by Rob Eastaway
It’s an interesting case study in how statistical forecasts are only as good as their weakest input. You might know certain details precisely (such as the number of cows diagnosed with BSE), but if the rate of infection could be anywhere between 0.01% and 100%, your predictions will be no more accurate than that factor of 10,000.
At least nobody (that I’m aware of) attempted to predict a number of victims to more than one significant figure. Even a prediction of ‘370,000’ would have implied a degree of accuracy that was wholly unjustified by the data.
DOES THIS NUMBER MAKE SENSE?
One of the most important skills that back-of-envelope maths can give you is the ability to answer the question: ‘Does this number make sense?’ In this case, the back of the envelope and the calculator can operate in harmony: the calculator does the donkey work in producing a numerical answer, and the back of the envelope is used to check that the number makes logical sense, and wasn’t the result of, say, a slip of the finger and pressing the wrong button.
We are inundated with numbers all the time; in particular, financial calculations, offers, and statistics that are being used to influence our opinions or decisions. The assumption is that we will take these figures at face value, and to a large extent we have to. A politician arguing the case for closing a hospital isn’t going to pause while a journalist works through the numbers, though I would be pleased if more journalists were prepared to do this.
Often it is only after the event that the spurious nature of a statistic emerges.
In 2010, the Conservative Party were in opposition, and wanted to highlight social inequalities that had been created by the policies of the Labour government then in power. In a report called ‘Labour’s Two Nations’, they claimed that in Britain’s most deprived areas ‘54% of girls are likely to fall pregnant before the age of 18’. Perhaps this figure was allowed to slip through because the Conservative policy makers wanted it to be true: if half of the girls on these housing estates really were getting pregnant before leaving school, it painted what they felt was a shocking picture of social breakdown in inner-city Britain.
The truth turned out to be far less dramatic. Somebody had stuck the decimal point in the wrong place. Elsewhere in the report, the correct statistic was quoted, that 54.32 out of every 1,000 women aged 15 to 17 in the 10 most deprived areas had fallen pregnant. Fifty-four out of 1,000 is 5.4%, not 54%. Perhaps it was the spurious precision of the 54.32’ figure that had confused the report writers.
Other questionable numbers require a little more thought. The National Survey of Sexual Attitudes has been published every 10 years since 1990. It gives an overview of sexual behaviour across Britain.
One statistic that often draws attention when the report is published is the number of sexual partners that the average man and woman has had in their lifetime.
The figures in the first three reports were as follows:
Average (mean) number of opposite-sex partners in lifetime (ages 16–44)
Men Women
1990–91 8.6 3.7
1999–2001 12.6 6.5
2010–2012 11.7 7.7
The figures appear quite revealing, with a surge in the number of partners during the 1990s, while the early 2000s saw a slight decline for men and an increase for women.
But there is something odd about these numbers. When sexual activity happens between two opposite-sex people, the overall ‘tally’ for all men and women increases by one. Some people will be far more promiscuous than others, but across the whole population, it is an incontravertible fact of life that the total number of male partners for women will be the same as the number of women partners for men. In other words, the two averages ought to be the same.
There are ways you can attempt to explain the difference. For example, perhaps the survey is not truly representative – maybe there is a large group of men who have sex with a small group of women that are not covered in the survey.
However, there is a more likely explanation, which is that somebody is lying. The researchers are relying on individuals’ honesty – and memory – to get these statistics, with no way of checking if the numbers are right.
What appears to be happening is that either men are exaggerating, or women are understating, their experience. Possibly both. Or it might just be that the experience was more memorable for the men than for the women. But whatever the explanation, we have some authentic-looking numbers here that under scrutiny don’t add up.
THE CASE FOR BACK-OF-ENVELOPE THINKING
I hope this opening section has demonstrated why, in many situations, quoting a number to more than one or two significant figures is misleading, and can even lull us into a false sense of certainty. Why? Because a number quoted to that precision implies that it is accurate; in other words, that the ‘true’ answer will be very close to that. Calculators and spreadsheets have taken much of the pain out of calculation, but they have also created the illusion that any numerical problem has an answer that can be quoted to several decimal places.
There are, of course, situations where it is important to know a number to more than three significant figures. Here are a few of them:
In financial accounts and reports. If a company has made a profit of £2,407,884, there will be some people for whom that £884 at the end is important.
When trying to detect small changes. Astronomers looking to see if a remote object in the sky has shifted in orbit might find useful information in the tenth significant figure, or even more.
Similarly in the high end of physics there are quantities linked to the atom that are known to at least 10 significant figures.
For precision measurements such as those involved in GPS, which is identifying the location of your car or your destination, and where the fifth significant figure might mean the difference between pulling up outside your friend’s house and driving into a pond.
But take a look at the numbers quoted in the news – they might be in a government announcement, a sports report or a business forecast – and you’ll find remarkably few numbers where there is any value in knowing them to four or more significant figures.
And if we’re mainly dealing with numbers with so few significant figures, the calculations we need to make to find those numbers are going to be simpler. So simple, indeed, that we ought to be able to do most of them on the back of an envelope or even, with practice, in our heads.
2
TOOLS OF THE TRADE
THE ESSENTIAL TOOLS OF ESTIMATION
For most back-of-envelope calculations, the tools of the trade are quite basic.
The first vital tool is the ability to round numbers to one or more significant figures.
The next three tools are ones that require exact answers:
Basic arithmetic (which is built around mental addition, subtraction and a reasonable fluency with times tables up to 10).
The ability to work with percentages and fractions.
Calculating using powers of 10 (10, 100, 1,000 and so on) and hence being able to work out ‘orders of magnitude’; in other words, knowing if the answer is going to be in the hundreds, thousands or millions, for example.
And finally, it is handy to have at your fingertips a few key number facts, such as distances and populations, that crop up in many common calculations.
This section will arm you with a few tips that will help you with your back-of-envelope calculations later on – including a technique you may not have come across that I call Zequals, and how to use it.
ARE YOU AN ARITHMETICIAN?
In the opening section there was a quick arithmetic warm-up. It was a chance to find out to what extent you are an Arithmetician.
Arithmetician is not a word you hear very often.
In past centuries it was a much more familiar term. Here, for example, is a line from Shakespeare’s Othello: ‘Forsooth, a great arithmetician, one Michael Cassio, a Florentine.’ That line is spoken by Iago, the villain of the play, who is angry that he
has been passed over for the job of lieutenant by a man called Cassio. It is an amusing coincidence that Shakespeare’s arithmetician Cassio has a name very similar to Casio, the UK’s leading brand of electronic calculator.
Iago scoffs that Cassio might be good with numbers, but he has no practical understanding of the real world. (This rather harsh stereotype of mathematical people as being abstract thinkers who are out of touch with reality is one that lives on today.)
Shakespeare never used the word ‘mathematician’ in any of his plays, though in Tudor times the two words were often used interchangeably, just as ‘maths’ and ‘arithmetic’ are today – much to the annoyance of many mathematicians.
So what is the difference between maths and arithmetic? If you ask mathematicians this question, they come up with many different answers. Things like ‘being able to logically prove what is true’ and ‘seeing patterns and connections’. What they never say is: ‘knowing your times tables’ or ‘adding up the bill’.
Arithmetic, on the other hand, is entirely about calculations.
Here’s an example to show what I mean:
Pick any whole number (789, say). Now double it and add one. By using a logical proof, a mathematician can say with absolute certainty that the answer will be an odd number, even if they are unable to work out the answer to ‘what is twice 789 add one?’1
On the other hand, an arithmetician can quickly and competently work out that (789 × 2) + 1 = 1,579, without needing a calculator.
The strongest arithmeticians can do much harder calculations, too. They can quickly work out in their head what 4/7 is as a percentage; can multiply 43 × 29 to get the exact answer; and can quickly figure out that in a limited-overs cricket match, if England require 171 runs in 31 overs they’ll need to score at a bit more than five and a half runs per over.
My mother, who left school at 17, was a strong arithmetician, as were many in her generation. That was almost inevitable. A large part of her schooling had been daily practice filling notebooks with page after page of arithmetical exercises. But she knew little about algebra, geometry or doing a formal proof, in the same way that many top mathematicians are hopeless at arithmetic.
There is, however, a huge amount of overlap between arithmetic and mathematics. Many arithmetical techniques and short cuts lead on to deep mathematical ideas, and most of the maths that is studied up until school-leaving age requires an element of arithmetic, even if it’s no more than basic multiplication and addition. Arithmetic and maths are both grounded in logical thinking, and both exploit the ability (and joy) of seeing patterns and connections.
And yet, although arithmetic crops up everywhere, after the age of 16 it is very rarely studied. Almost without exception, public exams beyond 16 allow the use of a calculator, and most people’s arithmetical skills inevitably waste away after GCSE.
A while ago, a friend who runs an engineering company was talking with some final-year engineering undergraduates about a design problem he was working on. ‘We have this pipe that has a cross-sectional area of 4.2 square metres,’ he said, ‘and the water is flowing through at about 2 metres per second, so how much water is flowing through the pipe per second?’ In other words, he was asking them what 4.2 × 2 equals. He was assuming that these bright, numerate students would come back instantly with ‘8.4’ or (since this was only a rough-and-ready estimate) ‘about 8’. To his dismay, all of them took out their calculators.
Calculators have removed the need for us to do difficult arithmetic. And it’s certainly not essential for you to be a strong arithmetician to be able to make good estimates. But it helps.
TEST YOURSELF
Can you quickly estimate the answer to each of these 10 calculations? If you get within (say) 5% of the right answer, you are already a decent estimator. And if you are able to work out exactly the right answers to most of them in your head, that’s a bonus, and you can call yourself an arithmetician.
(a) A meal costs £7.23. You pay £10 in cash. How much change do you get?
(b) Mahatma Gandhi was born in October 1869 and died in January 1948. On his last birthday, how old was he?
(c) A newsagent sells 800 chocolate bars at 70p each. What are his takings?
(d) Kate’s salary is £28,000. Her company gives her a 3% pay rise. What is her new salary?
(e) You drive 144 miles and use 4.5 gallons of petrol. What is your petrol consumption in miles per gallon?
(f) Three customers get a restaurant bill for £86.40. How much does each customer owe?
(g) What is 16% of 25?
(h) In an exam you get 38 marks out of a possible 70. What is that, to the nearest whole percentage?
(i) Calculate 678 × 9.
(j) What is the square root of 810,005 (to the nearest whole number)?
Solutions
BASIC ARITHMETIC
ADDITION AND SUBTRACTION
The classic written methods for arithmetic start at the right-hand (usually the units) column and work to the left. But when it comes to the sort of speedy calculations that are part of back-of-envelope thinking, it generally pays to work from the left instead.
For example, take the sum: 349 + 257.
You were probably taught to work it out starting from the units column at the right. The first step would be:
9 + 7 = 16, write down the 6 and ‘carry’ the 1.2
You then continue working leftwards:
4 + 5 + 1 = 10, write down the 0 and ‘carry’ the 1; 3 + 2 + 1 = 6.
Working this out mentally, however, it is generally more helpful to start with the most significant digits (i.e. the ones on the left) first.
So the calculation 349 + 257 starts with 300 + 200 = 500, then add 40 + 50 = 90, and finally add 7 + 9 = 16. The advantage of working from the left is that the very first step gives you a reasonable estimate of what the answer is going to be (‘it’s going to be 500 or so …’).
A similar idea applies to subtraction. Using the standard written method, working from the right, 742 – 258 requires some ‘borrowing’ (maybe you used different language). Here’s the method my children learned at school:
8 from 2 can’t be done, borrow 10, 12 – 8 = 4,
5 from 3 can’t be done, borrow 10, 13 – 5 = 8, 2 from 6 = 4.
Starting from the left, however, you can read it as 700 – 200 (= 500), then 40 – 50 (so subtract 10 from 500) and, if you want the exact answer, calculate the units 2 – 8 (subtract 6).
MULTIPLICATION AND TIMES TABLES
Calculators may be here to stay, but children are still expected to learn their times tables in primary school, just as they were one hundred years ago.
In the UK, this means learning all multiplications up to 12. In some countries, such as India, it’s not uncommon for this to be pushed to 20, so that some children might, for example, learn the answer to 13 × 17 off by heart.
When it comes to back-of-envelope maths, knowing your tables up to 10 is generally enough.
You may be a little rusty on your times tables. I’m guessing that the very fact that you are reading this means you can probably calculate 3 × 4 in your head, but many adults are out of practice when it comes to some of the harder multiplications. Notorious for tripping people up is 7 × 8, though according to one analysis of over a million calculations using times table that were done online,3 it is 9 × 3 that is answered incorrectly the most often.
Here are a few tips that can be handy when doing multiplications in your head. These apply to the times tables, but are also handy for multiplying larger numbers.
Tip 1
The order of multiplication makes no difference to the answer. For example, 3 × 5 is the same as 5 × 3. One way to convince yourself why this is true is to think of multiplication as counting eggs in a tray.
How many eggs are in the tray above? Three rows of five, or five columns of three? Either way, it comes to 15. What’s powerful about this truism is that you can think of any multiplication as being like counting eggs in a
tray. It means you can be confident that 7,431 × 278 is the same as 278 × 7,431, even if you don’t know what the answer is.
This idea is not just restricted to multiplying two numbers together: 5 × 13 × 2 is the same as 2 × 5 × 13. By rearranging the order of the numbers that you are multiplying, you can often make a calculation easier. In this case, since 2 × 5 = 10, we can arrange 5 × 13 × 2 to become 10 × 13 = 130.
Tip 2
Multiplying by 3 is the same as doubling a number, then adding the number again. Thus, 3 × 12 is the same as 2 × 12 (24), then add another 12.
Tip 3
Multiplying by four is the same as doubling, then doubling again. And to multiply a number by eight, you double it three times.
Tip 4
Instead of multiplying by nine, you can simply multiply your starting number by 10 and then subtract the starting number. For example, 9 × 8 is the same as 10 × 8 (= 80), minus 8 (= 72). Likewise, 9 × 68 is the same as 10 × 68 (680) minus 68 (= 612).
Tip 5
Multiplying by 5 is the same as halving the answer and then multiplying by 10. So, 468 × 5 looks hard. But it is the same as 468 ÷ 2 (= 234) × 10 (= 2,340), which is considerably easier. You can, of course, reverse the order and multiply by 10, then divide by two: 43 × 5 = 430 ÷ 2 = 215.
TEST YOURSELF
Trying working these out in your head (the short cuts mentioned above might help, or use your own method):