Two records with the same average that describe very different players

0:00Two players. Ten games each. Here are the twenty scores, in the order they were made. The first player: thirty, ninety-one, nought, sixty-four, forty-two, eighty, thirty, five, a hundred and seventeen, seventy-one. The second: fifty-three, forty-six, forty-eight, fifty, fifty-three, fifty-three, fifty-eight, sixty, fifty-seven, fifty-two. Look at them for a moment before anybody summarises anything. One of these players had a game where he scored nothing at all, and a game where he scored a hundred and seventeen.

0:38The other never dropped below forty-six and never got past sixty. Nobody would confuse those two careers. That is the point of this video, and everything that follows is about how easy it is to throw that difference away. Start with the average, and do the adding rather than looking the answer up. The first player's ten scores add to five hundred and thirty. The second player's ten scores also add to five hundred and thirty.

1:11Ten games each, so divide by ten twice, and both averages come out at fifty-three. Notice how differently the two totals were reached. One of them got there through a nought and a hundred and seventeen. The other got there without a single score leaving the forties and fifties. The adding does not care. It hands back the same number either way. The average is not the only way to name a middle, so try the other one.

1:44The middle value of a list is read off the list in order, so both records get sorted first. That sorting is the step everything here turns on. In order, the first player reads nought, five, thirty, thirty, forty-two, sixty-four, seventy-one, eighty, ninety-one, a hundred and seventeen. In order, the second reads forty-six, forty-eight, fifty, fifty-two, fifty-three, fifty-three, fifty-three, fifty-seven, fifty-eight, sixty. Ten is an even count, so there is no single entry in the middle. The rule takes the fifth and the sixth and averages them.

2:27For the first player that pair is forty-two and sixty-four. For the second it is fifty-three and fifty-three. Those are not the same pair. They are not even close to being the same pair. And both averages of them come out at fifty-three. So here is the whole summary. Two averages and two middle values, four numbers. Every one of them reads fifty-three. Put another way: the four numbers make two comparisons, and the count of comparisons that come out different is nought.

3:03A table with two columns, and nothing in it that can tell the two columns apart. That is not a coincidence anybody arranged to be spooky. It is a demonstration, and it works because both of these numbers were built to answer the same question. Both of them answer where. Neither of them was ever asked how far apart. Before going further, one thing has to be ruled out. Maybe the comparison is broken. Maybe it would say the same about any two records at all.

3:40So run two summaries that are deliberately not the rule. Take the fifth entry on its own, which is what dropping the even-count branch of the rule comes to. That reads forty-two for the first player and fifty-three for the second. Different. The comparison can say no. Take the point halfway between the lowest score and the highest, which is another perfectly reasonable way to name a centre. That reads fifty-eight and a half for the first player and fifty-three for the second.

4:15Different again. So the agreement at fifty-three is a fact about these two summaries, not a machine that says yes to everything. The two fifty-threes are not even the same kind of thing. Count how many of the first player's games ended on fifty-three exactly. Nought. He never once scored it. Count the second player's. Three. He scored exactly fifty-three three separate times. So one middle value is a score that was actually made, and the other is a number that was never on the board.

4:53That is not a quirk of these two lists. For an even count the middle value is the average of two entries, and an average of two different numbers is neither of them. Take four hundred ten-game records that nobody chose, generated and never looked at. In three hundred and eighty-two of them the middle value is not one of the scores. Do the same with records of nine games, where the rule takes a single entry instead of averaging two, and the count of middle values that are not one of the scores is nought. Every single time, it is a score somebody made.

5:36There is a third way to name a middle, and it is worth trying because it is the one that behaves differently. The commonest score. Whichever value comes up most often. For the first player that is thirty, which he made twice. For the second it is fifty-three, which he made three times. Thirty and fifty-three. Those are different, so this one does tell the two records apart. Hold on to that, because it is going to stop working in a few minutes, and how it stops working is more interesting than the fact that it does.

6:14Now stop summarising and just draw the twenty numbers. One line, marked from nought to a hundred and twenty. One dot per game. The first player on the upper row, the second on the lower, both on the same scale, because a comparison of two pictures drawn to different scales is not a comparison. The upper row is strewn from end to end. The lower row is a tight little block sitting between the forties and the sixties.

6:45Measure how much of the line each one covers. The first player's dots run from nought to a hundred and seventeen, which is ninety-seven per cent of the axis. The second player's run from forty-six to sixty, which is eleven per cent of it. Nobody needs a formula to see the difference. The eye gets it in about a second. And not one of the four summaries reported it. So put a number on what the eye just saw.

7:17Take each score, ask how far it is from fifty-three, and average those distances. For the first player that comes out at thirty-one point six. For the second, three point two. One of those is nine point eight seven times the other. Nine times the strewing, from two records that agreed on every summary we computed. Square the distances instead of taking them as they come, and average those: a thousand three hundred point six against seventeen point four.

7:52Whichever way the distances are handled, the gap is enormous, and the four numbers in the table missed all of it. Why is an average blind to this? Not through carelessness. Through what it is. Take each score, subtract the average, and add up those differences with their signs left on. For the first player, that total is exactly nought. For the second, exactly nought. That is not a fluke of these two. Over four hundred records nobody chose, the number that fail to add to nought at their own average is nought.

8:32The average is the place where the record balances. The overshoots and the undershoots cancel there, exactly, always. And it is the only such place. Try every whole number from nought to a hundred and twenty against the first player's record: one of them balances it, and the count that even come within a single point of balancing it is one. By comparison, the count of those four hundred records that balance at their middle value instead is one. Just one, out of four hundred.

9:08So the cancelling is the average's whole character. It is built to have the distances cancel, which means it is built to throw them away. Watch the cancelling do its work. Take any record, move one score down by some amount and another score up by the same amount. The total does not budge, so the average does not budge. Over four hundred records and seven different amounts each, that is two thousand eight hundred attempts. The number that end up with a different average is nought.

9:44Move both scores the same way instead, up by the same amount, and the number that keep their average is nought. So it really is the cancelling doing it, not just the moving. Now the middle value. If the two moves land on the lowest score and the highest score, the middle of the list has nothing to do with either of them. Over those same two thousand eight hundred attempts, the number that keep their middle value is two thousand eight hundred. All of them.

10:17Move an inner entry instead, one that sits just left of the middle, and cancel it against the highest score. The average is still safe: two thousand eight hundred keep it. But the middle value survives only a hundred and twenty-six times out of two thousand eight hundred. So what protects the middle value is not the cancelling at all. It is where the two moves land. Which means a third record can be built to order.

10:48Start from the second player, the tight one. Take his lowest score, forty-six, and drive it down to nought. Take his highest, sixty, and drive it up to a hundred and six. Forty-six down, forty-six up. Two entries changed out of ten, and the total is still five hundred and thirty. So the average is still fifty-three. The two moves landed on the ends, so the middle of the sorted list never moved, and the middle value is still fifty-three.

11:21And here is where the commonest score stops helping. He still scored fifty-three three times, and nothing else more than once. The commonest score is still fifty-three. All three centres, unchanged. All three. Now measure the strewing. The average distance from fifty-three has gone from three point two to twelve point four. Three point eight seven times as strewn. Same three summaries. A record that now runs from nought to a hundred and six.

11:57It is worth checking that this is a real construction and not a trick of arithmetic, so run it backwards. Bring the two ends in instead of driving them out. Same equal and opposite move, same cancelling, same protected middle. Do that by every amount until the two ends meet, which gives eight records including the one we started from. The number of them with a different average or a different middle value: nought. The centres are just as safe going in.

12:31The number of them with more strewing than the original: nought. Not one. So the direction is what decides the strewing, and the centres cannot see the direction. They are looking at the total and at the sorted middle, and neither of those knows which way the ends went. Push that all the way, and the argument stops being about three records and starts being about a family. Drive the ends out by nought, by one, by two, and on up to forty-six, which is as far as it goes before a score would have to be negative. That is forty-seven records.

13:12Every one of them has an average of fifty-three, a middle value of fifty-three, and a commonest score of fifty-three. The number with any centre other than fifty-three is nought. Their strewing runs from three point two up to twelve point four, and it takes forty-seven different values along the way. Every record in the family is differently strewn from every other. Now count the pairs. Out of the family, the number of pairs that any of the three centres can tell apart is nought.

13:48The number of pairs the average distance can tell apart is a thousand and eighty-one, which is every pair there is. And arithmetic puts no ceiling on it. Drive the ends out by a million instead of forty-six, and the average is fifty-three, the middle value is fifty-three, the commonest score is fifty-three, and the average distance is two hundred thousand and three point two. The centres are pinned. The strewing is free. That is the whole of it.

14:23Everything so far was built by hand, so try it on records nobody picked. Four hundred ten-game records, generated, never inspected. Widen each of them by five. The number whose average changes: nought. The number whose middle value changes: nought. The number that fail to get more strewn: nought. Four hundred for four hundred, in all three columns. And the commonest score turns out to be the weakest of the three, in a way that is worth naming.

14:58Of those four hundred records, two hundred and ninety-five have no commonest score at all, because two different values are commonest together and there is nothing to choose between them. Of the hundred and five that do have one, seventeen lose it to the widening. Of the eighty-eight that keep one, the number that end up naming a different score is nought. So the commonest score is not unreliable because it wanders. It is unreliable because three times out of four it is not there.

15:35One last thing, and it is a correction to something you might have started believing. Set aside the last game of each record, so there are nine scores instead of ten and the rule takes a single middle entry rather than averaging two. The first player's middle value is now forty-two. The second player's is fifty-three. They no longer agree. So the tie at fifty-three was chosen. Somebody picked two records that happen to land on the same two summaries, because that makes the failure obvious in one glance.

16:13What is not chosen, and what does not depend on anybody's data, is the family. Forty-seven records, three centres frozen, the strewing running free. The agreement was an illustration. The impossibility is structural. So a second number is needed, and its shape is already decided. It cannot be built out of positions, because every number built out of positions is a centre, and centres are exactly what cannot see this.

16:46It has to be built out of distances. Pick a centre, ask how far each observation is from it, and combine those distances into one figure. Average them as they come, and the two players read thirty-one point six and three point two. Square them first and then average, and they read a thousand three hundred point six and seventeen point four. Those are two different recipes and they will get separate names and separate treatment, but they are made of the same ingredient: a centre, and the distances to it.

17:25The property being measured is how widely the observations are strewn, and the single number that reports it is called a measure of dispersion. One number says where. The other says how far apart. It takes both of them to describe a record, and now you know exactly why.