Multiply every value by the same number
If every value is multiplied by a number, the mean is multiplied by that number too.
Everything doubles
The values 2, 3 and 7 have mean 4. Now every value is doubled.
What will the mean become?
What this lesson covers
The idea
If every value in a collection is multiplied by a fixed number, the mean is multiplied by the same number, as the distributive property shows.
Everything doubles
The values 2, 3 and 7 have mean 4. Now every value is doubled.
What will the mean become?
- It stays 4
- It goes up by 2, to 6
- It doubles, to 8
Multiply every value
Pick a number to multiply every value by. The weights spread out from 0, and the pivot moves too. Find the number that puts the mean where you are asked. Do all three.
Mean × the same number
If every value in a collection is multiplied by a fixed number, the mean is multiplied by the same number, as the distributive property shows.
Multiply 2, 3, 7 by 5: they become 10, 15, 35. (10 + 15 + 35) ÷ 3 = 60 ÷ 3 = 20 The mean went from 4 to 20: 4 × 5.
Why? Suppose there are n values with mean a. Multiply every value by 5. The distributive property pulls the 5 out of the total: 5 × 2 + 5 × 3 + 5 × 7 = 5 × (2 + 3 + 7) new mean = 5 × (old total) ÷ n = (old total ÷ n) × 5 = 5a
The distances on the two sides are multiplied too: 3 | 3 became 15 | 15. The two sides still balance.
Notes
If every value in a collection is multiplied by a fixed number, the mean is multiplied by the same number, as the distributive property shows.
Check yourself
The mean of 12 numbers is 6.5. Every number is multiplied by 4. What is the new mean?
Answer: 26
Every value is × 4, so the mean is × 4: 6.5 × 4 = 26.
The mean height of a class is 150 cm. Every height is written again in metres: every value is divided by 100. What is the new mean?
The mean of some values is 10. Every value is doubled and then 3 is added to it. What is the new mean?
Answer: 23
Doubling makes the mean 10 × 2 = 20. Adding 3 to every value adds 3 to the mean: 20 + 3 = 23.
Every value in a collection is multiplied by 5 and the new mean is 35. What was the old mean?
Answer: 7
The old mean × 5 = 35, so the old mean is 35 ÷ 5 = 7.
- 1.5 m — correct. Yes! Dividing every value by 100 divides the mean by 100: 150 ÷ 100 = 1.5.
- 150 m. The length is the same, but the number is not: 150 cm is written 1.5 m. Every value is divided by 100, so the mean is too.
- 50 m. Dividing by 100 is not subtracting 100. Every value is divided by 100, so the mean is divided by 100.
- 15,000 m. That multiplies by 100. A metre is bigger than a centimetre, so the numbers get smaller.