Level 3 · Ages 11–13
Read it again any time, including while you work through the exercises.
You know the simple mean: add the values and divide by how many there are. But in real life, values do not always matter equally. A test with coefficient 3 weighs three times as much as a small quiz with coefficient 1. To take that into account, we use the weighted mean.
The idea is simple: each value comes with a 'weight' (also called a coefficient or a frequency). A grade with coefficient 3 is as if you had received that grade three times. So we multiply each value by its weight, add everything up, then divide by the sum of the weights (and definitely not by the number of values!).
As a formula: weighted mean = (sum of value × weight) ÷ (sum of weights). For example, with a 12 at coefficient 2 and a 15 at coefficient 3: (12 × 2 + 15 × 3) ÷ (2 + 3) = (24 + 45) ÷ 5 = 69 ÷ 5 = 13.8.
This mean is useful everywhere: computing a mean from a frequency table (5 students got 10, 3 students got 14...), an average price in a shopping basket, or a term average with coefficients. Whenever a table gives you values AND their frequencies, think weighted mean.
A good reflex to check your work: the weighted mean always lies between the smallest and the largest value, and it leans toward the values with the biggest weights.
Zoe got 12 (coefficient 2) and 15 (coefficient 3). Compute her weighted mean.
12 × 2 = 24 ; 15 × 3 = 45
I multiply each grade by its coefficient: the 15 counts as three grades of 15.
24 + 45 = 69
I add these products to get the weighted total.
2 + 3 = 5
I add the coefficients: that is the number to divide by, not 2!
69 ÷ 5 = 13.8
I divide the weighted total by the sum of the coefficients. 13.8 is indeed between 12 and 15, closer to 15 which weighs more.
Result: Zoe's weighted mean is 13.8.
In a group, 4 people scored 10 points and 6 people scored 15 points. Compute the group's mean score.
4 × 10 = 40 ; 6 × 15 = 90
The frequencies act as weights: I compute each group's total points.
40 + 90 = 130
The whole group scored 130 points in total.
130 ÷ 10 = 13
I divide by the total number of people: 4 + 6 = 10. The mean score is 13, between 10 and 15 and closer to 15.
Result: The group's mean score is 13 points.