Understanding Numbers: Simplify life's mathematics. Decode the world around you.: 20 thought-provoking lessons

by Marianne Freiberger, Rachel Thomas

Overview

"Simple, clear explanations of twenty ways in which mathematics helps us to understand practical issues of everyday life. Suitable for teenagers and adults, and beautifully produced. Highly recommended!" - Professor Ian Stewart, bestselling author of The Cabinet of Mathematical Curiosities

Mathematics is an indispensable tool for life. From the systems that underpin our newsfeeds, through to the data analysis that informs our health and financial decisions, to the algorithms that power how we search online – mathematics is at the heart of how our modern world functions.

In 20 dip-in lessons, Understanding Numbers explains how and why mathematics fuels your world and arms you with the knowledge to make wiser choices in all areas of your life.

• Make sense of health statistics
• Understand the mathematics behind political voting systems
• Get to grips with how online search tools work
• Discover how mathematics can create the perfect city

Product Details

ISBN-13:	9781781318164
Publisher:	White Lion Publishing
Publication date:	04/09/2019
Series:	BUILD+BECOME
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	8 MB

About the Author

Rachel Thomas and Marianne Freiberger are the editors of Plus magazine, which publishes articles from the world's top mathematicians and science writers on topics as diverse as art, medicine, cosmology and sport. 

Before joining Plus in 2001, Rachel was a math consultant in Australia working for government and industry. She recently edited the Gazette of the Australian Mathematical Society and has developed and taught science writing workshops for graduate students. She also created mathematical walking tours of London and Oxford and a virtual mathematical tour guide of the world for Maths in the City. Rachel obtained her MSc in Semigroup Theory in 1998 from the University of Western Australia.

Marianne joined Plus in 2005 after completing a PhD and then a three year postdoc at Queen Mary, University of London. As a researcher she worked in complex dynamics and held various teaching engagements. In the world of math communication she has been Editor-in-Chief of the Mathscareers website and sometimes gives presentations to mathematicians about how to communicate their work to a wider audience, and to journalists about how to deal with math in the media.

Read an Excerpt

CHAPTER 1

PREVENTING DISEASE

Infectious diseases are frightening. The 2014 Ebola outbreak claimed over 11,000 lives. Over 12,000 people died from swine flu in the US during the 2009 pandemic. The total death toll of AIDS to date is about 35 million.

It's not hard to see why infectious diseases can spread so quickly. Suppose that an infected person goes on to infect two other people during the course of their disease — not an unrealistic assumption if you consider the coughing and spluttering that goes on on public transport. A single infected person will infect another two people, giving a total of 1 + 2 = 3 infected people. The two newly infected people will infect another two each, giving a total of 1 + 2 + 4 = 7 infected people. The four newly infected people then infect another two each, giving a total of 1 + 2 + 4 + 8 = 15 people, and so on.

Continuing in this vein, you see that the number of infected people grows very fast. In fact, it grows exponentially. If each infected person infects their two victims within the first day of catching the disease, then it will only take 26 days to infect a population larger than that of the UK. And that's starting with a single infected individual. (You might want to work out for yourself that the number of infected people after n days would be 20 + 21 + 22 + ... + 2n-1 Luckily, you don't need to tap this long sum into a calculator to get the result: a sum of this form is always equal to 2n - 1).

The number 2 obviously plays an important role in this example. If an infected person infected more than two people a day, then the disease would spread a lot quicker. And if an infected person infected fewer than two people a day, the disease would spread more slowly. In fact, it turns out that the number 1 is the watershed in this context. If an infected person infects, on average, more than one other person, the number of sick people will grow beyond any bound, as long as nothing bars the path of the disease. If, on the other hand, an infected person infects fewer than one other person on average, the spread will eventually come to a halt of its own accord.

Epidemiologists, those tasked with analyzing the spread of disease, have a name for the number of individuals that are, on average, infected by a person who has a particular disease, assuming that all the population is susceptible to catch the disease: it's called the basic reproduction number of the disease. Looking up basic reproduction numbers of common diseases gives you a good idea of how dangerous they are. The basic reproduction number of Ebola is between 1.5 and 2.5. For AIDS it lies somewhere between 2 and 5. For influenza (the 1918 epidemic strain) it's between 2 and 3. And for measles it's between 12 and 18!

What is to be done in the face of such ferocious exponential growth? Epidemiologists use complex mathematical models to see how a disease might spread. Importantly these models can be used to test the effect an intervention, such as vaccination or perhaps a travel ban, might have. The results don't always chime with intuition and can lead to outraged headlines. But rest assured: the advice epidemiologists come up with is based on thorough mathematical investigation.

WHY DOES VACCINATION WORK?

To give an idea of how even some basic maths can help, let's turn to the sometimes contentious subject of vaccination. The idea behind vaccination is to make people immune to a disease by injecting them with a pathogen, but it isn't without problems. It can be difficult and costly to get hold of everyone in a population; some people may be put at risk because of underlying health problems; and others may flatly refuse to be vaccinated. Luckily, though, you don't need to vaccinate everybody in a population to ensure the disease eventually fizzles out. Here's a short calculation to show why.

Suppose you have vaccinated a proportion p of the community, so these people are now immune to the disease. This means a proportion 1 - p is still susceptible to catching the disease. The basic reproduction number, call it R, gives the number of people a sick person infects, on average, in a totally susceptible population. Since after vaccinating only a proportion, 1 - p of the population, are still susceptible, the reproduction number is now only a proportion, 1 - p, of what it was in a totally susceptible population: the basic reproduction number of R turns into an effective reproduction number of R × (1 - p). In order for the disease to eventually fizzle out, we'd like the effective reproduction number to be less than 1, so

R × (1 - p) < 1

A bit of rearranging will show that p, the proportion vaccinated, must therefore be at least 1-1/R:

1 - 1/R < p

In other words, to ensure the disease dies out, you need to vaccinate a proportion of at least 1 - 1/R of the population.

For a basic reproduction number of 2, you only need to vaccinate 1 - ½ = ½ of the population. If R is 3, the upper bound for influenza, you should vaccinate 1 - 1/3 = 2/3 of the population. Importantly, our calculation shows that not only vaccinated people benefit from vaccination. People who haven't been vaccinated do as well, because their overall risk of catching the disease has decreased. In this way people who cannot have a vaccination for whatever reason can still be protected. Vaccination isn't just for you, it's for everyone!

CHAPTER 2

DETECTING DISEASE

It seems obvious that knowledge is better than ignorance, that it is better to find out if you are likely to get a disease, rather than wait until you experience symptoms. But the example of screening for diseases illustrates that this isn't always the case. Screening programmes have both benefits (saving lives) and harms (which we will discuss below) – and these have to be balanced using careful statistical analysis. This is why screening programmes are so carefully researched before they are approved, and rigorously monitored to ensure the balance between benefit and harm is preserved.

The first thing to remember is that screening is not the same as diagnosis. Screening programmes check for well-understood markers that are a clear indication that a person is at a high risk of having a disease. In almost all cases the test is not the same as the diagnostic test used when a patient goes to see the doctor with symptoms of the disease (one exception is the screening test for HIV, hepatitis B and syphilis in pregnancy, which is the same as the diagnostic test). Screening is used to pick up these disease markers in the general population who, otherwise, have no symptoms of the disease.

So if your screening result shows that you do have the markers for a disease, usually called an abnormal result, then this does not mean that you have the disease. You will be called in for more tests, to confirm if you do or do not have the disease. And, in fact, the majority of people who get abnormal results in a screening programme do not have the disease in question.

This might seem surprising at first, but it stems from the fact that screening tests are not 100% accurate in their prediction as to whether a person will go on to develop the disease.

UNDERSTANDING SCREENING NUMBERS

Before you had the screening test you would have believed you had a 1% chance of having the disease, represented by the area of the blue circle (people having the disease) out of the white circle (the whole population) in the picture opposite. The screening test gave you new information (that you are in the green circle), which allows you to update your probability of having the disease – it is the area of the intersection of the green and blue circles (you have the disease and had a positive result) compared to the area of the green circle. This picture illustrates an incredibly useful result from probability theory, called Bayes' theorem. It allows you to update your beliefs about a particular state of affairs in the light of new evidence.

Working through this example illustrates how important it is for screening programmes to test for markers that reliably indicate that a person with an abnormal result is very likely to either have, or go on to develop, the disease in question. And this is the reason why medical experts are very careful in their design, and ongoing monitoring, of screening programmes. For example, they did not extend cervical cancer screening to women younger than 25, despite public pressure and media campaigns. The cervical cancer screening programme looks for changes in cells in the cervix, which is a strong indicator that a woman over 25 has, or will develop, cervical cancer. However it is not a good indicator for younger women, who are more likely to have changes in the cells in their cervix without going on to have cancer. Screening of this younger age group would have resulted in many more false positives, without guaranteeing to save any more lives.

Ultimately it is your decision whether or not to participate in a screening programme. Health organizations around the world are continuously improving their information about screening to help make that an informed decision. You need to understand the limitations of screening as outlined here and elsewhere, and weigh up the evidence and what it means for you.

CHAPTER 3

TESTING TREATMENTS

How do you know that a new medicine or treatment works? It's tempting to think that if something worked for a friend, or a friend of a friend, or some blogger on the internet, that it will work for you. But there are many reasons why anecdotal evidence is not reliable as proof that a treatment or change of lifestyle will cure an illness.

The person in the story might have got better because of some other change in their life, or they might have got better regardless of any action they took. It's impossible to know what was responsible for someone's recovery from anecdotal evidence alone. Instead, the medical community requires treatments to be tested using randomized controlled trials (RCTs), and this takes some careful statistics.

How do we know?

RCTs have been developed over the last century to remove the possibility of intentionally or unintentionally biasing the results of testing treatments. They involve two groups of participants: the control group and the study group. The study group will be given the treatment in question. If the effectiveness of this treatment is being assessed against the treatment normally used for that particular condition, then the control group will be given the normal treatment. This allows researchers to assess the difference between the outcomes of these two types of treatment.

If there is no current treatment for the condition, however, the control group will be given a placebo – something that mimics treatment but has no physiological effect, such as a sugar pill. This allows someone receiving treatment to be compared to someone not receiving treatment, but it also allows for something called the placebo effect: there is a great deal of research showing that the mere idea that they are receiving treatment, or medical attention, can make some people better. Even if the new treatment was not effective, then some people in the study group may still improve thanks to the placebo effect, but this should be comparable to any similar improvement in the control group.

In order to get an objective view of any difference between the study and control groups, it's important that people involved in the study don't know who is in the control group and who is in the study group — if, for example, people in the control group know they are getting the placebo treatment then there is less chance of any placebo effect coming into play. To avoid this, RCTs are usually blinded. A single-blinded trial is one where the participants don't know which group they are in, and a double-blinded trial is one where both the doctors and participants don't know who is in the study group. A trial can also be triple-blinded if the researchers doing the final analysis of the results are also in the dark.

A blinded trial could still be influenced by the choice of who goes in which group. If the people in your study group were less sick, then you might get more favourable results for your treatment when compared to the sicker patients in the control group. The most powerful aspect of RCTs is that participants are randomly allocated to both groups. This balances out the healthier and sicker patients between the groups, and also evens out any other unknown factors that could affect the progression of a participant's illness.

HAS IT WORKED?

Given the parameters of an RCT, it would be surprising if we saw an improvement for those in the study group for a treatment that doesn't work (aside from the placebo effect). But rare and surprising things do happen, and maths provides the tools to quantifying these.

Suppose you have a pair of dice. If you rolled them 20 times you wouldn't be surprised if you rolled a double six once, or not at all. (The probability of rolling a double six is 1/6 × 1/6 =1/36.) But if you rolled several double sixes out of the 20 rolls, you might start to doubt the fairness of the dice, and think they were weighted in some way. A similar approach is used to statistically understand the results of an RCT.

Significance level

RCTs are designed so that there is only a 5% probability that the difference in outcomes observed between the control group (taking a placebo) and the study group would occur by chance alone. To use the technical term, they are designed at a 5% significance level. That is, if the treatment were no more effective than a placebo, and you ran 20 trials, at most one trial would have shown up this difference in observed outcomes by chance alone. Compare this to rolling our pair of dice: if the dice were both fair (which we can think of as our treatment being no more effective than a placebo) then we wouldn't be surprised if we rolled a double six out of 20 rolls. But we would be very surprised if we rolled several double sixes.

Point measure

The effect that is measured in one trial is called a point measure. This is the average of effects for that particular group of people at that particular time.

Confidence interval

What you are actually interested in is the underlying real effect of the treatment for the whole population who have that disease. This is why an RCT report refers to what is called a confidence interval. Usually, this is the point measure plus or minus an amount that depends on the variability in the trial's data.

The confidence interval is calculated so that researchers are 95% sure that the underlying real effect of the treatment being tested will be somewhere in that confidence interval. The level of confidence, 95%, is equivalent to the significance level of 5%. The precise definition of a confidence interval means that if we repeated the trial 20 times for different groups taken from the general population of people with the disease, then 19 of the confidence intervals arising from these trials would contain the true underlying improvement in that population.

If the confidence interval given for the effect detected in an RCT does not include 0 (no difference in outcomes between the two groups), then that indicates the treatment's effect was statistically significant: we can be confident that the effect was really there. But statistical significance isn't the final hurdle – the effect must also be clinically significant. That is, it must make a marked difference on the outcome of the patients.

Individual anecdotes about the effects of treatments don't allow us to understand fully what works and what doesn't: careful design of trials and statistical analysis are crucial in finding the best treatments.

CHAPTER 4

STATISTICALLY SPEAKING

We are faced with statistics on a daily basis. And when it comes to matters of health, these can be daunting, misleading and, well, confusing. Suppose you're a menopausal woman considering hormone replacement therapy (HRT). You've heard that HRT can increase your risk of getting breast cancer, so you seek further information. Here's what you're told:

HRT increases the risk of breast cancer by 6 in 1,000: out of 1,000 women on HRT, six will develop breast cancer that otherwise wouldn't. However, HRT also reduces the risk of colon cancer by 50%.

What would you make of this? It's very tempting to conclude that the benefit outweighs the risk. A huge 50% risk reduction for the very nasty colon cancer, versus a tiny 6 in 1,000 increase for breast cancer. Should you go for HRT, safe in the knowledge you've made an informed decision based on the evidence?

The answer is no, not yet. The information you have been given here is not enough to base a decision on. Let's start with the 50% reduction in the risk of colon cancer. That sounds great, it's a halving of the risk. The question is, however, how big was your risk to start with. For the sake of the argument, let's suppose that 80 in 100 women get colon cancer when they are at the age at which they might also take HRT (nowhere near as many women get colon cancer, but like we say, it's for the sake of the argument). The 50% reduction brings that down to 40 women in 100. In other words, your risk of colon cancer is reduced from 80% to 40% by HRT: a reduction that may well be worth putting up with other side effects, including an increased risk of breast cancer.

(Continues…)

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Excerpted from "Understanding Numbers"
by .
Copyright © 2019 Marianne Freiberger and Rachel Thomas.
Excerpted by permission of The Quarto Group.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

INTRODUCTION,
HOW TO USE THIS BOOK,
01 HEALTH,
01 Preventing disease,
02 Detecting disease,
03 Testing treatments,
04 Statistically speaking,
Toolkit 01–04,
Further Learning,
02 ENVIRONMENT,
05 Architecture,
06 Modelling traffic,
07 Predicting the unpredictable,
08 Changing climates,
Toolkit 05–08,
Further Learning,
03 SOCIETY,
09 Everyday averages,
10 Voting,
11 Evidence,
12 Debts and savings,
Toolkit 09–12,
Further Learning,
04 RELATIONSHIPS,
13 The evolution of human kindness,
14 How we search,
15 The statistics of sex,
16 Making relative sense,
Toolkit 13–16,
Further Learning,
05 COMMUNICATION,
17 Networks,
18 Staying secure,
19 Big data,
20 Knowing where you are,
Toolkit 17–20,
Further Learning,
Epilogue,

Editorial Reviews

“Simple, clear explanations of twenty ways in which mathematics helps us to understand practical issues of everyday life. Suitable for teenagers and adults, and beautifully produced. Highly recommended!”—Professor Ian Stewart

From the Publisher