# A Mean Calculation

What is the mean of this set of numbers?

## Problem

Can you work out the mean of this set of numbers without using a calculator?

97 97 98 99 99 99 100 100 101 101 103 104 104 105

## Student Solutions

**Subtracting 100 from each of the numbers**

The numbers are all around 100, so the mean will be close to 100.

If the numbers were all 100 smaller, then their mean would be 100 smaller. So finding the mean of the same set of numbers, but all 100 smaller, and then adding 100, will give the mean.

Subtracting 100 from each of the numbers gives

-3 -3 -2 -1 -1 -1 0 0 1 1 3 4 4 5

These numbers add up to 7, and there are 14 of them, so their mean is 7$\div$14 = 0.5.

So the mean of the original numbers is 100.5.

**Converting groups/pairs of numbers into 100s**

Notice that 97 + 103 = 200 = 100 + 100, so finding the mean of 97, 103 and some other numbers will be the same as finding the mean of 100, 100 and the other numbers.

So finding the mean of

97 97 98 99 99 99 100 100 101 101 103 104 104 105

is the same as finding the mean of

100 100

Pairing off the 99s with the 101s, it is the same as finding the mean of

100 100 100 100 100 100

97 and 98 are 3 and 2 less than 100, so will balance with 105 to leave three 100s. So we can find the mean of

100 100 100 100 100 100 100 100 100

99 and 104 do not balance, but can be replaced with 100 and 103, since 99 + 104 = 100 + 103.

So we can find the mean of

100 100 100 100 100 100 100 100 100 100 103

which is 1407$\div$14.

Notice that 1400$\div$14 = 100, and 7$\div$14 = 0.5,

so 1407$\div$14 = 100.5