Whole class project: Each student shares the average age of their family and the number of family members. Discuss among the class and come up with a way to find out the average age of all the families of the class.
Answer: Multiply each family’s average age by its number of members to get the family’s total age; add all the totals and divide by the total number of members: average age = a1n1 + a2n2 + …n1 + n2 + … (a weighted mean). Simply averaging the family averages would be wrong unless all families are the same size.
Step-by-step solution
Idea: This is an average of averages for collections of different sizes, so each family’s average must be weighted by the number of its members.
- Each student gives a (average age) and n (members). Family’s total age = a × n. Write these on the board.1 mark
- Average age of all family members = sum of all a × nsum of all n. Example with three families: (30 years, 4 members), (25, 5), (40, 3): 120 + 125 + 12012 = 36512 ≈ 30.4 years.1 mark
- Note: the plain mean of the averages, 30 + 25 + 403 ≈ 31.7 years, is different and wrong for ‘all family members’, because families have different sizes. (It would answer a different question: the average of the ‘typical family ages’, each family counted once.) Also make sure two students from the same family do not count it twice.1 mark
Find each family’s total age (average × number of members), add these totals for all families, and divide by the total number of members. This weighted mean gives the average age of all the families’ members.
Answer to write in the exam
Total age of a family = a × n
Average age = a1n1 + a2n2 + … + aknkn1 + n2 + … + nk
e.g. 30 × 4 + 25 × 5 + 40 × 34 + 5 + 3 = 36512 ≈ 30.4 years
∴ Weighted mean of family averages, weights = family sizes
Common mistakes that cost marks
- Adding the family averages and dividing by the number of families.
- Dividing the total age by the number of students instead of the number of family members.
- Counting a family twice when siblings are in the same class.
How this can come in the exam
MCQ (1 mark)
Family X (3 members) has an average age of 20 years and Family Y (5 members) has an average age of 36 years. The average age of all 8 people is
- 28 years
- 30 years
- 32 years
- 26 years
Show answer
(B) 30 years
60 + 1808 = 2408 = 30 years.
Try one yourself
Two families: 4 members averaging 22 years and 6 members averaging 32 years. Find the average age of all 10 people.
Show answer
88 + 19210 = 28 years.
More questions like this
- Given some data with corresponding weights, what would happen to the weighted average if all the weights are increased by a constant value, say 1? If required, experiment with some data. What do you observe? Justify your answer using algebra.
- A farm has some cows, sheep, and chickens. Last year the cows made up 60%, the sheep 25%, and the chickens 15%. There was a decrease in the number of all three animals’ population over the year. Choose the possibilities for the change in their respective shares of the population —
(i) % of cows decreased, % of sheep decreased, % of chickens decreased
(ii) % of cows increased, % of sheep increased, % of chickens increased
(iii) % of cows remained the same, % of sheep remained the same, % of chickens remained the same
(iv) % of cows decreased, % of sheep increased, % of chickens remained the same
(v) % of cows increased, % of sheep increased, % of chickens decreased. - In a badminton academy, there is a group of 11 trainees — 8 seniors and 3 juniors. Their heights and average heights (in cm) are given in the table below. Find the average height of the whole group.
Two students calculate the average height of the whole group in two ways:
Method 1: 165.5 + 149.332 = 314.832 = 157.415
Method 2: 165 + 169 + 164 + 167 + 170 + 159 + 164 + 166 + 146 + 149 + 15311 = 177211 = 161.09
Whose calculation gives the correct average height of the whole group? - Why did Method 1 not work?
- Shreyas calculated it differently as (165.5 × 8) + (149.33 × 3)8 + 3 = 1324 + 44811 = 177211 = 161.09. Do you understand why this also works? Can you see why 165.5 × 8 gives the sum of the heights of all the seniors and 149.33 × 3 gives the sum of the heights of all the juniors?