We saw earlier how Method 1 (badminton example) did not produce the correct value. Why does Method 1 give the correct answer in this case? When does Method 1 work and when does it not work?
Step-by-step solution
Idea: The correct combined average of collections with averages a, b, c and sizes p, q, r is ap + bq + crp + q + r. See what happens when p = q = r.
- Correct combined average of three collections = (sum in 1) + (sum in 2) + (sum in 3)(size 1) + (size 2) + (size 3) = ap + bq + crp + q + r.½ mark
- In the cycling data every week has 5 days, so p = q = r. Then ap + bp + cpp + p + p = (a + b + c)p3p = a + b + c3, which is exactly Method 1.1 mark
- So Method 1 is correct here: each week’s average comes from the same number of days, so treating the weeks equally is fair.½ mark
- In the badminton example the sizes were 8 and 3, not equal. Then 165.5 × 8 + 149.33 × 311 = 161.09 is not 165.5 + 149.332 = 157.415, so Method 1 fails.½ mark
- Rule: Method 1 works when all the collections have the same size. When sizes differ, weight each average by its size. (It also happens to give the right answer if all the averages are equal, whatever the sizes.)½ mark
Answer to write in the exam
Combined average = ap + bq + crp + q + r
Cycling: p = q = r = 5 days
ap + bp + cp3p = (a + b + c)p3p = a + b + c3 = Method 1
Badminton: sizes 8 and 3 are unequal, so 165.5 + 149.332 ≠ 165.5 × 8 + 149.33 × 311
∴ Method 1 works only when the collections are of equal size
Common mistakes that cost marks
- Saying Method 1 worked ‘by luck’. It is exact whenever the group sizes are equal; the algebra shows why.
- Cancelling wrongly: (a + b + c)p3p = a + b + c3, not a + b + c.
- Thinking the number of collections matters. What matters is whether the collections have equal sizes.
How this can come in the exam
Assertion (A): Three sections have 35 students each. The average of the three section averages equals the average of all 105 students.
Reason (R): When collections are of equal size, the combined average is the simple mean of their averages.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
With equal sizes p: ap + bp + cp3p = a + b + c3. R is true and explains A.
Two batches have averages 70 and 80. Find the combined average if (a) both batches have 10 students, (b) the batches have 10 and 30 students.
Show answer
(a) Equal sizes: 70 + 802 = 75 (1 mark). (b) 10 × 70 + 30 × 8040 = 700 + 240040 = 77.5 (1 mark).Try one yourself
Four relay runners each ran 3 laps. Their average lap times were 62, 65, 60 and 69 seconds. Can you find the average lap time of the team by averaging these four numbers? Find it.
Show answer
Yes, because each runner ran the same number of laps (3). Average = 62 + 65 + 60 + 694 = 2564 = 64 seconds.
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