Comparing two proportions that have different totals

0:00Two test scores. Forty-two out of fifty in one paper, seventy out of eighty in the other. Which paper went better? Three people look at those same four numbers and reach three different answers. The first says the fifty-mark paper: only eight marks dropped there, against ten in the other. The second says the eighty-mark paper, because seventy marks is more than forty-two. The third says neither, because the two papers are out of different totals and cannot be compared at all.

0:35One is right, one is right by accident, and one is half right in a way worth taking seriously. Start with the eight and the ten, the reasoning almost everybody finds first. Fewer marks lost ought to mean a better paper, and against one shared total it always does. Out of the same fifty, dropping eight beats dropping ten every time. But these two papers are not out of the same total.

1:06So draw them. A bar of fifty and a bar of eighty, and the second is visibly longer. Eight shaded off the short bar eats more of it than ten shaded off the long one. So the smaller loss is the bigger fraction, and the first verdict is backwards. The obstacle is real. The third person named it correctly, then drew the wrong conclusion from it. Different totals block the direct comparison. They do not block comparison itself.

1:40Rescale both papers onto one shared total of a hundred and it goes away. Forty-two out of fifty becomes eighty-four out of a hundred. Eighty-four per cent. Seventy out of eighty becomes eighty-seven and a half out of a hundred. The eighty-mark paper wins, and now it wins for a reason instead of by luck. That rescaling onto one base is the entire job a percentage does. Go back to the first person, whose method was not wrong in kind, only in execution.

2:16Count what was lost, by all means. Only count it as a share of its own paper. Eight out of fifty is sixteen per cent. Ten out of eighty is twelve and a half. Twelve and a half is less than sixteen, so the losses counted properly pick the eighty-mark paper too. And it always will. What a paper keeps and what it drops make up the whole of it, so the two shares add to a hundred.

2:45Reading the scores and reading the losses are one reading taken from opposite ends. They can never disagree. A method that gives the right answer once repaired teaches more than a method that is simply banned. So percentages settle comparisons. Now watch what the move costs. Two almond drink mixes on a shelf. A pouch and a carton. The pouch holds a hundred and fifty grams. The carton holds four hundred.

3:18Inside the pouch: ninety-nine grams of sugar, thirty of milk solids, twelve of almond powder, nine of additives. Inside the carton: two hundred and seventy-two, sixty-four, forty, and twenty-four. On both labels the weights account for the pack exactly, and that is what makes everything after this work. Two questions. Which has the larger share of almond, and which the smaller share of additives? Both labels are in grams and the packs are different sizes, so the grams answer neither question.

3:58Same obstacle as the two test papers. Same repair. Ninety-nine out of a hundred and fifty is sixty-six per cent. That is one cell of eight, and the other seven are empty. Fill them. The pouch is sixty-six, twenty, eight and six. The carton is sixty-eight, sixteen, ten and six. Each row totals a hundred, and that is not a happy accident. It is the weights accounting for the pack, in a different currency.

4:31The almond question has a clean answer, so take it first. Twelve grams out of a hundred and fifty is eight per cent. Forty out of four hundred is ten. The carton holds more almond by weight and a larger share of it as well. Both readings agree. Now the additives, where the topic actually lives. Nine grams in the pouch. Twenty-four in the carton, which is about two and two-thirds times as much.

5:03Nine out of a hundred and fifty is six per cent. Twenty-four out of four hundred is six per cent. Not close. The same number. So the question of which has the smaller share of additives has no winner, and any answer to it has been invented. That was one column of four, and the four do not behave alike. The carton is the heavier pack in all four of them, and carries the larger share in only two: the sugar and the almond.

5:37The milk solids go the other way outright. Sixty-four grams against thirty, and sixteen per cent against twenty. Heavier, and a smaller share. Both true, because they answer different questions. So the idea that more grams means a larger share is right on two of these four columns and wrong on two. Which is exactly the score that keeps a wrong rule alive for years. One more thing those rows do, and it is a gift to anybody checking their own work.

6:13Every row must total a hundred, because the weights total the pack. So a row that does not is a misread cell, and it tells you the size of the misreading. Read the pouch's milk solids as thirty-three instead of thirty, and the row comes to a hundred and two. Two over. And two per cent of a hundred and fifty grams is three grams, which is exactly the slip.

6:41A row that misses a hundred is not a rounding curiosity. It is a receipt, with the size of the mistake on it. Now the mirror image, because a percentage becomes a trap as easily as a tool. Two people eat biscuits. One person's biscuits are twenty-five per cent sugar, the other's are thirty-five. Who ate more sugar? If they had eaten the same weight of biscuit the answer is thirty-five, and the gap is ten grams in every hundred.

7:16They did not. The first ate a hundred and twenty grams, the second ninety-five. Twenty-five per cent of a hundred and twenty is thirty grams. Thirty-five per cent of ninety-five is thirty-three point two five. The larger percentage still wins, but the gap has collapsed to three and a quarter grams, under a third of what it was. And it is only a question of how hard the totals pull. At a hundred and thirty-three grams they are dead level; at a hundred and thirty-four the smaller percentage is ahead.

7:56One more case, where the totals are far too big to picture. About eight thousand two hundred million people are alive. Four countries, in millions: eighty-three, one thousand four hundred and sixty, a hundred and seventy-five, and three hundred and forty-seven. Those raw figures are almost unreadable against each other. As shares of everybody: about one per cent, about seventeen point eight, about two point one, and about four point two.

8:30Together, those four hold a shade over a quarter of the world. And the big one needs no calculator. Fourteen hundred and sixty against eight thousand two hundred is a bit under two tenths, so a bit under twenty per cent. That is the tool working as advertised: four numbers nobody can compare, turned into four that anybody can. One last picture. A survey chart. Seven age bands, two bars each, one for women and one for men.

9:05Fourteen bars, and each is a percentage: how many of that band can use a computer. Thirteen carry a number, and one is drawn and left unlabelled. Six statements are made about it, and the job is to say which ones it settles. The twenties top every band. True, and true in both series. The women trail the men. True in all six bands that carry both numbers, and unanswerable in the seventh.

9:35More than a quarter of the thirties. False. The men land on exactly a quarter and the women well below it. Half the twenties. False. The men reach thirty-seven, thirteen short of half. Fewer than one in ten of the oldest. True, and true wherever you decide the oldest begins, since nothing in the last two bands reaches ten. And one the chart cannot settle at all: that the twenties band holds more people than the teenage band.

10:08Nothing on that chart is a head-count. Every bar is a share, and a share is a number with its total taken away. Which is the whole topic in one line. Put two proportions on one base and you can compare them. Take the base away and you cannot count anybody.