Estimation: when an approximate answer is the right answer

0:00Somewhere in your school there is a register with an exact number in it. How many children are enrolled today, precisely. The head of the school knows that number. You almost certainly do not. And your book asks you: so how many children are in your school? Not the exact figure. About how many. Is it about a hundred and fifty? About four hundred? About a thousand? You might feel that answering about four hundred is a sort of failure. That the real answer is in the register, and you do not have it.

0:36That is exactly backwards, and this section exists to say so. An estimate is not a failed exact answer. It is the answer to a different question — coarser, and far easier. And what makes an estimate right is not how close it is, for its own sake. It is whether it settles the thing you were deciding. Suppose you want to know whether the school hall can hold everybody.

1:05About five hundred settles that. The hall seats six hundred, so yes, it can. Four hundred and ninety-seven adds nothing to that decision. You would do the same thing either way. So the question is never how accurate is this. It is: accurate enough for what? Your book works one through, from a pupil called Paromita. Her year has three sections. Hers holds thirty-two children. The other two hold twenty-nine and thirty-five.

1:38Add them. Thirty-two and twenty-nine is sixty-one. Plus thirty-five is ninety-six. And then Paromita says: my year group is about a hundred. Notice what she has just done. She has thrown away four children, on purpose. Not because she cannot add. Because a hundred is a number she can multiply in her head, and ninety-six is not. That trade is the whole technique. You swap an awkward number for a convenient one, and you pay for it in accuracy.

2:13Now the scaling. Her school also runs classes seven, eight, nine and ten. Three sections in each, the same as hers. So she assumes each of those year groups is about the same size as her own. About a hundred. Five year groups. A hundred in each. About five hundred children in the school. And that is genuine arithmetic, done in ten seconds, out of three numbers she happened to know.

2:43She counted nobody. She measured a small piece and multiplied it up. Which is the shape of every estimate in this section, and most you will ever make. But now let's do the thing the book invites and does not do. Let's attack it. Paromita made three assumptions, and each one is a place the answer could be wrong. One: that her three sections are typical. Two: that the older years are the same size as hers.

3:15And three, which is the interesting one: that rounding ninety-six up to a hundred will not matter. It does matter, and here is why. She did not round once. She rounded five times, and every time in the same direction. Four extra children per year group. Times five year groups. Twenty extra children. So her five hundred is not roughly four hundred and eighty, could be either side. It is guaranteed too high, by about twenty.

3:47And that is far more useful than a margin of error. She knows which way she is wrong. Right. The exercises. First, pacing. Your book asks for four distances. To the classroom door, across the school ground, to the school gate, and all the way home. Only the first one is countable. You can walk to the door counting your steps. Twelve, say. For the ground, and certainly for home, you would give up. But you can pace out ten metres and count the steps.

4:23If ten metres takes you fourteen steps, then one step is a bit over seventy centimetres. Now every distance you can walk is one you can measure. Count the steps, multiply by seventy centimetres. A small measurement, multiplied up. The same move Paromita made. Second, rates. How many times do you breathe in a day? You cannot count that. But you can count for one minute. Say fifteen breaths in a minute. Sixty minutes in an hour, so nine hundred an hour.

4:59Twenty-four hours in a day. Twenty-one thousand six hundred breaths. And notice how that number arrived. You measured for sixty seconds, and then multiplied twice. Your book asks the same for blinks. Harder, because you blink less when you concentrate, and not at all when you sleep. So you have to choose how many hours to count. Which is another assumption, and you should say it out loud. Now the best thing in this section, and it is a question your book asks twice.

5:35Somebody hands you a number. Do you believe it? An estimate that can be built forwards can be run backwards. You take their number and work out what it would require. Then you look at what it requires, and ask whether that is plausible. A skill you will use for the rest of your life. On advertisements, on headlines, on things people tell you confidently. Let me show you, with the book's own example.

6:05Sheetal says she has spent about thirteen thousand hours in school so far. Do you agree with her? Build it. Hours in a school day, times school days in a year, times years at school. Say six hours a day. Say two hundred and twenty school days. That is one thousand three hundred and twenty hours a year. Now, how many years? If Sheetal is in class six, about six or seven.

6:33Six years gives seven thousand nine hundred and twenty. Seven years gives nine thousand two hundred and forty. Thirteen thousand would need nearly ten years of school. So under those assumptions, no. It is about half as much again as it should be. But change the assumptions. Seven hours a day, two hundred and forty days, eight years counting nursery — and you get thirteen thousand four hundred. So the honest answer is not yes and not no. It is: thirteen thousand works if you count from nursery and the days are long. Say which, and we can check.

7:16Which brings us to Roshan, and the answer teachers are most suspicious of. Roshan reckons fruit custard for five people costs about a hundred rupees. Do you agree? And the correct answer here really is it depends. But only if you finish the sentence. It depends on which fruit. On how much. On whether the milk and custard powder are counted. Bananas and one apple, and a hundred rupees is generous. Out-of-season berries, and a hundred will not get you started.

7:51It depends, on its own, is a shrug. It depends on the fruit, and here is the sum for two different choices, is an answer. That difference is what the whole section is teaching. Last one, and it is my favourite, because you can test the method against a known answer. Your book asks for the distance from Gandhinagar in Gujarat to Kohima in Nagaland, and tells you to find them on a map.

8:21That instruction is the content. A map has a scale bar. Measure the gap, read the scale, multiply. You will get something near two thousand two hundred kilometres, in a straight line. But nobody travels in a straight line. Roads wander, usually by a quarter to a half again, so by road it is closer to three thousand. So even here you have to say which question you answered. And to test the method: the length of India, from the southern tip to the northern, measured the same way, comes out about three thousand two hundred and thirty kilometres.

9:02The surveyed figure is three thousand two hundred and fourteen. Agreement to better than one per cent, from a ruler and a scale bar. So, your turn, and this is the chapter's own closing exercise. Write an estimation question. Not one you know the answer to. One somebody else could attempt. A good one has three properties. It cannot be counted directly. It breaks into a measurement and a multiplication. And reasonable people would argue about the assumptions.

9:39How many words are in your maths textbook. How many litres the tank on your roof holds. How many days off you get in a year. That last one is worth doing, because you can check it. Estimate first, then count it on a real calendar and see the gap. Most estimates in life never get checked at all. Take the chance when you are offered it. Next time, a question a child can ask in one sentence, and nobody on Earth can answer.