Add the same number to every value

The idea

If a fixed number is added to every value in a collection, the mean increases by that same number, so its relative position stays the same.

Ten bonus marks for everyone

Five students scored 4, 5, 6, 9 and 11. The mean is 7. The teacher adds 10 bonus marks to every score.

What will happen to the mean?

• It stays 7
• It goes up by 2
• It goes up by 10

Slide every value

Use the slider to add the same number to every value. All the weights slide together. Find the number that puts the mean (the pivot) where you are asked. Do all three.

Mean + the same number

If a fixed number is added to every value in a collection, the mean increases by that same number, so its relative position stays the same.

Add 10 to 4, 5, 6, 9, 11: they become 14, 15, 16, 19, 21. (14 + 15 + 16 + 19 + 21) ÷ 5 = 85 ÷ 5 = 17 The mean went from 7 to 17: 7 + 10. The book shows the same with 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5, whose mean also goes up by 10.

Why? Suppose there are n values with mean a. Add 3 to every value. The total grows by 3 added n times, that is 3n: new mean = (old total + 3n) ÷ n = old total ÷ n + 3n ÷ n = a + 3

The pivot also keeps its place among the weights: the distances on the two sides stay the same.

Notes

If a fixed number is added to every value in a collection, the mean increases by that same number, so its relative position stays the same.

Check yourself

The mean of some marks is 52. Every mark is increased by 5. What is the new mean?

Answer: 57

Adding 5 to every value adds 5 to the mean: 52 + 5 = 57.

The mean of a set of values is 24. The same number is added to every value and the new mean is 31. What number was added?

Answer: 7

The mean went from 24 to 31, so the number added to every value is 31 − 24 = 7.

A teacher adds 3 marks to every student's score. Which statement is true?

The values 4, 5, 6, 9, 11 have total 35. If 4 is added to every value, what is the total of the new values?

Answer: 55

The total grows by 4 for each of the 5 values: 35 + 5 × 4 = 55. (Check: the new mean is 7 + 4 = 11 and 11 × 5 = 55.)

• The mean increases by 3 × (the number of students).. The total goes up by 3 for each student, but the mean is the total shared by the students: it goes up by only 3.
• The mean increases by 3, and the pivot keeps its place among the weights. — correct. Yes! Every weight slides 3 to the right and so does the pivot. The distances on the two sides are unchanged.
• The mean stays the same because only the total changes.. The mean is the total divided by the number of values, so it changes when the total changes.
• Only the biggest score changes the mean.. Every score changes by 3, so the whole beam slides by 3 and so does the mean.