# How Not to Be Wrong | Jordan Ellenberg

Summary of: How Not to Be Wrong: The Power of Mathematical Thinking

By: Jordan Ellenberg

## Introduction

Dive into the world of applied mathematics with ‘How Not to Be Wrong: The Power of Mathematical Thinking’ by Jordan Ellenberg. This book unveils the hidden mathematical formulas and principles that shape our everyday lives, from making decisions to identifying patterns. Explore how math empowers us to solve real-life problems, helps us navigate uncertainty with probability, and enhances our understanding of the world around us. Unravel the interconnected web of statistics, geometry, and reasoning that lies beneath seemingly mundane situations. With a blend of lucid explanations and engaging examples, Jordan Ellenberg demonstrates that math is not only essential but also exciting, enlightening, and often surprising.

## Math – The Science of Not Being Wrong

Math is a practical tool used in solving everyday problems and is a reflection of our intuition. The science of mathematics is based on common sense and helps us reason and make sense of things.

Remember the times when you thought convoluted mathematical formulas in school were useless? Well, think again. Math is an essential tool in solving everyday problems. The truth is, we all use math daily, but we seldom perceive it as such.

Mathematics is the science of not being wrong, and in essence, it is about using reason to solve problems. For instance, during World War II, military advisors were curious why a plane always had more bullet holes on its fuselage than the engine upon returning from tours in Europe. Exploring this phenomenon, a young mathematician suggested that planes that had bullet holes on the engine could not make it back. He proposed reinforcing better armor for the engine instead of the fuselage, which could save more planes. This was due to the mathematical phenomenon known as the survivorship bias. The concept relates to the logical error of concentrating on the things that survived a particular process. In this case, advisors ignored the planes that never made it back because they focused on the planes that did. Therefore, mathematics revolves around using logic to avoid making judgments based on incomplete data.

Moreover, math is based on intuition and common sense. For instance, if you add seven stones to five stones, it’s the same as adding five stones to seven stones, but it’s hard to explain why it works. This is commutative, and it’s what underlies math principles. Mathematics reflects the things we already know intuitively. Thus, we can’t solve equations entirely based on intuition, but math is derived from our common sense.

In summary, math is indispensable in solving everyday problems, and it is a reflection of our intuition. The science of mathematics is based on common sense and helps us reason and make sense of things.

## Linear Regression Simplified

The concept of linearity simplifies difficult mathematical problems by breaking them down to simpler ones. Linear regression in statistics uses this approach to measure how specific observations relate. By assuming linearity, we can use straight lines to approximate the area of shapes such as circles. This method of simplification is commonly used in statistical analyses to find trends in data. Rather than connecting every data point, linear regression offers an approximation of the overall trend represented as a straight line.

## Drawing Conclusions from Observational Data

Observational data can be misleading and drawing conclusions from it can be quite precarious. This is because data can arise by chance. Scientists use probability theory to address this issue through null hypothesis significance tests. The tests involve assuming that the data does nothing at all, then calculating the probability that the observed data came about by chance. If the probability is less than a certain p-value, the data is considered statistically significant. In this way, scientists can make conclusions with 95 percent certainty about the proposed effect.

## Understanding Probability

Probability theory is an invaluable tool in situations of uncertainty. It can predict what we should expect to happen, such as the value of a lottery ticket or a stock option. Expected value is calculated by multiplying the chance of each outcome by the value given that outcome and adding up the results. However, expected value doesn’t take into account the risk involved in a given bet. This means that it’s essential to examine the risks involved before making a bet or investment. For instance, a risky investment is only a good idea if one has enough money to cover possible losses.

## The Regression Effect in Artistic Success

Have you ever wondered why a novelist’s second book tends to not be as successful as the first? The answer may lie in a mathematical concept called the regression effect, which states that an unlikely outcome will likely regress to the mean in the next attempt. This phenomenon affects anything that involves randomness, including genetics, which explains why the children of very short and very tall parents tend to have heights closer to the average. However, regression is often unrecognized, leading researchers to mistake it for a biological phenomenon. For instance, when studying the effects of bran on digestion, participants who reported a fast digestion rate followed by slow digestion were likely experiencing the regression effect instead of bran’s actual effects. Similarly, when a famous writer’s second book is not as well-crafted as their first, literary critics often attribute it to “exhaustion,” when it’s just mathematics at work. Understanding the regression effect on artistic success is essential for creators and critics to avoid mistaking it for other reasons, which could lead to incorrect conclusions.