How can a “mathematician’s toolkit” help you spot misleading statistics in the news?

Why is understanding exponential growth crucial for making better predictions in uncertain times?

Discover how Oliver Johnson’s Numbercrunch equips you with essential math skills to interpret data visualization, understand viral growth, and avoid cognitive biases. Learn to navigate the modern world’s information overload with confidence.

If you are tired of being misled by confusing charts and headlines, keep reading to master the simple mathematical rules that will sharpen your critical thinking and decision-making skills.

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Does your job require you to, say, create a budget or describe a societal trend such as unemployment? If so, a grasp of numbers is a must. Alas, many people lack confidence in their numeracy skills. Mathematician Oliver Johnson offers several tips and tricks to help you strengthen your math muscles. Though many of his insights will boggle the uninitiated, Johnson strives to make the field of mathematics more accessible to laypeople. Deepening your understanding of numbers will help you better navigate uncertainty, assess information, and make informed predictions.

Take-Aways

• Make sense of numbers by expanding your understanding of data visualization.
• Gauge the accuracy of big numbers by making your own approximations.
• Learning to calculate probabilities can help you work with random data.
• Your learning flatlines when you consume repetitive information within a self-affirming social bubble.
• Gain insight into the viral spread of diseases or campaigns by studying networks.
• Viruses spread through a network following an S-shaped pattern.
• Game theory can help you navigate competitive interactions.
• Check your biases and treat mistakes as opportunities for learning and growth.

Summary

Make sense of numbers by expanding your understanding of data visualization.

While many people feel intimidated by numbers, finding ways to visualize data can help you better grasp the story they tell. Learn a few simple tips and tricks to help you cast a critical eye over statistics, so you will know when governments or corporations are manipulating data to serve their own needs by, say, cherry-picking data, manipulating axes on graphs, or presenting data from convenient time periods to tell the narrative they want to spin.

“Numbers can sometimes be used to obfuscate an issue or can be presented without context. I would like to help you make sense of numbers, to feel less daunted by them, and to understand the rules that govern their ebb and flow.”

One of the first concepts you need to grasp when plotting data is the difference between different types of “functions,” or mathematical rules that can serve as models. For example, in the simplest function, known as a “constant function,” the numerical output is the same every time the model makes a calculation, creating a horizontal line when you plot the points on a graph.

By contrast, in a linear function, your output describes a more dynamic relationship between your x-axis variable (for instance, date) and your y-axis variable (for example, distance). Your line might slope upwards or downwards, and its degree of steepness visually conveys information that enables you to make predictions about the future. For example, if you were to use Isaac Newton’s laws of motion to calculate the trajectory of a hypothetical space probe as it drifted in space — “under its own momentum, without power or friction” — and you measured the probe’s distance from Earth at the same time every day, the probe, traveling at a constant speed, would travel away from Earth by the same distance each day. Mapping this data generates a straight line that slopes upward toward the right of your graph. However, if the space probe were powered by rocket engines that provided a constant amount of acceleration, the speed would increase each day, resulting in a graph with a curved line bending upward in a “quadratic function.”

While a quadratic function adds a fixed amount to the distance traveled each day, an “exponential function” multiplies the previous value by a fixed amount. An exponential curve increases slowly at first and then accelerates quickly. Consider a bacterium that splits into two parts every hour. After three hours, you will have eight bacteria, which may appear innocuous. The growth might seem linear or quadratic initially. But after 24 hours, you will have 16,777,216 bacteria. The exponential growth model can explain patterns in nuclear chain reactions or football transfer fee records.

While understanding models can help you make better sense of your data, don’t put too much blind faith in them: You may run the risk of “overfitting” your model by failing to account for some degree of randomness in numerical variation, which results in inaccurate predictions.

Gauge the accuracy of big numbers by making approximations.

Visualizing huge numbers — those in the millions, billions, and trillions — can be difficult without context. Determining whether a numerical value sounds as though it’s in the “right ballpark” will help you avoid misinformation and errors. Perform a simple “smell test” when examining big numbers: Compare them to the value of other large numbers in your working memory to see if they make intuitive sense. For example, if a report claimed that the interest on the UK’s national debt amounted to £60 million each year, you could ask whether it makes sense that that figure is equivalent to, say, the cost of a Premier League footballer. Often, when figures don’t pass such smell tests, someone has made a mistake. In this case, someone likely has confused “billion” and “million.”

Big numbers often become more relatable if you calculate them on a per capita basis. For example, in December 2023, the US national debt was roughly $34,000 billion, while its population was just shy of 330 million. Thus, the country’s debt amounted to about $100,000 per person.

“Models that explain current data too well can be less trustworthy than ones that provide a looser fit.”

Another way to wrap your head around big numbers is to use a “Fermi estimation” — a series of rough deductions to inform a sensible guess. For example, if someone asked you, “How many piano tuners are there in Bristol?” you might first consider several factors, including the population of Bristol, the percentage of Bristolians with a piano, the frequency that pianos require tuning, the length of time it takes to tune a piano, and so on. These questions will help you conceive of a reasonable ballpark figure.

Learning to calculate probabilities can help you work with random data.

People tend to see patterns where none exist. For instance, if a number appears in two consecutive lottery draws, the odds of it appearing in the next draw neither increase nor decrease. By getting to grips with the rules that govern randomness — especially the notion of expected behavior and the extremes of what is likely — you can deepen your understanding of numbers:

• “Expected value” — While events may be random, you can still determine the likelihood of probable outcomes by determining a random “sample average.” For example, if you roll a die 10 times, tot up the total of all the rolls (say, 35) and divide by the number of rolls (10) for a sample average value of 3.5. Your expected value is likely to be close to your sample average.
• “Variance” — Variance is the quantity by which your results differ from your expected value. “The more spread out the values, the higher the variance.”
• “Law of large numbers” — The more you repeat an experiment, the closer your sample average will approach your expected value. If you toss a coin one million times, it’s unlikely to land on heads exactly 500,000 times, but neither will it land on heads zero times.
• “Central limit theorem” — This theorem holds that when you conduct several experiments using random samples, plotting the probability of outcomes will form a bell-shaped curve. However, this rule only applies if your sample data is reasonably independent — the value of one outcome can’t be contingent on the value of another; for instance, if you flip a coin, your outcome of heads or tails isn’t influenced by the previous coin flip. Failure to understand this can have catastrophic results. The 2007–2008 subprime mortgage crisis occurred in part because those making mathematical models treated the financial risks of default in Florida and California as though they were independent when in reality, these risks were connected to US economic performance.
• “Extremes” — Averages provide useful information, but they don’t offer a complete picture. Calculating the extremes of what is possible is also important. For example, meteorologists need to be able to forecast the probability of rare devastating weather events. As global climate change intensifies, these “once in a hundred years” events will become more frequent.

Your learning flatlines when you consume repetitive information within a self-affirming social bubble.

Mathematics can explain the dangers of getting stuck in a “filter bubble” — a social group that shares only self-reinforcing news with one another — when consuming information. Glean the most information possible from a communication system by seeking out diverse, independent sources, according to Claude Shannon, whose 1948 paper “A Mathematical Theory of Communication” established the field of information theory. If you consume only repetitive information from similar sources, your information becomes less valuable, as it becomes difficult to learn something new when “entropy,” or uncertainty, shrinks. With low entropy in a filter bubble, gaining fresh insights is difficult, leaving you more susceptible to bias and misinformation.

“All things being equal, we learn the most from two successive pieces of information that are truly independent.”

Shannon demonstrated that information could be distilled into a series of zeros (0s) and ones (1s). For example, when flipping a coin, heads could be represented by 1, while tails could be represented as 0. Shannon applied his notion of entropy to demonstrate that “some random quantities are more random than others.” Consider the theoretical possibility of a “biased coin” that is more likely to land on heads than tails. Real-world situations in which you have a higher likelihood than 50/50 of guessing an outcome have lower entropy and function similarly to a biased coin. For example, a bookmaker might have information about a game, giving an unfair advantage in predicting who is most likely to win.

Gain insight into the viral spread of diseases or campaigns by studying networks.

Mathematical models can help you better understand distribution throughout a system, whether you’re interested in predicting, say, the spread of a virus or a political campaign. If, for example, you’re trying to understand how to go viral on X [formerly Twitter], imagine each user as a chess piece moving around the squares of a chessboard. While each player’s movement is random, players are more likely to move their piece to a square that’s connected to other squares, as opposed to a square at the edge of the board. On X, users are most likely to engage with posts that are “well-connected” (that is, reposted) to others in their social network.

“We can think mathematically about the way things spread, whether that be a virus in a city or a campaign on a social network.”

The spread of a virus through a population is similar to the spread of information on a social network. Visualize individuals as vertices. The virus would likely spend more time with the most well-connected individuals, making those with the most social contacts the most likely disease vectors. The key difference in studying the spread of a virus, as opposed to understanding the movement of a chess piece, is that the virus can move in several directions at once, exhibiting a pattern of activity known as a “branching random walk.” There are other factors to consider when predicting viral spread: For example, if you assume individuals develop immunity after getting infected once, then these immune individuals can be viewed as “roadblocks” within the network, giving rise to theories such as “herd immunity.” However, don’t trust such models completely, as it’s impossible to accurately predict the frequency with which individuals see their contacts or interact with others.

Viruses spread through a network following an S-shaped pattern.

To understand contagion, familiarize yourself with a common pattern present in viral processes — including launching a new product — known as the “sigmoid curve.” This S-shaped curve illustrates a period of slow initial growth followed by a period of rapid growth, after which a saturation phase occurs and growth slows. A curve might flatten after a rapid growth period for numerous reasons: For example, there may be an initial period of buzz when you launch a new product, as consumers scramble to try it, but factors such as income limitations or age may later slow adoption of your product.

“The coronavirus…spread according to an internal logic that can be studied with the same mathematical tools that allow us to study properties of objects in space, or dice rolls, or random walkers staggering home from the pub.”

The spread of COVID-19 variants, such as delta and alpha, exhibited sigmoid growth. As these variants rapidly gained prevalence, they initially spread slowly within populations before accelerating, ultimately reaching a point of saturation (where most individuals had been infected or vaccinated), causing the curve to flatten. Graphing the sigmoid function can help you to visualize the dynamics of contagion and to understand how external factors limit exponential growth, which is not indefinitely sustainable.

Game theory can help you navigate competitive interactions.

One of the classic problems of game theory is the “prisoner’s dilemma”: Imagine both you and a friend are being held in custody for the same crime, and you’re given a choice: Betray and blame the crime on your friend or remain silent. The stakes are as follows: If only one of you betrays the other, the betrayer who collaborated with the authorities will walk free, while the other will serve five years in prison. If you both remain silent, the authorities won’t have enough evidence to convict either of you of the full crime, but you’ll both get one-year prison sentences (they can still convict you of a less serious offense). And if you both cooperate with the authorities and betray one another, you’ll both serve three years in prison.

“The prisoner’s dilemma is an example of what we refer to as a game. It has two players who each have clearly defined choices, according to which they both receive a pay-off.”

In the prisoner’s dilemma, the prisoners are separated and thus don’t know what the other will choose. Because you both lack information, you are powerless to select the best possible choice for yourself. If you were perfectly logical, you’d betray your friend, because it ensures you’ll never serve a full five-year sentence; you will either serve three years or walk free, regardless of what your friend chooses. However, sometimes pure logic doesn’t yield the best possible choice (trusting each other and getting the lowest possible sentence). The prisoner’s dilemma plagued leaders of the United States and Russia during the Cold War. Both countries increased defense spending because they failed to trust each other. It was only through a series of talks, during which both sides shared information and showed the other evidence that they were slowly disarming, that the nations were able to escape the prisoner’s dilemma.

Check your biases and treat mistakes as opportunities for learning and growth.

When working with numbers, examine your own biases. Perhaps you think you are completely unbiased, but most people have strong opinions and look for information that confirms those beliefs, especially when discussing heated topics, such as health and safety measures during a pandemic. Avoid taking overly extreme positions, and endeavor to embrace a more moderate, centrist approach.

“Mathematical techniques can give insights in a way that is divorced from emotion and personal biases.”

Be mindful of any social pressure to conform to “groupthink,” as failing to perceive a situation accurately can lead to complacency or the adoption of faulty solutions. Be wary, too, of the tendency to look for a “magic bullet,” or a singular factor you can use to explain a complex situation, as the reality is often more messy. Finally, have the humility and courage to admit when you’re wrong and to learn from any mistakes you’ve made. Taking a U-turn is always better than continuing down a path that you know to be wrong.

About the Author

Mathematician Oliver Johnson is a professor of information theory at the University of Bristol.