A weighted mean is an average in which each value counts according to its importance or its group size. It appears when marks carry different percentages, when prices are mixed by quantity, and when two group means must be combined. It builds directly on choosing an average.
Why is a simple average sometimes wrong?
A simple mean treats every value as equally important. A weighted question tells you they are not: a final exam may be worth more than a quiz, and a class of 30 should count for more than a class of 10.
The idea behind both cases is the same. Turn each value into its contribution to the total, then divide by how much the total represents.
How to calculate a weighted mean, step by step
- List each value with its weight (percentage, count, quantity).
- Multiply each value by its weight.
- Add all the products.
- Divide by the sum of the weights.
- Check that the answer lies between the smallest and largest value.
For combining group means, the “value” is the group mean and the “weight” is the group size. Multiplying them simply recovers the group total.
Worked example
A student’s final mark has three parts: coursework 80 (weight 20%), a test 65 (weight 30%) and an exam 72 (weight 50%). Find the overall mark.
Step 1, products: 80 × 0.20 = 16, 65 × 0.30 = 19.5 and 72 × 0.50 = 36.
Step 2, add: 16 + 19.5 + 36 = 71.5.
Step 3, divide by the weights: the weights are 0.20 + 0.30 + 0.50 = 1, so the answer stays 71.5.
Check: 71.5 lies between 65 and 80, and it is closer to 72 than to 80 because the exam carries the most weight.
A second case: combining two classes
Class A has 20 students with a mean of 60. Class B has 30 students with a mean of 70. Find the mean of all 50 students.
Totals: 20 × 60 = 1200 and 30 × 70 = 2100. Together that is 3300 marks for 50 students, and 3300 ÷ 50 = 66.
The mistake to watch for
A common slip is to average the averages.
Mistaken answer: (60 + 70) ÷ 2 = 65.
This treats both classes as the same size. Class B is larger, so the combined mean should sit closer to 70 than to 60.
The correction is to ask “how many are behind each mean?”. The answer of 66 is closer to 70 than 65 is, which matches the larger class. Whenever your combined mean is exactly halfway between two group means, check that the groups really are equal in size.
Check yourself
Try these, then open each answer.
1. A module has a quiz marked 50 (weight 1), a project marked 70 (weight 2) and an exam marked 90 (weight 3). Find the weighted mean to 1 decimal place.
Show answer
Products: 50 × 1 = 50, 70 × 2 = 140 and 90 × 3 = 270. Sum = 460. Sum of weights = 1 + 2 + 3 = 6.
460 ÷ 6 = 76.666…, so 76.7 to 1 decimal place.
2. 15 students have a mean mark of 40 and 5 students have a mean mark of 60. Find the mean of all 20.
Show answer
Totals: 15 × 40 = 600 and 5 × 60 = 300. Together 900 for 20 students.
900 ÷ 20 = 45. It is nearer 40 than 60 because the first group is three times larger.
3. A shop mixes 3 kg of nuts at RM12 per kg with 2 kg of nuts at RM20 per kg. What is the mean price per kg of the mixture?
Show answer
Cost: 3 × 12 = 36 and 2 × 20 = 40, total RM76 for 5 kg.
76 ÷ 5 = RM15.20 per kg.
Where this leads next
Next, see how averages are estimated when the raw values are hidden, in interpreting grouped data. The same idea of “value times amount” is what makes grouped means work.
The statistics and distribution explorer shows the contributions visually, and the non-calculator working trainer helps with the multiplication and division. Go back to the module overview to see the full route.
Students who can follow a worked weighted mean often still stall on a fresh context. A teacher watching you set it up is how the habit of starting from totals becomes automatic in online one-to-one Mathematics tuition.