Turning a vague comparison into a question with a data answer

0:00Two friends are coming round this evening. One of them is five feet tall, the other is six. One is a woman and one is a man. Which is which? You have an answer. Almost everybody has the same answer, and almost everybody gives it straight away. Now here is the awkward part. You cannot prove it, you have never met either of them, and you would still bet on it.

0:27That is not sloppy thinking. That is a whole way of reasoning, and this video is about what it is and when you are allowed to use it. Start with why the guess is any good at all. It is good because five foot men and six foot women are uncommon. Uncommon. Not impossible. And the distance between those two words is the whole thing. Take two groups where one really is taller. Taller on the average, taller in the middle, tallest tallest, and even tallest shortest. Every way of measuring it agrees.

1:07Now pick one from each group at random and compare just those two. There are a hundred such pairings here, and the rule holds for eighty of them. It fails for fifteen, and five are a dead tie. So a rule can be right about the groups and wrong about the pair in front of you. One group being taller never meant every single one is taller. It could not have meant that, and it never did.

1:36Keep that, and listen to six perfectly ordinary things people say. A batter has been steady all season, so a century is expected tomorrow. The cycle ride to school takes about a quarter of an hour. This pen will probably last another fortnight. The village has lost around a hundred people over ten years. More fruit and vegetables added two kilometres to a daily run. A boy spends about seven hours a day at school.

2:08Sport, travel, stationery, population, health, school. Six subjects with nothing whatever in common. Except that they have something in common, and it is worth finding. Every one of them has a number in it. That is not it — plenty of sentences have numbers in them. In every one of them, the number is standing in for a run of cases. About a quarter of an hour is not one ride to school. It is many rides, squashed down into one number.

2:42Probably another fortnight is not this pen. It is pens. Not one of those six is a fact about a single occasion, even though every one of them sounds like one. And that is why each of them can be perfectly true and still be wrong tomorrow. Sentences like that have a name. They are statistical statements. A statistical statement is a claim about a group of cases whose content is a number.

3:13A count, a share, a chance, or a prediction. And it comes with a property no ordinary fact has. It can be true about the group and false about any particular member of it, at the same time, with no contradiction anywhere. Which is uncomfortable, and is the price of being able to say anything at all about things that vary. Now turn it round. If that is a statistical statement, what is a statistical question?

3:46It is one that collecting data can settle, and that nothing less than collecting data can settle. Notice what that does not say. It does not say a question with a number for an answer. What does one tennis ball cost in the shop on the corner? That has a number for an answer, and it needs no data at all. Ask the shopkeeper. You are finished. The number is not the test. So what is?

4:15The test is whether the thing varies. How tall are the students in our year? You cannot answer that by measuring one student, and not because measuring is hard. You cannot answer it because the next student is a different height, and the one after that is a different height again. There is no single value sitting there waiting to be found. There is a collection to be gathered, and then squashed down into something sayable.

4:45And notice that the varying is not a fault in the question. The varying is the entire reason the question needs data. If everyone were the same height you would measure one person and go home. Second question. Are onions typically dearer in one town or the other? Prices move from month to month, so again there is no one number to look up. Here is a year of them for two towns. These figures are made up, to show the shape of the thing.

5:19Add them up and average them, and the first town is dearer. That is the answer, and it is a real answer. But look along the rows. In two of those twelve months, the first town was the cheaper one. So somebody who walked into both towns on one particular day would get the wrong answer, one time in six. And that is precisely what the word typically is doing in the question.

5:47It is not vagueness. It is carrying the fact that the individual months disagree with each other, on purpose. So now you can sort questions, and it is worth doing seven of them. What does a tennis ball cost across the whole country? Shops differ. That needs data. How old are the dogs living on this street? Dogs differ. That needs data. What share of the class enjoys walking uphill? People differ. Data.

6:20How much rain fell here month by month last year? Months differ. Data. Four out of seven so far, and the reason has been the same reason every time. Now the other three. Do you like reading? How many bricks are in this wall? Who bowled best in yesterday's match? One answer each. No collecting required. Nobody has marked this for us, so that sorting is ours, and every line of it has to be defended.

6:52And two of those three will not stay where I put them. Do you like reading. Asked of you, that is one fact about one person. Ask it of the whole class and the words have not changed at all, but now it is a spread. Six out of ten, say. So the wording of a question does not decide this. Who is being asked about decides it. The other one is awkward for the opposite reason. Who bowled best in yesterday's match?

7:23That match is over. The data is finished, it is written down, and there is not one more number to collect. What is not settled is which measure everybody agrees to use. Most wickets can end in a tie between two bowlers. Fewest runs given away can pick a third one entirely. That is not missing data. That is an unsettled question about what the word best means. Which is a good moment to say what the word statistics actually covers, because it is wider than people expect.

8:01Five activities. Collecting. Organising. Analysing. Interpreting. Presenting. Exactly one of those five is arithmetic. Deciding what is worth collecting is on the list, and it comes first. So is showing the result in a way somebody else can actually see. If you thought statistics meant working out averages, that is one fifth of it, sitting in the middle. So, an argument. Two batters, four matches each. The first scored nought, seventeen, twenty one, and ninety.

8:38The second scored sixty seven, fifty five, eighteen, and thirty five. Who batted better? Add them up. One hundred and twenty eight against one hundred and seventy five. The second one. Averages. Thirty two against forty three point seven five. The second one again. Middle score. Nineteen against forty five. The second one. Best single innings. Ninety against sixty seven. The first one. And match by match, two apiece. A dead heat.

9:13Four different ways of asking, and they do not agree. Nobody is disputing the eight numbers. What is in dispute is what better is supposed to mean, and no amount of further collecting will settle that for you. Which is where this goes next.