These days, keeping up with the news can sometimes feel like a maths exam. We face a constant barrage of figures, whether through national budgets, coronavirus data, hospital waiting lists or football transfer fees. It can be very easy to switch off and ignore all this, but being able to put these numbers into context and understand what they really mean is vital to our role as informed citizens. Here are some mathematical tricks and ideas that can help you make better sense of the world.
The budget: don’t be blinded by big numbers
Jeremy Hunt’s budget will feature bewilderingly large sums of money, comprehension of which is not helped by the fact that the words millions, billions and trillions all sound the same but have vastly different effects. One way to make remote figures more comprehensible is by thinking about national budgets on a per capita basis – even if everyone will not pay the same. Divided among the population of the UK, something costing a million pounds would cost us about 1.5p each. A billion-pound proposal works out at £15 per head. A trillion pounds, the unit by which we measure annual gross domestic product (GDP), or national debt, equates to £15,000 each.
These comparators let us think in a more informed way about the costs of various items. For example, the annual salary bill for 650 MPs (each paid just over £84,000) is about £55m, or approximately 80p per head. It’s not nothing, but it might suggest that cutting these salaries would provide little in the way of direct cost savings overall. By contrast, in January 2022, the most expensive month of the coronavirus test and trace system, just over £3bn was spent (largely on the tests themselves). This makes them about £45 each, which may calibrate your opinion about the costs and benefits of providing Covid tests free at the point of use. Furthermore, when comparing MPs’ salaries with test and trace, it would be wise to bear in mind Parkinson’s law of triviality, which states that the discourse can be unduly dominated by minor issues.
View image in fullscreenJeremy Hunt’s budget will contain a lot of big numbers: knowing that every £1m of spending will cost each member of the UK population 1.5p makes these costs more relatable. Photograph: Tayfun Salcı/ZUMA Press Wire/REX/Shutterstock
Some ballpark figures are just as useful as accurate ones
The budget figures above are only rough partly because there is little to be gained by adding more detail. For instance, using the mid-2021 Office for National Statistics mid-2021 UK population figure of 67,026,292, my million-pound item actually costs us 1.4919518p each. However, this would probably not change your opinions about value for money! Additionally, it would be reasonable to consider the accuracy of that quoted population figure. At best, it might be more than 18 months out of date, but even then the last few digits were questionable, given missing and inaccurate census responses, an uncertain number of births and deaths on the day itself and so on.
Indeed, you should imagine that all figures quoted in the news have some degree of uncertainty associated with them. For instance, we are used to the idea that opinion polls are based on random statistical samples and hence come with a certain margin of error. For this reason, you should not read too much into small changes in vote shares between polls, to avoid building a false narrative around random fluctuations.
Large numbers of small multiplicative changes can combine into a very significant effect
However, the same problems of uncertainty and precision affect figures such as GDP growth, which are again based on sampling and estimation to some extent, and subject to later revision. It would be wise not to overweight the February news stories that the UK had narrowly avoided a recession by reporting zero GDP growth in the fourth quarter of 2022. The quoted figure of zero could as easily be plus or minus a few tenths of a per cent in reality, and feeding it into a binary classification “Are we in recession or not?” ignores that. Rather, it would be better to say that growth is essentially flat, and whether the technical definition of recession is satisfied makes little difference to most people’s everyday lives.
Beware of exponential errors
Another important mathematical concept that became more familiar through the pandemic but is important elsewhere as well, is exponential behaviour. This describes a process that is multiplied by the same amount at every step in time – such as a debt that accrues compound interest at a fixed rate every day. The key thing is that large numbers of small multiplicative changes can combine into a very significant effect. For example, a £100 debt that increased by 1% every day would grow to £3,778 in a year.
It is worth bearing this effect in mind when hearing long-term budget forecasts or cost predictions for projects such as HS2, the high-speed rail line. Estimates of the rate of growth that are consistently wrong in the same direction can combine into very large errors when projected far into the future. Similar effects were seen when some Covid models incorrectly estimated the rate of exponential growth and ended up being exponentially wrong as a result.
Indeed, the effect of the UK pensions triple lock is similar. Since the annual increase is guaranteed to be at least the maximum of the rate of inflation and wage growth, pensions will generally grow faster than either prices or wages in the long run. In fact, it’s plausible that they will roughly grow exponentially faster than both, which might make the triple lock financially challenging in the long run, albeit politically sensitive to remove.
Extreme events are important
Another vital concept is the effect of randomness, and of extreme effects in particular. Consider building houses near a river. Normal behaviour, such as the fact that the river is usually not in flood, isn’t the most important thing – what really matters is the frequency of severe flooding. In that sense, to consider processes such as changes in climate in terms of average values can be misleading – a 2C change in temperature on any particular day probably wouldn’t seem too dramatic. The danger is that these climate changes will increase the frequency and severity of extreme events. Buildings designed for a level of flooding that occurs, say, once every 100 years will probably not be manageable if it happens every five years instead, and if even more severe events become feasible.
In that sense, to understand the news we need a better understanding of extremes, and to understand that events such as pandemics or market crashes can be highly improbable but have very significant consequences when they do occur. This leads on to another mathematical point: that not all modelling errors should be treated the same. In technical language, we would say that the loss can be very asymmetric.
View image in fullscreenWhile there is uncertainty about the potential for bird flu to infect humans, investing in new vaccines now could avoid the ruinous cost of another pandemic. Photograph: Ian Forsyth/Getty Images
For example, early in the pandemic the UK government developed the Nightingale hospitals, the majority of which were actually never used for patients. Of course, this overestimate of required capacity had a financial cost associated with it. But consider the opposite scenario, where the dangers had been underestimated, such resources had not been created and existing hospitals had exceeded even their surge capacity. As we saw in India’s Delta variant wave of spring 2021, such a collapse in healthcare would have caused a catastrophic number of deaths.
Similarly, the result of a false positive Covid test (someone wrongly isolating and potentially losing income) and a false negative (someone wrongly being given a clean bill of health and potentially infecting more people) were very different. In that sense, while we seek to balance the overall effect of different kinds of errors, it is important to remember that the outcomes associated with each may not be the same.
As a result, it can be wise and necessary sometimes for governments to follow strategies that may well seem wrong in hindsight. For instance, it feels prudent to invest now in the capacity to manufacture H5N1 vaccines even if a bird flu pandemic in humans may well not occur, because we know that the potential outcome would be so severe if it did happen.
Expected goals
Mathematical ideas can help in understanding sport. Recent years have seen a huge growth in teams using analytic methods to improve their performances, most famously in the Moneyball story of the Oakland Athletics baseball team in the US. In the UK, Brentford football club has used data science techniques to find and develop players and to adapt their tactics, and the team now lie in the top half of the English Premier League, despite having the smallest wage bill.
However, there are limits to this data-driven approach to sport. In particular, it is valuable to understand the uses and limitations of “expected goals”, one of the most visible new statistics. Expected goals are calculated using a huge database of past matches, analysing the outcomes of shots from different positions on the field and in different circumstances. For example, if a shot from the corner of the penalty area gives a goal 10% of the time, then creating such an opportunity gives a team 0.1 of an expected goal. These fractions of goals are added up over the course of the game.
The word “expected” here is mathematical terminology that refers to an idea of a long-run average if the shot was taken many times. But it’s important to understand that there is no guarantee of the result, even when these shots are made. Much of the joy of sport is in its intrinsic unpredictability – that Saudi Arabia could beat the eventual winners Argentina 2-1 in the World Cup, despite “losing” by 2.29 expected goals to 0.15. Long shots do sometimes go in – sometimes a goalkeeper has a brilliant day. Equations can help us do better on average, but sport will always have unexpected events – otherwise, why watch it?
Oliver Johnson is professor of information theory and director of the Institute for Statistical Science in the school of mathematics at Bristol University
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Numbercrunch: A Mathematician’s Toolkit for Making Sense of Your World by Oliver Johnson is published by Bonnier Books (£22). To support the Guardian and Observer order your copy at guardianbookshop.com. Delivery charges may apply
