Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything
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Two and a quarter pounds of jam
weigh about a kilogram
A metre measures three foot three,
it’s longer than a yard, you see.
BACK-OF-ENVELOPE CONVERSIONS
If it’s too much of a faff to use the accurate ratios above, you can use a Zequals-style approach to give you conversions that will suffice in most situations. And conveniently, all of the common rough-and-ready conversions only require doubling or halving.
Accurate conversion Rough conversion Example
Litres to pints × 7/4 (or 1.75) Double 10 litres ~ 20 pints
Litres to (UK) gallons × 7/32 Quarter 20 litres ~ 5 gallons
Kilometres to miles × 5/8 Halve 200 km ~ 100 miles
Metres per second to miles per hour × 2¼ Double 10 m/s ~ 20 mph
Centimetres to inches × 2/5 Halve 6 cm ~ 3 inches
Metres to yards × 13/12 (add 1/12) Equal 70 metres ~ 70 yards
Kilograms to pounds × 2¼ Double 10 kg ~ 20 pounds
Celsius to Fahrenheit × 9/5 and add 32 Double and add 30 20 °C ~ 70 °F
There are of course other more obscure imperial measurements that you might encounter, such as acres (land), furlongs (in horse racing) and fluid ounces (cooking), but these rarely crop up in everyday encounters and you’re unlikely to have to deal with converting them on the hoof.
TEST YOURSELF
Do rough conversions of the following in your head:
(a) 70 miles in kilometres.
(b) 40 kilograms in pounds.
(c) 150 metres in yards.
(d) 100 kilometres in miles.
(e) 25 °C in Fahrenheit.
(f) 10 stones in kilograms (one stone is 14 pounds).
Solutions
A QUIRKY METHOD FOR MILE–KILOMETRE CONVERSION
About 800 years ago, a mathematician called Leonardo of Pisa (who was nicknamed Fibonacci) wrote about a curious sequence of numbers. Starting with 0 and 1, the sequence goes as follows:
0 1 1 2 3 5 8 13 21 34 55
Each number in the sequence is obtained by adding the previous two terms. So, after 55, the next term will be 34 + 55 = 89.
Now here is the remarkable thing. From the number 3 onwards, if you take any two consecutive terms in the Fibonacci sequence, their ratio is very close to 1.6. For example, 13 ÷ 8 = 1.625, and 34 ÷ 21 = 1.619. This isn’t just a fluke; it turns out that as you go further along the sequence, the ratio of successive terms in the Fibonacci sequence gets closer and closer to a number known as the ‘Golden Ratio’, which is roughly 1.618.
The coincidence is that the Golden Ratio is very close to 1.609, which is the ratio of miles to kilometres. So if you want to convert 13 miles to kilometres, then, just by glancing at the Fibonacci sequence, you can estimate that the answer is going to be about 21 km, and you’ll be correct to within 1%.
It works in reverse, too. Travelling around Europe, you spot that your destination is 34 kilometres. ‘That’s 21 miles,’ you can state, with remarkable accuracy.
ESTIMATION AND STATISTICS
AVERAGES AND UNCERTAINTY
The word ‘average’ is used in everyday speak to mean ‘typical’ or ‘somebody in the middle’. In many situations it’s fine to use this general word, but it’s worth being reminded that there are three different averages that are commonly used.
The mean is the most commonly used average. It’s found by adding up all the values or measurements, and dividing by the number of items you are measuring. The mean is what’s used when referring to average adult height, batting averages in cricket and also people’s income.
The median is the middle value, if you were to line up all the data from smallest to largest.
The mode is the data value that crops up the most often. For example, the ‘modal’ shoe size for an adult woman in the UK is 6.
We’ve seen earlier that most statistics have an element of uncertainty, so that statistics that you are presented with might be an over- or under-statement of the true figure.
The cause of this ‘error’ will be one of two things: either the method you use to measure the statistic isn’t reliable (a weighing scale that gives different readings each time, for example), or the thing that you are measuring tends to vary (for example, if you are looking to find the height of a typical person).
Either way, the ‘true’ figure is going to be somewhere on a spread of possible values. Most often this spread (more formally known as a distribution) will look something like this:
This shape is known as the Normal distribution (so called for the banal reason that it’s not abnormal), though it’s often called a bell curve (because it is shaped like a bell). Points in the central, higher region of the curve represent values or readings that arise most frequently, while the lower regions left and right are the more extreme and less frequent values. The heights of children in a class, the time it takes for daffodils to bloom and many other everyday phenomena follow this sort of pattern. It’s handy that this spread is symmetrical, because it means the average (mean) value is right in the middle. In distributions like this, it’s just as likely that a statistic or measurement will be higher than the quoted figure as lower, so it’s fine to call the highest point the ‘average’.
However, not all statistics follow this pattern. For example, if you were monitoring the amount of time that people spend in a toilet cubicle at a rock concert,7 the distribution would look something like this:
Most people might spend three or four minutes, but a few take 10 minutes or more, and one or two will exceed 20 minutes. This distribution is known as a ‘log normal’ (cue lavatorial jokes). Looking at this graph, you’d say that the typical time spent is two to three minutes, but the ‘mean’ (the total time spent divided by the number of people) is going to be higher than that because of the few extremes.
Adult income, similarly, has a skewed distribution, with many people clustered in the range £20,000–£30,000, but with a long tail to the right that includes a few on multi-million salaries. So, while the middle (median) income is around £25,000, the ‘mean’ income will be to the right of the peak. The majority of people earn less than the average (mean) income – which can make the choice of which average to quote very political.
There are two other distributions that are also quite common. If you take a random sample of tweets on Twitter, the number of ‘likes’ for individual tweets will look like this:
The most common number of likes is zero, followed by 1, then 2, each becoming less likely. This is known as an exponential distribution.
Finally, if you were to pick a more economically developed country and gather together everyone between the age of 20 and 50, the distribution of their ages will typically look like this:
Yes, there will be bumps from year to year, and there might be a slight trend going up or down, but broadly speaking it’s a flat line. If you pick somebody at random from this group, the chance of the person you choose being 23 will be about the same as the chance of them being 33.
You’ll get a similar pattern with, for example, the time you expect to wait for a London Underground train if you turn up at a station at random. Trains might be spaced every couple of minutes, and you are as likely to arrive on the platform at the beginning of the two-minute gap (just after a train left) as you are to arrive at the end of the gap (as a train is pulling in). And, of course, you could arrive anywhere in between.
Knowing which of these distributions a particular statistic belongs to can be helpful in making estimates.
WORKING OUT PROBABILITY
‘Probability’ is the formal way of saying ‘the chance of a something happening’. Probabilities range from absolutely certain (i.e. it will happen 100% of the time, such as the sun rising tomorrow) and impossible (0% of the time), and can be anything in between. The chance of flipping a head on a fair coin, for example, is 50%, while the chance of picking a Heart from a regular pack of cards is 25%, since a pack is divided equally into four suits.
Although percentages are the most common way of expressing everyday probabilities, there are other ways of saying the same thing. The chance of picking a Heart can be represented as:
A fraction (¼, or ‘1 in 4’).
As a decimal (0.25).
As bookmakers’ odds (selecting a Heart is 3:1 against, so that for every time you pick a Heart, you can expect to pick a card that isn’t a Heart three times).
But some probabilities can’t just be stated by simple inspection. Sometimes, to find (or estimate) a probability, you need to take a different approach. For example:
To estimate the probability that a car will be yellow, you could do a survey. Count the next 100 cars that you pass on the open road. If one of them is yellow, that suggests the chance of a car picked at random being yellow is about 1 in 100. The bigger the sample, the more accurate your estimate of the probability will be.
To estimate the chance that it will be warmer than 25 °C in Paris next 1 July, you can use history. If it has been warmer than 25 degrees on 1 July in seven of the last 10 summers, in the absence of a weather forecast it’s a reasonable guess that there’s a 7/10 (70%) chance that it will exceed that temperature this year too.
Sometimes we just have an instinct for the probability of something happening based on a hunch. When answering the quiz question: ‘Who was older when they became US President: George W. Bush or Bill Clinton?’ you will have a gut feeling of what the answer is, to which you could put a value: ‘I reckon it’s 90% Bill Clinton’ or ‘a bit more than 50:50 it’s Bush’. If you have absolutely no idea, then the chance you will get the right answer out of a choice of two is exactly 50%.
The more vague the probability that you are using in your calculations, the less reliable your estimates based on it will be. So, while you can work out precisely the odds on you getting a ‘flush’ of five cards in poker, you can only estimate very approximately the chance that your team will win a pub quiz.
BACK-OF-ENVELOPE SURVEYS
Many of the statistics that are fed to us are the result of surveys. If we’re told that 7% of the public plan to vote for the Green Party at the next election, or that 65% of children spend over three hours a day at home in front of screens, those figures aren’t based on a census of the whole population, but on a sample of perhaps 1,000 people, chosen carefully to be ‘representative’ of the population by age, gender, social background and so on. Thanks to the large sample, the pollsters and market researchers can give us a figure that they know is quite reliable.
But surveys don’t have to be restricted to the professionals. You can do your own back-of-envelope versions. The experts would pull their hair out if you conducted a sample based on, say, 10 people, but even a tiny sample can begin to give you a sense of the bigger picture.
Several years ago, I was at a committee meeting where concern was expressed at the falling membership of a national body that I was working with. There was a proposal to hire some consultants to help draft a survey that could be sent out to all of the several thousand members, canvassing their views.
I had an alternative suggestion. We needed information quickly. Why didn’t the 10 of us around the table each take a couple of copies of a short questionnaire, and the next time we met a potential member over the coming week or two, ask them to fill it in. I reckoned that even a handful of responses would give us a huge pointer to what the big issues were. I was voted down, but I decided to do my guerrilla research anyway. Out of five people that I talked to, two said that the reason they were not members was that they had no time to enjoy the benefits, and three said they had once been members but now met their needs using free alternatives that were available via social media.
With that tiny sample, it would be bogus to claim with confidence that ‘40% of the target group no longer have time to enjoy the benefits of membership’ or that ‘60% now get their resources elsewhere’. But, in truth, we didn’t need to know the results to the nearest 10%. If the result of the mini survey had been 90% and 30%, we would still have come to the same conclusion, which was that new threats had emerged, from time pressure and from social media, and that these had to be addressed. (That national body never did get round to its big survey – but it did make some positive changes, including engaging in social media.)
Back-of-envelope surveys are nothing special. We all do them, all the time. We ask a few friends which plumber they’d recommend, or what the going rate is for the tooth fairy when a child loses a tooth. A survey of three people will produce a result with a huge margin of error. And yet, in the end, we can still glean valuable insights, and make better decisions, if our survey tells us that 67% of the public (OK, in truth it’s two out of three of our friends) say that in their house the tooth fairy pays £1.
Of course, you should do big, statistically rigorous and representative surveys when you can, and when accuracy is important. But when you don’t have the time or the money to do it, don’t rule out the value of un-rigorous, biased, back-of-the-envelope alternatives – as long as you remember not to set too much store by the numbers in the results.
HOW LONG WILL WE BE IN THIS QUEUE?
It’s October half-term. I’m at Legoland theme park. Again. The kids are desperate to go on the ‘Pirate Falls’ ride. My heart sinks when I see the sign saying the wait is currently one hour. That will be one hour shuffling slowly along the queue, with the prospect of a five-minute boat ride and a soaking at the far end.
Luckily, the queue gives good views of people setting off on the ride, so this is a chance to check out if Legoland’s prediction of one hour is right. If it’s really going to take that long, we’ll go and find something else to do.
We decide to do a survey. We watch the boats setting off at the start of the ride, and over a period of five minutes we count how many people have gone past. Some boats have four people in them (Hooray, that will deplete the queue!), a couple have none (Boo! What a waste of a good boat).
Over five minutes we count 36 people, so we work out that the average throughput is about seven people per minute. Then we estimate the length of the queue – about 150 people.
One hundred and fifty people at seven people per minute – 150 divided by 7: that’s about 20. Clearly Legoland’s prediction of one hour is massively wrong; it will be more like 20 minutes. This cheers me up no end, though it turns out that there’s a factor I hadn’t allowed for: Q-bots, which allow premium customers to go straight to the front of the queue through a separate entrance. It turns out that Q-Bots slow the queue down by about 25%, so it’s nearer 25 minutes before we get to the front of the queue.
Still, back-of-envelope thinking ensured we were better informed than those who trusted the waiting-time sign, and it also helped occupy us for a few minutes. I was happy. Until we got to the end of the ride, when we left the boat looking like drowned rats.
THE CHANCE OF MULTIPLE EVENTS
You can work out the probability of two or more independent events happening by multiplying their probabilities together.
It’s often convenient to work out the probability of more than one event happening by using fractions – indeed, this is perhaps the most important application of the methods that you learned at school for adding and multiplying fractions. For example, the chance of getting two sixes when rolling two dice is worked out as:
That’s a lot easier than working out 16.7% × 16.7%.
The events are independent if one of the events is in no way influenced by the other: rolling a dice and flipping a coin are independent events, but living in Wales and having the surname Jones are not. One in 20 people in the UK live in Wales, and around 1 in 100 in the UK have the surname Jones – but the chance that the next winner of the UK Lottery is a person who lives in Wales and has the surname Jones is not 1/20 × 1/100 (= 1 in 2,000). The proportion of Welsh residents called Jones is around 1 in 17,8 so the chance of a Lottery winner being Welsh and called Jones is going to be roughly
Knowing whether or not
two events are independent is partly common sense and partly experience, but for back-of-envelope purposes, as a starting point you can generally treat two events that don’t have an obvious connection as being independent.
For example, suppose you’re running a bit late, and need to catch a bus to the station where you want to catch a train. Suppose about four-fifths (80%) of trains leave your local station on time, and your hunch tells you that the chance you’ll have to wait more than five minutes for a bus (and hence be late for the train) is about one in two (50%). Now, the chance of the bus being late and the train being late aren’t entirely independent: bad weather would affect both of them, for example. But that connection probably isn’t significant. The chance that you’ll have to wait more than five minutes and that the train will be on time is therefore going to be about 4/5 × 1/2 which is 2/5, or 40%.
SPOTTING TRENDS
Much of our modern life is underpinned by statistics. They dominate our news, and we use them to form opinions, make judgements and, most importantly, make decisions. It’s a statistician’s job to wade through data and to spot important patterns and connections. And in an era where ‘big data’ is used to help advertisers, political parties and other institutions to understand our behaviour in frightening detail so that they can understand and perhaps influence us, it should come as little surprise that the best statisticians can earn huge salaries.
The maths involved in statistics can get quite advanced. If you’ve got a set of data (such as the fictitious points in the graph below) and are looking for the straight line that most tightly passes through it (the so-called ‘best fit’), there are sophisticated mathematical techniques you can use to find it.9 But in many cases, the human eye is good enough. A bit like in a spot-the-ball competition, I’ve used my judgement, and a hunch, to draw a straight line through the points below. It suggests there’s a slight trend upwards. Your line might be different, but it’s unlikely to be that different.