The mean as the point where the distances balance

0:00Take two numbers, three and seven, and average them. Add them and divide by two. You get five. Now look at where five sits. It is exactly halfway between the two. Try eight and nine: the average is eight and a half, and again it is exactly halfway. That is not luck, and it is not an example. For any two numbers at all, the average and the halfway point are the same number.

0:29So with two values you can find the average by eye, without dividing anything. Which is a problem, because it teaches a rule that is about to stop working. Add a third value and the two part company immediately. Take two, four and nine. Halfway between the smallest and the largest is five and a half. The average is fifteen divided by three, which is five. Close, but not the same - and the difference matters, because halfway between the extremes ignores everything in between. Move the four anywhere you like and halfway does not move at all.

1:11The average moves. It is paying attention to every value. So the halfway point is not a rough version of the average. It is a different measurement, and it can land on either side: four, eleven and fifteen average to ten, while halfway between the extremes is nine and a half. People call the average the centre of the data. What would that actually mean? Not halfway between the extremes - we have just seen that is something else.

1:45Not the middle value either. That is a different measurement again. Here is a claim worth making, because it can be tested. Stand at the average and look both ways. Measure how far it is out to each value below you, and add those distances up. Then measure out to each value above you, and add those up. The claim is that the two totals are the same. Every time.

2:14Test it. Ten, ten, eleven and seventeen. The total is forty-eight, over four values, so the average is twelve. Below twelve: ten is two away, the other ten is two away, eleven is one away. Two plus two plus one is five. Above twelve: seventeen, and only seventeen, five away. Five against five. And notice what is not being claimed. The arrows themselves are not equal - one long one on the right against three short ones on the left. It is the totals that match.

2:52Six, seven and eight balance at seven with the longest arrow on neither side. Two, four and nine balance at five with the longest on the right. But two, six, ten and twelve balance at seven and a half with the longest arrow on the left. Three diagrams would have taught you a rule. The fourth breaks it. Balance also does not mean the same number of values on each side.

3:21Look again at ten, ten, eleven, seventeen. Three values sit below twelve and one sits above it, and it balances anyway. The one value five away pays for three values that are two, two and one away. Counting values and measuring distances are different tests, and they have different answers. Split that collection so the numbers on each side match and you land at ten and a half - the middle value.

3:52Measure the distances from there and you get one on the low side against seven on the high side. Nowhere near balanced. So why should the two totals ever agree? Pick any point on the line. Take the total of the distances up to the values above it, and subtract the total of the distances down to the values below it. Write that out and something collapses. Every value contributes itself minus the point, whichever side it is on.

4:25So the whole difference is just the total of the values, minus the point counted once for every value. That is one line of arithmetic, and it decides everything. It is zero exactly when the point, multiplied by the count, equals the total. Which is to say, exactly when the point is the total divided by the count. The average was never given the balancing property. Dividing the total by the count is what balancing means, written the other way round.

5:02That same line says something stronger: there is only one such place. Slide your candidate above the average and watch. Try twelve and a half on our four values: the low side totals six and a half against four and a half. Try eleven instead: two against six. The imbalance is two one way and four the other, and each time it is exactly four - the number of values - times the distance you moved.

5:32You can see why from the arrows. Move up half a step and every value below gets half a step further away, while every value above gets half a step closer. Three below gain half each; one above loses half. The sides move at their own head counts, and the gap opens at their sum. So the imbalance grows the moment you leave, and it never comes back. One balancing point. No second one anywhere.

6:04Now the sentence you started with, in the other language. Sharing fairly means levelling: take from whoever has more than the share, give to whoever has less, until everyone has the same. How much has to come off the top? For our four values, five. How much has to go on at the bottom? Five. Those are not two facts. They are the same two totals that were on the balance diagram, because the surplus above the average is measured by exactly the arrows on one side, and the shortfall below it by the arrows on the other.

6:43Try levelling to eleven instead and it falls apart: you would take six off the top and put only two on at the bottom. Two short, every time. Fair share and balance point are one statement said twice. Two things the balancing point is not. It need not be one of your values. Two, six, ten and twelve balance at seven and a half, which is not in the collection and is not close to anything in it. It is a position, not a member.

7:17And it is not where the arrows are shortest. At the average of ten, ten, eleven and seventeen, the arrows add up to ten. Move to the middle value, ten and a half, and they add up to eight. Shorter. The average makes the two totals equal, not small - and those are genuinely different requests. The middle value answers the second one. Here is what the balance reading buys you.

7:47Twenty-five throws, plotted as a column of marks above each score. Someone claims the average is seventeen. Is it? You could add all twenty-five and divide. Or you could weigh the two sides against seventeen. Below: the distances add up to thirteen. Above: thirty-one. So no. The high side wins by eighteen, and the average has to sit above seventeen. And now the arithmetic finishes itself. That surplus of eighteen, spread over twenty-five throws, is eighteen twenty-fifths - so the average is seventeen point seven two.

8:28The failed test handed you the answer. Sometimes you do not even need the distances. Three albums, each plotted by song length. One of them averages five point five seven minutes. Which? On the second album, every single song is shorter than five point five seven. On the third, the same. So for either of them, every value sits below that number. The low side has all the arrows and the high side has none.

9:02Nothing can balance there, so neither album can have that average, and no dividing was done at all. That leaves the first, and it checks out: seven songs totalling thirty-nine minutes. One last warning, and it is about how much a balance point tells you. Here are two collections. Three, five, ten, twelve. And two, six, ten, twelve. Both balance at seven and a half. Both have totals of seven on each side. Draw only the rule and the two sums and the diagrams are identical.

9:40But the arrows are different lengths, and so are the collections. The balancing point is a real property, argued and not asserted, and it is one number. It tells you where the collection balances. It does not tell you what the collection is.