80 (30%), final 90 (50%). Final grade?
Why isn't GPA just the average of your course grades?
Courses have different credit hours — a 4-credit course should count more than a 1-credit one. GPA is a weighted mean with credit hours as the weights.
Forgetting to divide by the total weight. Σ(wᵢxᵢ) alone isn't the answer — you must divide by Σwᵢ. (If the weights sum to 1, that division is trivial — but you still have to remember it.)
Sanity check: a weighted mean must lie between the smallest and largest values. If yours doesn't, you forgot to divide by the total weight.
- Weighted mean = Σ(weight × value) ÷ Σ(weights).
- Equal weights → ordinary mean.
- Used for grades, GPA, indices — anywhere data points matter unequally.