The balancing act

You already know two ways to find the centre of some data. The mean is the sum of all the values divided by how many values there are. The median is the middle value once the data is sorted.

Start with two numbers. The mean of 3 and 7 is (3 + 7)/2 = 5. The mean of 8 and 9 is 8.5. On a dot plot, the mean sits exactly halfway between them.

With two numbers, 'halfway' and 'the mean' are the same thing. With three or more numbers they are usually different, and this is where many people go wrong.

Take 10, 10, 11 and 17. Halfway between the smallest and the largest (10 and 17) is 13.5. But the mean is (10 + 10 + 11 + 17)/4 = 48/4 = 12. Not the same at all.

So in what way is 12 the centre? Measure distances from it.

• Below 12: 10 is 2 away, the other 10 is 2 away, 11 is 1 away. Total 5.
• Above 12: 17 is 5 away. Total 5.

The total distance is the same on both sides.

That is what the mean really is: the balance point. Think of the dots as equal weights on a ruler. The mean is the point where the ruler balances.

Is there any other balance point? No. Move to the right of 12: every distance on the left grows and every distance on the right shrinks, so the balance breaks. Move left and the opposite happens. There is exactly one centre.

This also tells us what happens when we include a new value:

• A value above the mean makes the right side heavier, so the mean moves right: it increases.
• A value below the mean makes it decrease.
• A value equal to the mean changes nothing: the mean stays the same.