Rob Eastaway - Maths on the Back of an Envelope - Clever ways to (roughly) calculate anything

And, indeed, there are times when even saying ‘roughly’ isn’t enough.

Every month, the Office for National Statistics publishes the latest UK unemployment figures. Of course this is always newsworthy – a move up or down in unemployment is a good indicator of how the economy is doing, and everyone can relate to it. In September 2018, the Office announced that UK unemployment had fallen by 55,000 from the previous month to 1,360,000.

The problem, however, is that the figures published aren’t very reliable – and the ONS know this. When they announced those unemployment figures in 2018, they also added the detail that they had 95% confidence that this figure was correct to within 69,000 . In other words, unemployment had fallen by 55,000 plus or minus 69,000. This means unemployment might actually have gone down by as many as 124,000, or it might have gone up by as many as 14,000. And, of course, if the latter turned out to be the correct figure, it would have been a completely different news story.

When the margin of error is larger than the figure you are quoting, there’s barely any justification in quoting the statistic at all, let alone to more than one significant figure. The best they can say is: ‘Unemployment probably fell slightly last month, perhaps by about 50,000.’

It’s another example of how a rounded, less precise figure often gives a fairer impression of the true situation than a precise figure would.

We’ve already seen that the statistics should really carry an indication of how much of a margin of error we should attach to them.

An understanding of the margins of error is even more important when it comes to making predictions and forecasts.

Many of the numbers quoted in the news are predictions: house prices next year, tomorrow’s rainfall, the Chancellor’s forecast of economic growth, the number of people who will be travelling by train … all of these are numbers that have come from somebody feeding numbers into a spreadsheet (or something more advanced) to represent this mathematically, in what is usually known as a mathematical model of the future.

In any model like this, there will be ‘inputs’ (such as prices, number of customers) and ‘outputs’ that are the things you want to predict (profits, for example).

But sometimes a small change in one input variable can have a surprisingly large effect on the number that comes out at the far end.

The link between the price of something and the profit it makes is a good example of this.

Imagine that last year you ran a face-painting stall for three hours at a fair. You paid £50 for the hire of the stand, but the cost of materials was almost zero. You charged £5 to paint a face, and you can paint a face in 15 minutes, so you did 12 faces in your three hours, and made:

£60 income – £50 costs = £10 profit.

There was a long queue last year and you were unable to meet the demand, so this year you increase your charge from £5 to £6. That’s an increase of 20%. Your revenue this year is £6 × 12 = £72, and your profit climbs to:

£72 income – £50 costs = £22 profit.

So, a 20% increase in price means that your profit has more than doubled. In other words, your profit is extremely sensitive to the price. Small percentage increases in the price lead to much larger percentage increases in the profit.

It’s a simplistic example, but it shows that increasing one thing by 10% doesn’t mean that everything else increases by 10% as a result. 7

There are some situations when a small change in the value assigned to one of the ‘inputs’ has an effect that grows dramatically as time elapses.

Take chickenpox, for example. It’s an unpleasant disease but rarely a dangerous one so long as you get it when you are young. Most children catch chickenpox at some point unless they have been vaccinated against it, because it is highly infectious. A child infected with chickenpox might typically pass it on to 10 other children during the contagious phase, and those newly infected children might themselves infect 10 more children, meaning there are now 100 cases. If those hundred infected children pass it on to 10 children each, within weeks the original child has infected 1,000 others.

In their early stages, infections spread ‘exponentially’. There is some sophisticated maths that is used to model this, but to illustrate the point let’s pretend that in its early stages, chickenpox just spreads in discrete batches of 10 infections passed on at the end of each week. In other words:

N = 10 T,

where N is the number of people infected and T is the number of infection periods (weeks) so far.

After one week: N = 10 1= 10.

After two weeks: N = 10 2= 100.

After three weeks: N = 10 3= 1,000,

and so on.

What if we increase the rate of infection by 20% to N = 12, so that now each child infects 12 others instead of 10? (Such an increase might happen if children are in bigger classes in school or have more playdates, for example.)

After one week, the number of children infected is 12 rather than 10, just a 20% increase. However, after three weeks, N = 12 3= 1,728, which is heading towards double what it was for N = 10 at this stage. And this margin continues to grow as time goes on.

Sometimes the relationship between the numbers you feed into a model and the forecasts that come out are not so direct. There are many situations where the factors involved are inter-connected and extremely complex.

Climate change is perhaps the most important of these. Across the world, there are scientists attempting to model the impact that rising temperatures will have on sea levels, climate, harvests and animal populations. There is an overwhelming consensus that (unless human behaviour changes) global temperatures will rise, but the mathematical models produce a wide range of possible outcomes depending on how you set the assumptions. Despite overall warming, winters in some countries might become colder. Harvests may increase or decrease. The overall impact could be relatively benign or catastrophic. We can guess, we can use our judgement, but we can’t be certain.

In 1952, the science-fiction author Raymond Bradbury wrote a short story called ‘A Sound of Thunder’ in which a time-traveller transported back to the time of the dinosaurs accidentally kills a tiny butterfly, and this apparently innocuous incident has knock-on effects that turn out to have changed the modern world they return to. A couple of decades later, the mathematician Edward Lorenz is thought to have been referencing this story when he coined the phrase ‘the butterfly effect’ as a way to describe the unpredictable and potentially massive impact that small changes in the starting situation can have on what follows.

These butterfly effects are everywhere, and they make confident long-term predictions of any kind of climate change (including political and economic climate) extremely difficult.

In 1995, Stephen Churchill, a 19-year-old from Wiltshire, became the first person to die from Variant Creutzfeldt–Jakob disease (or vCJD). This horrific illness, a rapidly progressing degeneration of the brain, was related to BSE, more commonly known as ‘Mad Cow Disease’, and caused by eating contaminated beef.

As more victims of vCJD emerged over the following months, health scientists began to make forecasts about how big this epidemic would become. At a minimum, they reckoned there would be at least 100 victims. But, at worst, they predicted as many as 500,000 might die – a number of truly nightmare proportions. 8