Numb and Number E-Book

William Hartston

• Herausgeber: Atlantic Books
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 325

Veröffentlichungsjahr: 2020

Beschreibung

'A wise, witty and insightful guide to clear thinking amid a deluge of percentages and probabilities.' Ian Stewart Like it or not, our lives are dominated by mathematics. Our daily diet of news regales us with statistical forecasts, opinion polls, risk assessments, inflation figures, weather and climate predictions and all sorts of political decisions and advice backed up by supposedly accurate numbers. Most of us do not even pause and question such figures even to ask what they really mean and whether they raise more questions than they answer. In this simple guide for anyone numbed by numbers, William Hartston reveals with clarity and humour why the figures being flung at us may not tell the whole story. Along the way he explains commonly misused mathematical terms, solves everyday mathematical problems and shows how to steer a safe path through the minefield of mathematics that surrounds us.

Leseprobe

By the same author

How to Cheat at Chess

The Penguin Book of Chess Openings

Soft Pawn

The Ultimate Irrelevant Encyclopedia

The Kings of Chess

Chess – The Making of the Musical

The Drunken Goldfish and Other Irrelevant Scientific Research

How was it for you, Professor?

The Guinness Book of Chess Grandmasters

Teach Yourself Chess

Teach Yourself Better Chess

The Book of Numbers: The Ultimate Compendium of Facts About Figures

Mr Hartston’s Most Excellent Encyclopedia of Useless Information

Forgotten Treasures: A Collection of Well-Loved Poetry (Vols 1, 2 and 3)

The Things That Nobody Knows

Even More Things That Nobody Knows

The Bumper Book of Things That Nobody Knows

Sloths

A Brief History of Puzzles

First published in Great Britain in 2020 by Atlantic Books,

an imprint of Atlantic Books Ltd.

Copyright © William Hartston, 2020

The moral right of William Hartston to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

1 2 3 4 5 6 7 8 9

A CIP catalogue record for this book is available from the British Library.

Hardback ISBN:  978-1-83895-084-2

E-book ISBN:      978-1-83895-086-6

Paperback ISBN: 978-1-83895-085-9

Printed in Great Britain

Atlantic Books

An Imprint of Atlantic Books Ltd Ormond House

26–27 Boswell Street

London

WC1N 3JZ

www.atlantic-books.co.uk

Do not worry about your difficulties in Mathematics.

I can assure you mine are still greater.

(Albert Einstein, 1943, in a letter replying to a little girl

who was having problems with maths at school)

Contents

Introduction

1. The Number of Our Days

How life expectancy confounds our expectations

2. Surveying the Scene

How to mislead with opinion polls

3. Risk and Behaviour

Our illogical attitudes towards risk

4. The Mathematics of Sport

The randomness of winning and losing

5. Saved You!

How governments try to look good

6. Numbers Large and Small

How huge and tiny numbers confuse us

7. The Insignificance of Significance

The misleading language of statisticians

8. Cause and Effect

Common logical confusions

9. Percentages and More Misleading Mathematics

More natural mistakes

10. Chaotic Butterflies

The mathematics of chaos, catastrophe and complexity

11. Torpedoes, Toilets and True Love

A difficult problem with many applications

12. Formula Milking

Unbelievable formulae in newspapers

13. Monkey Maths

An evolutionary perspective on numeracy

14. Pandemic Pandemonium

The world’s reaction to coronavirus

Bibliography

Acknowledgements

Index

‘It has become almost a cliché to remark that nobody boasts of ignorance of literature, but it is socially acceptable to boast ignorance of science and proudly claim incompetence in mathematics.’

(Richard Dawkins, BBC Richard Dimbleby Lecture, 1996)

One morning in December 2019, amid the deluge of opinion polls and electoral statistics in the British newspapers, TV current affairs programmes and my email inbox, I was particularly intrigued by two news reports and one press release.

The press release informed me that 10 million people in the UK suffer from headaches regularly, while the news stories carried the information that 37% of children in Croydon live in relative poverty and that exports from China to the United States in November fell 23% from a year earlier.

We live in a world in which the news has become increasingly dominated by data and numbers, but all too often they are flung at us with insufficient information to interpret them properly. The figures may cause some concern for people with headaches, for child welfare services in Croydon and for Chinese exporters, but most of us, including those groups, rarely stop to think about what they really mean and how they were calculated. We numbly let the numbers wash over us and do not pause to consider such questions as the following:

What do they mean by ‘suffer from headaches regularly’? Do they mean ‘frequently’? Once a year is regular but it is not frequent, as headaches go. This reminds me of a sign I have occasionally seen at railway stations advertising a ‘regular service to London’. One train a week may be regular, but that’s hardly a good advert for the service. And if those headaches are suffered frequently, does this mean every day, or every morning after a heavy drinking session, or whenever work piles up, or what?

And what about those Croydon children living in relative poverty? What is ‘relative’ meant to mean? Surely the 37% of children who are poorest are relatively poorer than the 63% who are richer than them? And are they poor relative to other children in Croydon or are we comparing them with the rest of the country? Without defining what is meant by ‘relative poverty’, giving a precise figure such as 37% is meaningless.

Finally, what should we make of that drop in exports from China to the US? When they say ‘fell 23% from a year earlier’ does that mean comparing the monthly figures for November 2019 and 2018, or comparing the figures for the whole year ending in those months? Also, there have been considerable fluctuations in exchange rate between the US dollar and Chinese yen over the year. Do we get the same 23% result if we compare the figures in Chinese currency? And what about exports from the US to China?

While we are on the subject, we should always pause to consider what it really means when one year’s figures, whether they are measures of trade, profits, turnover or anything else, are compared with those of the previous year. Were that year’s figures particularly good or very disappointing? Was this year’s 23% fall an expected counterbalance to last year’s rise, or did it just continue, or even accentuate, a path that had been declining for some years?

Most of us have neither the time nor the inclination to seek the answers to such questions but we should at least be aware that the figures flung at us may not tell the whole story. Particularly at election times, politicians, advertisers and others who seek to influence us carefully select the numbers that support their arguments.

As Lewis Carroll put it, ‘Long and painful experience has taught me one great principle in managing business for other people: viz., If you want to inspire confidence, give plenty of statistics. It does not matter that they should be accurate, or even intelligible, as long as there is enough of them’ (Lewis Carroll, Three Years in a Curatorship by One Whom It Has Tried, 1886). When people detect a deluge of numbers about to fall on them, they all too often put up their numbrellas to shelter from the figures and formulae and leave them falling unheeded to the ground. This book is an attempt to remove the numbness that numbers often induce while retaining a proper sense of scepticism when figures and statistics are flung in our direction.

Every science goes through three stages in its evolution: superstition, empiricism and finally mathematics. Chemistry had its roots in the theory, first proposed by Empedocles in the fifth century BC, that everything is composed of the four elements: earth, air, fire and water. Medieval alchemists introduced experimentation, and even Isaac Newton called himself an alchemist when looking for the mathematical rules behind physics and chemistry. When Aristotle, in the fourth century BC, explained why children look like their parents, he put it down to the mother’s imagination: ‘If in the act of copulation, the woman earnestly look on the man, and fix her mind on him, the child will resemble its father. Nay, if a woman, even in unlawful copulation, fix her mind upon her husband, the child will resemble him though he did not beget it.’

This mistaken explanation may have laid the foundations for genetics, which only entered the age of empiricism with the experimental breeding of peas by the Augustinian friar Gregor Mendel in the 1860s. It took almost another century before the discovery of DNA turned genetics into a proper mathematical discipline.

Until the year 1834, the word ‘scientist’ was unknown in English, and the first appearance of ‘genetics’ was in 1872. Yet, despite the developments of modern science and the public’s increasing dependence on it, far too many of us are content to remain in the realm of superstition, with occasional forays into the empiricist’s experimental methods, without even being tempted to venture into the realm of mathematics which can explain it all. We’d like to understand but are put off by the abstraction of mathematics, which makes the whole subject a no-go zone.

I suspect that the real trouble is that we process language so fast that, when numbers come along, we need to slow down in order to take them in and we are reluctant to do so. Mathematics in general and numbers in particular have become mere adornments, included in news stories and policy documents to add apparent weight to whatever is being said and convey a feeling of scientific validity.

When we hear reports of primitive societies that have no words to express numbers greater than three, we smile smugly and think that ‘one, two, three, many’ is no way to count, yet we have our own problems as far as large numbers are concerned. We are mostly all right with four and five, and have a fair grasp of hundreds and thousands, but once we get up to millions, or billions, or trillions, we become bemused. What’s the difference between a billion and a trillion anyway? We see both of them as ‘many’ so shrug it off when we hear that the UK national debt is over £1.8 trillion. Only when we learn that £1.8 trillion is more than £27,000 for every person in the country do we begin to understand just how much it is.

In this book I try to explain some of the ideas behind the figures and formulae we encounter every day and some of the maths behind them. We shall reveal many dubious ways in which statistics may be used by politicians and advertisers to mislead us, and the manner in which writers of film scripts and the press may mangle beautiful mathematical ideas such as Chaos Theory to inflict on us a horribly simplified version that gives the wrong idea entirely.

My underlying message is that mathematics is beautiful and provides the necessary tools to help understand the world around us and adopt a rational approach to the problems of life. I hope that this book will help readers to overcome any fear of figures they may harbour, and the next time they are caught in a deluge of numbers, statistics and mathematical symbols they will keep their numbrellas tightly furled and welcome all the numbers with an open and better-informed mind.

For me, I suppose, many of the ideas in this book really began with walruses. Those marine mammals wobbled their way into the picture long after my ill-spent youth studying mathematics and playing too much chess. After spending my formative years in that way, I started becoming interested in the rest of the world and began reading newspapers. That was when I discovered that the vast majority of people have a different reaction to numbers from that of mathematicians. Whenever I came across a statistic or other numerical data in a news story, I paused and asked what it really meant and that question more often than not left me with a feeling of dissatisfaction or even bemusement. The verbiage around the number usually provided insufficient information about the data they were relying on for me to assess or even understand the results.

I came to realize that most people, including the journalists responsible for the newspaper stories and almost all the people reading them, just saw the numbers as background adornments, letting them float past without consideration, trusting that they told the same story as the words. The numbers induced a comforting numbness that reassured the reader. But when I read about the walruses, I was far from reassured.

The story, which appeared in several UK newspapers, concerned an intriguing piece of research published in the journal BMC Ecology in 2003 under the title ‘Feeding Behaviour of Free-Ranging Walruses with Notes on Apparent Dextrality of Flipper Use’. The sentence in the paper that all the reports latched on to was one reporting that during feeding behaviour, ‘the walruses used their right flipper 89% of the time’ but interpretations of that sentence differed.

Some concluded that it meant that 89% of walruses were right-flippered and commented on the fact that this was much the same as the percentage of humans that are right-handed. Others reported that the research had found that all the walruses in the study were right-flippered and used that flipper 89% of the time compared with the left flipper 11% of the time.

Quite apart from wanting to know which of those was the case, I also became intrigued by that 89% figure for right-handedness in both walruses and humans and I asked myself whether that was really what the figures showed. If 89% of walruses are right-flippered, would we expect 89% of recorded flipper uses to involve the right flipper?

I did a quick calculation and found that this was surprisingly not the case. Let’s make some reasonable assumptions to keep the sums easy and see what happens.

Various pieces of research have come up with slightly different figures for the proportion of humans who are right-handed and the results vary between about 85% and 92%. So, to keep the calculation simple, let us suppose that 90% of us are right-handed and all of us use our dominant hand 90% of the time.

Then, in a sample of 100 people, we have on average 90 right-handers and 10 left-handers. Giving each of them ten trials, our right-handers will each use their right hand nine times and left hand once, while the left-handers will do the reverse. Our 90 right-handers will therefore give a total of 810 uses of the right hand and 90 uses of the left hand, while the 10 left-handers will have 90 uses of the left hand and 10 of the right hand. Our observations will therefore comprise a total of 820 right-hand uses and 180 left-hand uses. Paradoxically, it may seem, our assumption of a 90%–10% split in handedness has led to an 82%–18% split in observations. So walruses and humans, as far as handedness is concerned, are not the same after all.

I read the paper on which all these walrus reports were based and I contacted one of the authors to ask what the results had really shown. It turned out that the research was based on video footage of walruses feeding in Greenland, each video cut into fixed-length segments and each segment examined for flipper use. Only segments showing preferential use of one flipper over the other were included in the analysis, which formed the basis for the 89% figure. The number of walruses involved in the study was unknown, though the researchers told me that it was at least five.

Five walruses, however, do not constitute a significant sample (as we shall see when we discuss sample size and significance later in this book) and, as the researchers said, further research was clearly needed.

Sadly, I have not come across any later research into handedness in walruses, other than a 2014 report of the examination of the tusks of seventeenth-century walrus skeletons in Nova Scotia which concluded that walruses may have been predominantly left-handed. The mystery of handedness in walruses thus remains unresolved but at least we now know a little better what questions we should be asking of any observational or experimental results.

Finally, however, and to add to the confusion, I should add something about another piece of research into the handedness of a different sea creature that I learnt about just as I was writing about walruses. At the end of November 2019, the Guardian newspaper ran a story about newly published research under the headline: ‘Most dolphins are “right-handed”, say researchers’. A BBC World Service news report used exactly the same words and this was eagerly reported in other media too, but quite apart from the fact that dolphins do not have hands, the conclusion is not as evident as they made it sound.

The research on which it is based was published in November 2019 under the title ‘Behavioural Laterality in Foraging Bottlenose Dolphins (Tursiops truncatus)’ in the Royal Society Open Science journal and detailed the results of observing foraging behaviour by dolphins off Bimini in the Bahamas. This behaviour, the authors explain, involves dolphins swimming slowly along the ocean floor using echo location to identify a potential food source. When they have found one, they turn sharply, bury their rostrum (beak) in the seabed and dig out the food.

Analysing 709 such turns by at least 27 distinct dolphins, the authors report that 705 were to the left, with their right side and right eye downwards. Furthermore, all four examples of right turns were produced by the same dolphin which, the researchers point out, had ‘an abnormally shaped right pectoral fin’ which may have affected its turning behaviour.

The news reports of the research did not go into the reasons for concluding that left turns indicated right-handedness, but the researchers point out that previous research had indicated that a dolphin’s sight is better through its right eye and the echo-location clicks it makes are produced better from their right-hand ‘phonic lips’ than those on the left.

It may therefore make sense for a dolphin to keep its right side facing downwards where the food is located, but it seems to me it is just as easy, if you happen to be a dolphin, to keep your right side downwards when turning right as it is when turning left.

As the researchers point out, many animals have been shown to prefer one side to the other. Chimpanzees and gorillas show a significant right-hand bias, but orangutans are mainly left-handed. Herds of reindeer tend to circle in an anticlockwise direction while giraffes when splaying their legs tend to move their left leg first. For many more species, however, including lions, bats, chickens, parrots and toads, individual animals show a preference for using their right or left limbs, but there is no significant overall species preference for either right or left.

A good deal of research shows that newborn babies have a greater tendency to turn their heads to the right and this is seen by some as due to our greater right-handedness (though it may also be a sign of preferential muscle development). It is not clear to me why turning to the left is seen as a sign of left-handedness in babies but right-handedness in dolphins.

Despite reputable research showing that polar bears use their right and left paws equally, many collections of trivia or supposedly surprising information include the ‘fact’ that ‘all polar bears are left-handed’. This particular piece of disinformation can apparently be traced back to a single account of one polar bear seen by one Native American chieftain covering its nose with its right paw when sneaking up on a sea lion before battering it with its (preferred) left paw. Five walruses was a small sample in the earlier study but a sole polar bear is an even more extreme example.

Show me, Lord, my life’s end and the number of my days.

(Psalms 39:4, The Bible, New International Version)

Trying to estimate the number of our days is an essential part of life insurance and pension planning, but the calculation of life expectancy is something most people do not think about, and even among those who do think about it, very few understand.

Life is invariably fatal: 100% of people die. Or do they? It is frequently claimed, by people who do not pause to think about what they are saying, that more people are alive today than have ever died. That’s nonsense. There are currently, according to UN estimates, just over 7.7 billion people on Earth. Lack of accurate data, or indeed any data at all for much of the time, makes it difficult even to guess the number of people in past times, but the table gives plausible estimates for the years in which landmark figures were reached.

Year

World Population

2011

    7 billion

1999

    6 billion

1987

    5 billion

1974

    4 billion

1960

    3 billion

1927

    2 billion

1804

    1 billion

1700

610 million

1600

500 million

1500

450 million

1400

350 million

1100

320 million

  800

220 million

  600

200 million

From these figures, we may estimate that in the 500 years between 600 and 1100, well over a billion people (5 200 million) died, as very few people lived more than a hundred years. Another billion deaths would have been exceeded between 1100 and 1400, followed by more than 2 billion from 1400 to 1800. When we add the billion alive in 1804 and the 2 billion in 1927, we are already well over 7 billion.

Homo sapiens emerged between 50,000 and 300,000 years so, which adds a large number of dead people to our collection. We can only guess at population sizes and average lifespans in the early years but the best-informed estimates have reached the conclusion that around 108 billion people have ever lived. This means that about one-fourteenth of the people who have ever lived are alive today.

Looking on the bright side, this means that only 13 out of every 14 people who have ever been born have died so we might optimistically conclude that we have a 1 in 14 chance of living forever. Right?

Er, no. Just wait another 100 years or so and we’ll have the complete data on almost all of us. And that introduces the real problem of estimating life expectancy.

Having complete data is a problem often ignored by overeager users of medical statistics. What should we make, for example, of a recent report that deaths from breast cancer have been going down by almost 2% per year? Since everyone dies, a reduction in deaths from one cause must be matched by an increase in others. A decrease in premature deaths means something; an overall decrease needs further probing. And what does it mean when we read that life expectancy at birth in the UK is 79.6 years? It certainly doesn’t mean that the average age at which British people die is 79.6, because anyone who is 79 years old was born 79 years ago, not today. To predict how long a person born today will live involves making a prediction of medical progress for the next century. The spurious accuracy of that 79.6 figure covers a calculation based on a variety of assumptions, many of which are not easy to justify. Yet published figures of life expectancy have a huge economic effect on pension funds and government planning. Before considering the present state of life expectancy calculations, however, let us go back to its beginnings.

Extrapolation

In bygone years, if we wanted to predict the future we would look at stars in the sky, or deal out Tarot cards, or look for patterns in tea leaves or animal entrails. Now we collect vast amounts of data, plot them on graphs, try to detect patterns and work out what will happen if those patterns continue into the future. That’s extrapolation. On the whole, it gives more reliable results than Tarot cards or entrails, but as we see with life expectancy figures, trying to predict long into the future is a far from exact procedure. One cannot, after all, extrapolate progress.

Sometimes statisticians, especially those of an ultra-cautious disposition, refer to their suggestions of future trends not as predictions but projections. Predictions based on statistics are always made, to some extent at least, on the assumption that the future will work in much the same way as the present. Using the word ‘projection’ stresses that point and, in the case of life expectancy calculations, it is an admission that any attempt to extrapolate medical progress a lifetime ahead is little more than educated guesswork.

The earliest known collection and publication of data in a form that resembled a life expectancy table was by the Roman jurist Ulpian around AD 220. Much admired at the time as a legal authority, he advised on a system of taxation and inheritance payments that involved a death tax of around 5% on any legacy, with the remaining 95% funding an annual payment to the recipient of the legacy at a prescribed rate in a manner similar to modern annuities. To determine a fair rate, however, an estimate was needed of the life expectancy of the recipient and that was what Ulpian’s figures set out to provide.

Where his figures came from is not known, and great doubts have been expressed concerning their reliability and the statistical methods used for their calculation, but they suggest a female life expectancy at birth of 22.5 years and a male life expectancy of 20.4. For anyone reaching their late 30s, however, Ulpian predicts another 20 years of life while the over-60s could count on another five years on average. Ulpian himself lived to his early 50s. He was murdered in AD 223 in a riot between the soldiers and the mob by members of the Praetorian Guard, whom he had annoyed by reducing their privileges some years earlier.

Ulpian’s tables remained the last word in life expectancy predictions in the Roman Empire for several hundred years and were not surpassed significantly until the seventeenth century when an Englishman, whose name is now associated with a less earthly, more celestial observation, had a very bright idea. That man was to become England’s second Astronomer Royal and his name was Edmond (sometimes spelt Edmund) Halley.

Quite apart from his discovery of the comet named after him and his correct prediction of its return in a 76-year cycle, Halley made prodigious contributions to a number of scientific fields from an early age. He went to Queen’s College, Oxford, at the age of 16 and published papers on sunspots and the Solar System while still an undergraduate. He left Oxford after four years, without having taken a degree, to set up an observatory on the island of Saint Helena. Oxford made up for his formal failure to graduate by giving him an MA degree when he was 22, at which age he was also elected to be a Fellow of the Royal Society. He died at the age of 86, in 1742, allegedly after drinking a glass of wine against his doctor’s orders.

Halley’s great contribution to the science of life expectancy came in 1693 when he had been working in the Austrian town of Breslau (now Wrocław in Poland). He had come across data recording the annual numbers of births and deaths in the town over a five-year period and, most importantly, the sex and age of those who had died. Since Breslau was a small, tightly knit community far from the sea, and the number of births was roughly equal to the number of deaths over the five-year period, Halley reasoned that the number of people joining or leaving the town each year, other than through births and deaths, was small. This meant the total population (for which the records gave no precise figure) was reasonably stable, and that assumption enabled Halley to draw far-reaching conclusions.

His method was simple: the assumption of a constant birth rate told him the number of people of any age who would be alive if none of them had died, and by summing the mortality numbers for all lower ages, he calculated how many were still alive. This technique enabled him to calculate the odds an individual of any specified age had of reaching his next birthday. His table, for example, gave the number of 25-year-olds living in Breslau as 567 while the number of 26-year-olds was 560, so in his own words: ‘As for Instance, a Person of 25 Years of Age has the odds of 560 to 7 or 80 to 1 that he does not die in a year: Because that of 567, living 25 years of Age there die no more than 7 in a year, leaving 560 of 26 Years old.’

Halley did not specifically calculate life expectancy at birth, but we can see from his figures that around 50% of people died before they were 34. He did, however, devote a good deal of space to the calculation of sensible rates of annuities, as he made clear in the full title of his report: ‘An estimate of the degrees of the mortality of mankind; drawn from curious tables of the births and funerals at the city of Breslaw; with an attempt to ascertain the price of annuities upon lives.’

In fact, much of the motivation behind Halley’s work in this respect lay in the English government’s policy of raising funds for the war against France by selling annuities but the rates they offered were considerably less justifiable than those of Ulpian back in ancient Rome.

At one stage, the English government offered annuities that paid back their full price in only seven years; even when that time frame was doubled to 14 years, no account was made for the age of the purchaser. Nowadays, annuity rates are based on mortality tables. The total amount invested is usually built up over an individual’s working life, then used to purchase the annuity, which guarantees to pay a certain amount each year. The word ‘annuity’ comes from annus, the Latin for ‘year’, and the annual amount depends on the age of the individual and the number of further years they can expect to live. Back in the seventeenth century, the economist William Petty and the statistician John Graunt had made valiant efforts to draw up mortality tables for London some 30 years before Halley came along, but they lacked the precise data to calculate life expectancy that the town of Breslau offered, so their figures offered only limited help in making financial decisions. Without their contributions, however, Halley would probably not have had the idea of using the data in the way he did.

Today, it is clear that statisticians still rely on the basic techniques introduced by Halley, though the pace of change in modern life requires some significant changes and additions to those techniques.

When the Office for National Statistics tells us that life expectancy at birth in the UK is 87.6 years for a man and 90.2 years for a woman, what exactly does that mean and how did they calculate it?

Averages: Mean, Median or Mode?

‘Average’ is potentially a misleading word, for there are three common mathematical types of average: mean, median and mode. Which one is being used is rarely specified, but it can make a big difference.

When most people mention the average of a group of numbers, they are referring to the mean, which is what you get when you add all the numbers together then divide by the number of values in the group.

For the median, we put all the values in order from smallest to largest and the median is the one in the middle.

Finally, the mode (or modal value) is the most common value in the group.

The mean age at which men died in the UK in 2019 was 79. The median age of those deaths was 82 (so, of those who died, the same number of people were under 82 as over 82). The mode (the most common age) was 85. (For all three averages, the figures for women were around three years higher.)

Until around 1840, the modal age for people dying in the UK was 0, because of very high mortality rates shortly after birth. Remarkably, even in most developed countries, until the second half of the twentieth century it remained the case that more people died before their first birthday than during any other specific year of their life.

Life expectancy, you may think, is exactly what it says: the number of years we can expect to live. Well no: it’s not really that at all. Nor is it the mean average age at which people die, the median age at which people die (meaning that about half of all deaths occur before that age and half after) or the most common (modal) age at which people die. We shall come to what life expectancy really means in a moment, but first let’s look at a few figures.

In the three-year period 2016–18 in the UK, the mean age at which people died, which is what most people mean when they refer to an ‘average’, was 79.3 for men and 82.9 for women but the median age at death was 82.5 for men and 85.9 for women.

At first sight, this discrepancy between mean and median may seem strange, but it is just what one would expect when you consider the nature of the data. The ages at which people die spread from zero upwards, and the lower figures have a strong influence in pulling down the mean. A person may die 80 years before reaching the mean, but nobody reaching it lives another 80 years. All death ages count the same towards the median figure, so the median ends up higher than the mean.

This is further emphasized by the figures for the modal age of death – the age at which most deaths occur, which one could consider the ‘typical’ or most likely age of death. In 2018, for men in the UK this was 86 and for women it was 88. The fact that the difference in average age at death between men and women is only 2 years for the mode, while for the median it is 3.4 and for the mean it is 3.6, seems mainly due to higher infant mortality figures in males.

In 1974, UK mortality figures registered modal ages of 81 for women and 74 for men. The large decrease in the difference since then has been put down to improvements in working conditions for men and considerable decrease in smoking, both of which have greatly increased the lifespans of men.

While all these figures have undergone strong and steady improvement in the last half-century, the greatest change for men has been in the modal age of death – the age at which most deaths occur. In 1967, it was 67, rising to 72 the following year and 74 in 1969. It remained in the mid-seventies until 2000 when it increased to 79, then the following year it was 80 and it has slowly crept up to 86 in the most recent figures.

Perhaps most remarkable of all, in the UK the modal age of male deaths until 1966 was zero: infant mortality was responsible for more male deaths before the baby had reached his first birthday than were registered at any later year of life. For reasons that are still unclear, girl babies have a better survival rate than boy babies: the modal age at death for UK females has not been zero since the 1940s.

However, it has never been considered advisable to tell people that their modal life expectancy is zero or that that is the age at which people are most likely to die: starting with Edmond Halley, more useful measures were developed to give a figure that was believable, useful and not so depressing. Whether these measures are generally understood or even intelligible to most people is another matter.

When actuaries draw up tables intended for the use of life assurance companies or long-term governmental planning concerning matters such as pensions and retirement age, such tables usually take the form of lists giving the number of years of life a person may expect at various ages. The Office for National Statistics in the UK (www.ons.gov.uk) has even developed an online Life Expectancy Calculator into which anyone can enter his or her age and sex, click on the ‘Calculate’ button and it will tell them what age they can expect to live to. A 73-year-old male, for example, will discover that he can expect to live to around the age of 87, which is the same age predicted for a 45-year-old female. A woman aged 97 will discover she has an even chance of living to 100.

Looking at these figures quickly reveals the real problem: they involve seeing into the future. In the case of the 73-yearold man and the 97-year-old woman, we are looking only 14 and three years ahead respectively, so there is a fair chance that our estimates are not too far off. For the 45-yearold woman, however, the prediction looks 42 years into the future and we really have little idea of what global disasters or medical breakthroughs may happen over that period to throw our estimates way off balance.

The UK Office for National Statistics stresses that such figures are ‘projections’ not ‘forecasts’, meaning that they project current data into the future, and they have two distinct ways of doing this.

The first is called ‘period life expectancy’ and is the simpler and probably less accurate of the two. Period life expectancies use current mortality rates as calculated from existing data and assume that those rates apply throughout the remainder of a person’s life. This is essentially what Halley did in Breslau. Knowing how many babies were born and how many died in their first year of life enables you to work out the chance that a person will survive from nought to one. Similarly, you work out how many one-yearolds survive to see their second birthday, and how many of those reach the age of three and so on. Eventually you reach an age to which 50% of people have survived and that is taken to be the life expectancy.

Period life expectancies calculated in this manner are precise and based on current data, but possible future changes to mortality rates are not taken into account.

The second method, called cohort life expectancy, is more complex and probably more likely to be realistic, but it involves making certain assumptions about how long into the future historical trends will continue. Such judgements are always liable to have a subjective element.

To assess the cohort life expectancy at birth of a person born in 2020 we would ideally like to know the chance of such a person reaching their first birthday, then the chance of a one-year-old in 2021 reaching the age of two, and a two-year-old in 2022 reaching the age of three and so on. We do not yet, of course, have any of those figures but we have them for previous years. The ‘cohort’ that gives this system its name is the group of people born in the same year.