The arithmetic mean as fair-share

0:00An argument, and a real one. Two batters, four matches each. The first scored nought, seventeen, twenty one and ninety. The second scored sixty seven, fifty five, eighteen and thirty five. Add them up and it is one hundred and twenty eight against one hundred and seventy five, so the second man is ahead. But the best single innings was ninety against sixty seven, so the first man is ahead.

0:29And match by match it is two apiece, a dead heat. Four ways of asking, and they do not agree. This is about the one people reach for most, and about what it is actually doing. Because there is a detail that breaks the easy answer, and it takes only one match to break it. A second season. The first batter played five: twenty three, seven, ten, fifty two, eighteen.

0:58The second was down for five as well, but one was rained off before he ever came in. Twenty six, fifty three, two, nothing at all, and fifteen. Four matches of runs, and a dash where the fifth should be. Add up what there is. One hundred and ten against ninety six. So the first batter wins on the total. And that answer is worthless. He had five goes at it. The other man had four.

1:28The total is not measuring how well anybody batted. It is measuring how well they batted and how many chances they got, added together, with no way left to pull the two apart. Give a poor batter enough matches and his total will pass anybody's. When the counts match, totals are perfectly fine, and that is why nobody complained in the first season. The moment the counts differ, the total stops being about batting at all.

2:01Now a tempting repair, and it is wrong. Why not write a zero in the empty match and carry on as normal? Watch what that actually does. It does not change the runs by anything — ninety six and nothing more is still ninety six. What it changes is how many matches you divide by. Four becomes five. And that turns twenty four into nineteen point two, which hands the whole season to the other man.

2:31So a blank and a zero are not the same thing, and the difference is a verdict. A zero is a score. It goes in the top. A blank is a match that never happened, and it comes out of the bottom. So divide properly. Runs, over matches actually played. One hundred and ten shared across five matches is twenty two. Ninety six shared across four is twenty four. Twenty four beats twenty two. The second batter had the better season, and the total had said the exact opposite.

3:09That number is the average. Its proper name is the arithmetic mean. But what is it? Not the recipe. The thing the recipe produces. Two groups go out picking guavas. The first group is five people, and they bring back three, eight, ten, five and four. The second is six people: five, four, six, three, four and eight. Both groups brought back exactly thirty. The total cannot separate them, because on that measure there is nothing there to separate.

3:45And yet five people carrying thirty is plainly not the same as six people carrying thirty. So here is what the dividing does, drawn out. Pour all thirty into one heap. Then hand it back out until everybody is holding the same amount. In the first group everybody ends on six. In the second, everybody ends on five. That is the average. What each one would have if the whole lot were shared out equally.

4:17And look at what the surplus did. The ones who were above six gave up exactly as much as the ones below six were short by. Six units up, six units down. That is not a coincidence. It is the job. The average is the level a heap settles to. Which leads somewhere that catches people out. Five days of hibiscus flowers: two, seven, nine, four, three. Twenty five in all, over five days, so the average is five.

4:52But no day had five. Not one of them. The average is not one of the readings, and it never promised to be. Now a ball, bounced on eight days: six, two, nine, five, four, six, three, five. Forty over eight. The average is five again. And this time a five does appear, twice over. Same answer, and the two facts have nothing to do with each other. Whether the average lands on one of your numbers is luck.

5:24There is one thing it always does, though, and it is worth having. It always lands between the smallest reading and the largest. It has to. Below all of them and the heap is too small to go round; above all of them and there was never that much to hand out. A school's enrolment over six years: fifteen fifty five, sixteen seventy, seventeen fifty, two thousand and thirteen, two thousand and forty, twenty one twenty six.

5:56The average is eighteen fifty nine. No year had eighteen fifty nine students. No year came close. But it sits inside the range, and it says how big a typical year was in a way that no single year can. Now a race, and please do not guess. Two runners, timed for seven days. The first: seventeen, eighteen, seventeen, sixteen, nineteen, seventeen, eighteen. The second: twenty, eighteen, eighteen, seventeen, sixteen, sixteen, seventeen.

6:32One of them starts badly and finishes strongly. The other is scrappier in the middle. Have a verdict ready. Both of them add up to one hundred and twenty two. Seven days each. So the two averages are not close, they are the same number. And that number is not even whole. One hundred and twenty two over seven is seventeen and three sevenths. The one real difference is the spread: three seconds from best to worst against four seconds.

7:05Which is a genuine question about those runners, and it is not the question we asked. The relationship runs both ways, and that is useful. Average times count gives back the total. Told the average and how many there were, you have the sum. And once that is in your hand you start hearing averages everywhere. Thirty seven point two millimetres of rain a day through July. Forty five kilometres to a litre of fuel. Fifty eight glances at a phone in a day.

7:41Four point seven tonnes of wheat off a hectare in one farming region, against two point nine in another. Every one is a heap shared out, and every one is quietly hiding the group it was shared over. So ask it every time. Averaged over what? Over days, over vehicles, over people, over fields? A number with no group underneath it is not an average of anything. So, back to the batters.

8:12The average answers exactly one question, and it answers it completely. If the runs had been spread evenly over the matches actually played, how many would each match have got? Twenty two. And twenty four. It does not tell you either man ever scored twenty two. It does not tell you what happened in any one match. And on its own it does not tell you who is better. It tells you who is better per match, which is the same thing only if everybody agrees that per match is what better means.

8:48None of that is a weakness in the arithmetic. The arithmetic is finished and it is right. It is that one number, pulled out of a whole season, cannot carry everything the season had in it. So the next question is what else you can pull out.