When an approximate answer is the better answer

0:00A hundred thousand people came to the fair. That is what the headline said. One lakh. And somebody in the crowd points out something annoying. If she had stayed at home, the number would have been ninety nine thousand nine hundred and ninety nine. Which is true. But no headline in the world would have said that. It would still have said a lakh. And it would have been just as useful, because nobody reading it is trying to picture ninety nine thousand nine hundred and ninety nine separate people.

0:34That is not sloppiness. A headline is not counting people, it is telling you the size of the thing. One lakh answers the question that was actually being asked, which is roughly how many. So an approximation is not a worse answer to the same question. It is the right answer to a different one. Here is a town. In one place it is reported as having about seventy five thousand people.

1:06In another, the exact figure. Seventy six thousand and sixty eight. Both are correct. Neither one is a mistake. One of them is what a count found. The other is what a person would actually say out loud. The gap between them is one thousand and sixty eight people. As a share of the town, that is one point four per cent. So the rounded figure gets you to within about one and a half people in every hundred.

1:37Whether that is close enough depends entirely on what you wanted it for. Now look harder at that seventy five thousand, because something has been slipped past us. Round seventy six thousand and sixty eight to the nearest thousand, and you get seventy six thousand. Round it to the nearest ten thousand, and you get eighty thousand. Neither of those is seventy five thousand. Seventy five thousand is the nearest five thousand, which is not a place at all.

2:12It is a step that somebody chose. And that is the first real idea here. Before you round anything, you decide how rough to be. Nearest thousand keeps more of the truth. Nearest ten thousand is easier to carry in your head. That choice is about your reader, not about the number. Now the direction. Up, or down? A school has seven hundred and thirty two people in it, and somebody has to order sweets.

2:43Seven hundred is a nice round number. So is seven hundred and fifty. Order seven hundred, and thirty two people get nothing. Order seven hundred and fifty, and you have eighteen left over. Eighteen spare sweets is a small waste. Thirty two disappointed people is not a small problem. So the two options are not symmetric, and that is the whole point. One kind of error costs far more than the other.

3:14So this one rounds up. Not because of a digit. Because of what happens if you are short. Now the other direction. Something has a price of four hundred and seventy, and you are describing it to somebody who might buy it. You could say around four hundred and fifty. You could say around five hundred. Four hundred and fifty understates it by twenty. Five hundred overstates it by thirty.

3:45Twenty out and thirty out are nearly the same size of mistake, but they are not the same kind of mistake. And if you say five hundred, and they decide it is too expensive, you have talked them out of it for no reason at all. So this one rounds down. Notice as well that four hundred and fifty is the closer of the two. Here the careful direction and the accurate one agree. They will not always.

4:16Now a story about where rounding is not allowed at all. A father asks his son to ring the railway enquiry line, and find out whether a train is running on time. On the wall there is a poster of numbers to call. Police, one hundred. Fire, one hundred and one. Ambulance, one hundred and two. Disaster services, one hundred and eight. Emergencies, one hundred and twelve. Railway enquiry, one hundred and thirty nine.

4:48The boy has just learned about rounding, and he is rather pleased with himself. One hundred and thirty nine, to the nearest hundred, is one hundred. A few minutes later there is a police officer at the door. The joke is the whole lesson. One hundred and thirty nine is not a quantity. It is a name that happens to be written in digits. So put those three side by side.

5:17The sweets went up. The price came down. The phone number did not move at all. None of that was decided by looking at a digit. There is a rule you have probably been taught, about five and above going up. That rule is for one particular job, and this is not that job. Here the direction comes from a completely different question. What goes wrong if I am wrong?

5:44If being short is the disaster, round up. If overstating is the disaster, round down. That is a judgement about the world, not a calculation about the number. And if there is no such thing as being close, do not round at all. Now let's use this on something real. Here are the populations of some large cities, ten years apart. These are big numbers, and reading them exactly is genuinely hard work.

6:16So read them roughly instead. Mumbai, about a crore and a quarter. Delhi, about one point one crore. Bengaluru, about eighty four lakh. Hyderabad, about sixty eight lakh. Notice what just happened. You now know the shape of this table, and you did no arithmetic whatsoever. You could not tell me any of those figures exactly, and you did not need to. That is what approximation is for. Here is a question worth asking. Which of them grew the most?

6:51Mumbai is the biggest, so Mumbai is the obvious guess. Mumbai grew by just under five lakh. Bengaluru grew by about forty one lakh. That is nearly nine times as much growth, in a city that started at about a third of the size. Bengaluru is the answer, and it is not close. Pune, for comparison, grew by about six lakh. And one city here did not grow at all. Kolkata went down, by about eighty six thousand.

7:25So be careful with the sentence every city is growing. One of these was not. Now change the question, and watch the table rearrange itself. Not which city grew most. Which city almost doubled? Bengaluru very nearly did. About one point nine six times. Hyderabad, about one point eight seven. Surat, about one point eight four. And Vadodara went past double. Two point one times. But look where Vadodara sits on the other list. Fifth.

8:01It added under nineteen lakh people, while Bengaluru added over forty one. Both answers are right. They are answers to different questions. A difference asks how many more. A ratio asks how many times. Never let one of them answer for the other. So when is approximation not allowed? Whenever the number is a label rather than an amount. A phone number. An account number. A house number. None of those measure anything, so there is no such thing as being nearly right.

8:38And there are real amounts that need exactness too. The money in an account. The dose on a medicine bottle. The number of seats on an aircraft. Everywhere else, the honest question is not how precise can I be. It is how precise does this need to be. Here is something to try. Find the population of the place you live, and say it three ways. Exactly, to the nearest thousand, and to the nearest lakh.

9:08Then decide which of the three you would put in a headline, and why.