How Not to Be Wrong

How Not to Be Wrong Plot Summary

Introduction

Have you ever stared at a math problem and wondered, "When will I ever use this in real life?" You're not alone. This common frustration masks a profound misunderstanding about what mathematics truly is. Mathematics isn't just about solving equations or memorizing formulas—it's a powerful way of thinking that helps us navigate a complex world filled with uncertainty, statistics, and logical pitfalls. Mathematical thinking extends far beyond calculations. It provides us with mental tools to see through misleading statistics in news headlines, understand when correlation doesn't imply causation, and make better decisions when faced with uncertainty. Throughout this book, we'll explore how concepts like non-linear relationships, statistical significance, and Bayesian reasoning can transform the way you approach problems. You'll discover why extreme performances tend not to last, how crowds can be surprisingly wise under the right conditions, and why expected value is crucial for making smart choices. By the end, you'll see mathematics not as a collection of abstract puzzles, but as an extension of common sense that helps us avoid being wrong when it matters most.

Chapter 1: Linear vs. Non-linear: When More Isn't Always Better

We humans naturally think in straight lines. When something works well, we assume more of it works better. When something causes harm, we assume less of it is always better. This linear thinking is so intuitive that we often don't even realize we're doing it. Politicians argue that if some regulation helps the economy, more regulation must help more. Athletes might believe that if training six hours a week improves performance, training twelve hours will double the improvement. But reality rarely follows such simple patterns. Non-linear relationships are everywhere in the real world. Consider medication dosages: too little medicine has no effect, the right amount cures you, and too much becomes toxic. This creates what mathematicians call a non-linear curve—one that doesn't follow a straight line but instead bends, peaks, or changes direction. The relationship between sleep and health works similarly. Too little sleep impairs cognitive function and physical health, but too much sleep is also associated with negative outcomes. The optimal amount lies somewhere in the middle, creating an inverted U-shaped curve rather than a straight line. The Laffer curve in economics provides another classic example of non-linear thinking. Economist Arthur Laffer observed that at both 0% and 100% tax rates, government revenue would be zero—at 0% because no taxes are collected, and at 100% because people would stop working if all their income went to taxes. This means the relationship between tax rates and government revenue must form a curve, with some optimal rate that maximizes revenue. Moving away from that optimal point in either direction—whether raising taxes too high or cutting them too low—reduces revenue. Understanding non-linearity helps us avoid significant errors in judgment. When we see obesity rates rising over time, linear projection might suggest that "everyone will be obese by 2048"—a headline-grabbing but mathematically naive conclusion. Reality tends to follow curves that bend, plateau, or oscillate rather than continuing indefinitely in a straight line. Similarly, when we observe that a small amount of something (like exercise) produces benefits, we shouldn't automatically assume that massive amounts will produce massive benefits. Non-linear thinking reminds us that context matters, optimal solutions often lie between extremes, and that relationships between variables can change dramatically at different scales or intensities. By recognizing when relationships are likely to be non-linear, we can make more nuanced predictions and avoid the pitfalls of oversimplified linear thinking that pushes us toward unnecessary extremes.

Chapter 2: Correlation and Causation: Untangling Relationships

When two phenomena occur together, our minds instinctively create a story connecting them. Ice cream sales and drowning deaths both increase in summer months. Students who receive tutoring often improve their grades. Countries with higher chocolate consumption tend to produce more Nobel Prize winners. These correlations are real, but jumping to conclusions about causation can lead us astray. Understanding the relationship between correlation and causation is essential for making sense of a world filled with data and statistics. When we observe a correlation between variables A and B, several possibilities exist. A might cause B (exercise improves mood), B might cause A (feeling sick leads to taking medicine), or a third factor C might cause both A and B (warm weather increases both ice cream sales and swimming, leading to more drownings). Another possibility is that the correlation is simply coincidental, especially if you're examining many potential relationships at once. The more relationships you check, the more likely you are to find correlations by pure chance. The history of smoking research illustrates the challenge of establishing causation. By the 1950s, studies had established a strong correlation between cigarette smoking and lung cancer. Yet some scientists, including the renowned statistician R.A. Fisher, remained skeptical about causation. Fisher suggested alternative explanations: perhaps a genetic factor predisposed people both to enjoy smoking and to develop lung cancer. Or maybe—as he provocatively suggested—the early stages of lung cancer somehow made people want to smoke. While these theories sound far-fetched today, they illustrate an important point: correlation alone cannot prove causation. Scientists use several approaches to move beyond correlation toward causation. Randomized controlled trials, where subjects are randomly assigned to treatment or control groups, help eliminate many alternative explanations. Natural experiments, where circumstances create random variation in exposure to a potential cause, offer another approach. Researchers also use statistical techniques to control for confounding variables and build causal models that test specific pathways of influence. Understanding the distinction between correlation and causation isn't just academic—it has real-world consequences. Misattributing cause can lead to ineffective treatments, wasteful policies, and missed opportunities to address the true causes of problems. When we hear that two things are correlated, we should always ask: what's the evidence for causation, and what alternative explanations might exist? This skeptical approach helps us avoid being misled by spurious correlations and make better decisions based on a deeper understanding of causal relationships.

Chapter 3: Statistical Significance: What the Numbers Really Tell Us

When scientists announce they've found a "statistically significant" result, what exactly does that mean? Despite its authoritative sound, statistical significance is a specific technical concept with important limitations. At its core, it addresses a simple question: How likely is it that the pattern we observed could have occurred by random chance alone? The traditional approach uses p-values, where p < 0.05 means there's less than a 5% probability the observed result would occur if only random chance were at work. This threshold became a somewhat arbitrary standard for publication in scientific journals. But this approach has serious flaws. For one, it creates a false binary—results are deemed either "significant" or "not significant," when reality involves degrees of evidence. More problematically, it answers a narrow question about randomness, not whether a finding is important, large in magnitude, or reliable. Consider a medical study testing whether a new drug reduces recovery time for a common illness. If the drug reduces recovery by just 1% on average, but the study includes thousands of patients, this tiny effect might be statistically significant simply because the large sample size allows researchers to detect even minimal differences from chance. Conversely, a potentially important effect might be deemed "not significant" in a small study, leading researchers to incorrectly conclude the treatment doesn't work. The misuse of significance testing has contributed to the "replication crisis" in science. Researchers might analyze their data multiple ways until finding a "significant" result (p-hacking), or file away studies with "non-significant" findings (publication bias). One famous study demonstrated these problems by showing "statistically significant" brain activity in a dead salmon—a result achieved by testing so many brain regions that random noise eventually produced a "significant" pattern in one location. A more nuanced approach to evidence considers multiple factors: the size of the effect, not just its statistical significance; whether the finding has been replicated; whether it aligns with existing knowledge; and the quality of the study design. Bayesian statistics offers an alternative framework that explicitly incorporates prior knowledge and expresses results as degrees of belief rather than binary significance. Understanding these concepts helps us evaluate scientific claims more thoughtfully. When reading about a new study showing that coffee prevents cancer or that a certain personality trait predicts success, we should ask not just "Is it statistically significant?" but "How strong is the evidence overall?" This approach helps us avoid both dismissing real effects and being misled by statistical flukes.

Chapter 4: Regression to the Mean: Why Extremes Don't Last

Imagine a baseball player who hits an astonishing .360 in his rookie season, far above the league average of .270. The next season, his batting average drops to .310—still excellent, but lower than his spectacular first year. Sports commentators might attribute this to the "sophomore slump" or suggest he's facing more pressure. But often, the simplest explanation is a statistical phenomenon called regression to the mean. Regression to the mean occurs whenever an outcome involves both skill and luck. Extreme performances—whether exceptionally good or bad—typically combine skill with unusually good or bad luck. When measured again, the luck component is likely to be less extreme, pulling the result closer to the average. This doesn't mean skill has changed; it's simply that extraordinary luck rarely strikes twice in the same way. The statistician Francis Galton first formalized this concept in the 1880s while studying heredity. He noticed that the children of exceptionally tall parents tended to be shorter than their parents (though still taller than average), while children of very short parents tended to be taller than their parents. This wasn't due to biological mechanisms, but to the mathematics of correlation when measurements contain random components. This phenomenon appears everywhere. Companies with exceptionally high profits one year tend to perform less spectacularly the next. Students who score particularly well or poorly on one test typically move toward the average on subsequent tests. The most successful mutual funds in one period rarely maintain their top performance in subsequent periods. None of these patterns requires a special explanation—they're mathematical inevitabilities when outcomes combine stable factors with random variation. Failing to recognize regression to the mean leads to numerous errors. We create false narratives to explain natural statistical fluctuations. We implement interventions after extreme events, then incorrectly attribute subsequent improvements to our actions rather than regression. The Sports Illustrated cover jinx—where athletes featured on the magazine cover often perform worse afterward—isn't supernatural but a predictable consequence of regression to the mean. Athletes typically appear on the cover after exceptional performances, which are unlikely to be sustained. Understanding regression to the mean helps us avoid overreacting to extreme outcomes, whether in business, education, sports, or our personal lives. It reminds us that unusual results should be viewed with appropriate skepticism, and that sustained excellence is more impressive than occasional brilliance. This concept doesn't mean that genuine improvement or decline is impossible, but it does suggest we should be cautious about attributing meaning to changes that might simply reflect the natural statistical tendency of extreme results to moderate over time.

Chapter 5: Expected Value: Making Smarter Decisions Under Uncertainty

When faced with uncertain outcomes, how should we make decisions? Expected value provides a powerful framework by calculating the average outcome over many repetitions. It's not what you expect to happen in any single instance, but rather what would happen on average if you faced the same situation repeatedly. The concept is straightforward: multiply each possible outcome's value by its probability, then add these products together. Imagine you're deciding whether to bring an umbrella today. The forecast says 30% chance of rain. If you bring the umbrella and it doesn't rain, you're slightly inconvenienced by carrying it (let's say -$2 worth of hassle). If it rains and you don't have an umbrella, you get soaked (perhaps -$20 worth of discomfort). The expected value of bringing the umbrella is (0.7 × -$2) + (0.3 × $0) = -$1.40, while the expected value of leaving it home is (0.7 × $0) + (0.3 × -$20) = -$6. Since -$1.40 > -$6, bringing the umbrella has higher expected value. This approach explains why economist George Stigler famously advised, "If you never miss a plane, you're spending too much time in airports." The optimal strategy always involves some risk of failure. If you arrive at the airport three hours early for every flight, you'll never miss one, but you'll waste countless hours. By calculating expected values, you might determine that arriving closer to departure time is better overall, even if it occasionally means missing a flight. Expected value calculations appear throughout life, from insurance decisions to gambling strategies. The concept explains why casinos always win in the long run—each game has a negative expected value for players. It also explains why the lottery is generally a bad investment, except in rare circumstances when jackpots grow extremely large. Even then, the expected value calculation must account for the possibility of sharing the prize with other winners. The concept has limitations. Expected value works best for decisions you'll face repeatedly, where the law of large numbers can work its magic. For one-time, life-changing decisions, other considerations may dominate. Additionally, the utility of money isn't linear—gaining $1,000 feels better than losing $1,000 feels bad—which is why we need the more sophisticated concept of expected utility. Nevertheless, expected value remains an essential tool for making smarter decisions under uncertainty and avoiding the common pitfall of focusing too much on unlikely but dramatic outcomes.

Chapter 6: Bayesian Thinking: Updating Beliefs with Evidence

Bayesian thinking offers a powerful framework for updating our beliefs as new evidence emerges. Named after Thomas Bayes, an 18th-century minister and mathematician, this approach treats probability not just as a frequency of events but as a degree of belief that changes as we gather information. The core of Bayesian reasoning is simple: we start with a prior belief (what we think before seeing new evidence), examine the likelihood of observing our evidence given different possible truths, and then calculate a posterior belief (what we should think after seeing the evidence). This process mirrors how rational thinking should work—we don't discard our previous knowledge when confronted with new data, nor do we ignore new information that challenges our existing views. Consider a medical example: suppose a test for a rare disease (affecting 1 in 10,000 people) is 99% accurate. If you test positive, should you panic? Most people intuitively think the chance of having the disease is 99%, but Bayesian analysis reveals otherwise. We must consider both the accuracy of the test and the rarity of the disease. The calculation shows that even with a positive result, your chance of actually having the disease is less than 1%. This counterintuitive result emerges because false positives vastly outnumber true positives when testing for rare conditions. Bayesian thinking helps explain why different people can rationally reach different conclusions from the same evidence. If two people start with different prior beliefs—perhaps due to different background knowledge or experiences—they may legitimately arrive at different posterior beliefs even after seeing identical new information. This doesn't mean all beliefs are equally valid, but it does explain why reasonable people can disagree. The approach also illuminates the concept of extraordinary claims requiring extraordinary evidence. If someone claims they saw a dog on their morning walk, you might believe them without much evidence—dogs are common. But if they claim they saw a unicorn, you'd demand much stronger proof. Bayesian analysis formalizes this intuition: when a claim contradicts strong prior knowledge, we need correspondingly strong evidence to overcome our initial skepticism. Bayesian methods have practical applications across many fields. In spam filtering, your email program calculates the probability a message is spam given the words it contains. In medical diagnosis, doctors implicitly use Bayesian reasoning when interpreting test results. And in science, Bayesian statistics provides tools for weighing competing theories based on experimental data. By explicitly acknowledging our prior beliefs and systematically updating them with new evidence, we can avoid many common reasoning errors and make better decisions in an uncertain world.

Chapter 7: The Wisdom and Madness of Crowds

The wisdom of crowds describes the remarkable phenomenon where the collective judgment of many individuals often outperforms that of even the smartest experts. This concept, far from being merely a democratic platitude, has solid mathematical foundations that explain when and why groups make better decisions than individuals. The classic demonstration comes from a 1906 county fair where statistician Francis Galton observed a contest to guess the weight of an ox. While individual guesses varied widely, Galton discovered that the average of all 800 estimates was astonishingly accurate—within one pound of the true weight. This wasn't a fluke. Under the right conditions, aggregating many independent judgments can cancel out individual errors and home in on the truth. For the wisdom of crowds to work effectively, three conditions are generally necessary. First, the crowd needs diversity of opinion—people must bring different perspectives and background knowledge. Second, individuals must make independent judgments, not influenced by others' opinions. Third, there needs to be a mechanism to aggregate these diverse viewpoints into a collective answer, such as taking an average or median. The mathematics behind crowd wisdom connects to the law of large numbers. Random errors in individual judgments tend to cancel out when averaged, leaving the signal (the truth) more visible. If errors are truly random and independent, the accuracy of the group estimate improves with the square root of the group size—doubling the number of participants makes the collective estimate about 40% more accurate. However, crowds can also go maddeningly wrong. When judgments aren't independent—when people influence each other or share the same biases—crowds can amplify errors rather than correct them. This explains why crowds sometimes make catastrophically bad decisions, from financial bubbles to mob violence. The key distinction is between statistical aggregation of independent judgments (which often works well) and social influence processes (which can lead groups astray). Modern applications of crowd wisdom include prediction markets, where people bet real money on outcomes like election results or product sales. These markets often forecast events more accurately than individual experts because they aggregate diverse information held by many participants. Similarly, citizen science projects harness collective intelligence by having volunteers classify galaxies or identify protein structures, often achieving remarkable accuracy through aggregation. Understanding when crowds are wise and when they're not helps us design better systems for collective decision-making and knowledge creation. It also reminds us to value diversity of thought and independence of judgment, not just in formal settings but in everyday problem-solving. The wisdom of crowds isn't magic—it's mathematics, and it works best when we understand and respect its underlying principles.

Summary

Mathematical thinking offers us powerful tools to navigate a world filled with uncertainty, misleading statistics, and cognitive traps. The concepts explored in this book—from non-linear relationships to Bayesian reasoning—aren't just academic curiosities but practical mental equipment for making better decisions and avoiding common errors. They help us recognize when correlation doesn't imply causation, understand why extreme performances tend to moderate over time, and appreciate why expected value calculations can lead to counterintuitive but optimal choices. Perhaps the most valuable insight is that mathematics isn't just about calculation—it's about clear thinking. When we approach problems with mathematical rigor, we learn to question our assumptions, consider alternative explanations, and recognize the limits of our knowledge. This doesn't mean we need complex equations to think well, but rather that we should adopt the mathematician's habit of precise reasoning and careful analysis. For anyone seeking to navigate an increasingly complex world filled with data and statistics, developing this mathematical mindset isn't just helpful—it's essential for seeing through misleading claims and making decisions based on sound evidence rather than cognitive biases or statistical illusions.

About Author

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Jordan Ellenberg

Jordan Ellenberg is the John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison. His writing has appeared in Slate, the Wall Street Journal, the New York Times, the Washington Post, the Boston Globe, and the Believer.

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