TeaThese days, keeping up with the news can sometimes feel like a math test. We are constantly faced with a flood of data, whether through the national budget, coronavirus data, hospital waiting lists or football transfer fees. It can be all too easy to brush off and ignore this, but being able to put these numbers in 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 better understand the world.
Budget: don’t be blinded by the big number
jeremy hunt Budget That would appear to be a surprisingly large sum of money, the understanding of which is not helped by the fact that the words millions, billions, and trillions sound the same, but have very different implications. One way to make the outlandish figure more perceptible is to think of the national budget on a per capita basis – even if not everyone pays the same. Divided amongst the UK population, something costing one million pounds would cost us around 1.5p. The £1 billion proposal works out to £15 per head. One trillion pounds, the unit by which we measure annual gross domestic product (GDP), or national debt, equals £15,000 each.
These comparators allow us to think about the costs of different items in a more informed way. For example, the annual wage bill for 650 MPs (paid just over £84,000 each) is around £55m, or around 80p per person. It’s nothing, but it may suggest that these pay cuts will provide little in the way of direct cost savings overall. In contrast, January 2022 is the costliest month of the coronavirus test and trace system, just over £3bn was spent (mostly on the tests themselves). This makes them around £45 each, which may test your opinion about the costs and benefits of providing Covid testing for free at the point of use. Also, while checking and comparing the salaries of MPs with that of Trace, it would be wise to keep this in mind Parkinson’s law of insignificanceWhich states that discourse on minor issues can be unnecessarily dominated.
Some ballpark figures are only as useful as they are accurate
The above budget figures are only rough in part because there is little benefit from adding more detail. For example, using the middle of 2021 national statistics office With a UK population of 67,026,292 in mid-2021, my million-pound item actually cost each of us 1.4919518p. However, this probably won’t change your opinion about value for money! In addition, it would be appropriate to consider the accuracy of that quoted population figure. At most, it could be more than 18 months old, but even then the last few digits were questionable, given missing and incorrect census responses, uncertain numbers of births and deaths on the same day, and so on.
In fact, you should imagine that all figures quoted in the news have some degree of uncertainty attached to them. For example, we are used to the idea that opinion polls are based on random statistical samples and therefore come with a certain margin of error. For this reason, you shouldn’t read too much into small changes in vote share between elections, to avoid building a false narrative around random fluctuations.
However, the same problems of uncertainty and precision affect statistics such as GDP growth, which are again based on sampling and estimation to some extent, and are subject to subsequent revision. It would be wise not to give too much weight to UK February news narrow escape from recession By reporting zero GDP growth in the fourth quarter of 2022. The quoted figure of zero could easily be plus or minus a few tenths of a percent in reality, and feed this into a binary classification of “are we in a recession or not?” ignores him. Rather, it would be better to say that growth is essentially flat, and whether or not the technical definition of a recession is satisfied makes little difference to most people’s daily lives.
Beware of Exponential Errors
Another important mathematical concept that became more familiar through epistemology, but is also important elsewhere, is exponential behavior. It describes a process that is multiplied by the same amount at each step in time – such as a loan that accrues compound interest at a fixed rate every day. The key point is that a large number of small qualitative changes can add up to a very significant effect. For example, a £100 loan that grows at 1% per day will grow to £3,778 in one year.
It is worth keeping this effect in mind when listening to long-term budget forecasts or cost forecasts for projects. HS2, High-speed rail line. Growth rate estimates that are consistently wrong in the same direction may add up to much larger errors in the future. Similar effects were observed when some covid models wrongly predicted the rate of exponential growth and got exponentially wrong as a result.
In fact, the effect of UK Pension Triple Lock are equal. Since the annual increase is guaranteed to be at least the rate of inflation and the maximum of wage growth, pensions will generally grow faster than prices or wages in the long run. In fact, it is plausible that they will grow roughly faster than either, which may make triple lock economically challenging in the long run, though. politically sensitive to remove,
extreme events are important
Another important concept is the effect of randomness, and in particular the extreme effect. Consider building a home near a river. General behaviour, such as the fact that the river is not usually in flood, is not the most important thing – what really matters is the frequency of severe floods. In this sense, it can be misleading to consider processes such as climate change in terms of mean values – a 2C change in temperature on a particular day might not seem very 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 occurs every five years instead, and even more severe events become possible. .
In that sense, we need a better understanding of extremes to understand the news, and to understand that events such as pandemics or market crashes may be highly improbable, but when they do occur they have very important consequences. . This leads to another mathematical point: that not all modeling errors should be treated equally. In technical language we would say that the loss can be very asymmetric.
For example, at the start of the pandemic the UK government developed Nightingale Hospitals. most of which was never actually used for patients. Of course, this overestimation of required capacity had a financial cost associated with it. But consider the opposite scenario, where risks were underestimated, such resources were not built in, and existing hospitals exceeded their growth potential. As we saw in India’s delta variant wave of spring 2021, such a drop in healthcare would have resulted in a catastrophic number of deaths.
Similarly, a false positive Covid test (someone being incorrectly isolated and potentially losing income) and a false negative (someone being health billed incorrectly and potentially infecting more people doing) had a very different result. In that sense, while we seek to balance the overall impact of the different types of errors, it is important to remember that the consequences associated with each may not be the same.
As a result, it can sometimes be wise and necessary for governments to pursue strategies that may seem wrong in hindsight. For example, it seems prudent to invest now in the capacity to manufacture H5N1 vaccines, even though a bird flu A human pandemic may not occur, because we know that if it did, the potential consequences would be so severe.
Mathematical ideas can help to understand the game. Recent years have seen a huge increase in teams using analytical methods to improve their performance, most famously moneyball The story of the Oakland Athletics baseball team in America. In the UK, Brentford Football Club has used data science techniques to find and develop players and optimize their strategy, and the team is now in the top half of the English Premier League despite having the lowest wage bill.
However, there are limits to this data-driven approach to sports. In particular, it is important 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, analyzing different positions on the field and the results of shots in different situations. For example, if a shot from the corner of the penalty area scores a goal 10% of the time, creating such a chance gives the team 0.1 of the expected goal. These fractions of goals are added up during the game.
The word “expected” here is mathematical terminology that refers to the idea of running averages over a long period of time if the shot is taken multiple times. But it is important to understand that there is no guarantee of the result, even if these shots are made. Much of the joy of the game lies in its intrinsic unpredictability – that can beat saudi arabia Despite “losing” to eventual World Cup winners Argentina 2–1, 2.29 expected goals of 0.15. Long shots sometimes go in – sometimes the goalkeeper has a great day. Equations can help us do better than average, but there will always be unexpected events at play – otherwise, why watch it?
is oliver johnson Professor of Information theory and Director of the Institute for Statistical Science School of Mathematics at the University of Bristol
NumberCrunch: A Mathematician’s Toolkit for Making Sense of Your World is published by Bonnier Books (£22) by Oliver Johnson. to endorse Guardian And Observer order your copy at guardianbookshop.com, delivery charges may apply