Summary of How Not to Be Wrong by Jordan Ellenberg

BookSummaryClub Blog Summary of How Not to Be Wrong by Jordan Ellenberg

Have you ever stopped to think about how often you use mathematics in your daily life? As much as you thought you would have no use for it after school, you cannot deny how much you use it. Whether you are simply adding up the cost of your groceries, or calculating how far you need to drive before you fill gas in your car or maybe even calculating how many tiles you need to buy to renovate the bathroom, there is no denying that mathematics comes in handy. 

Mathematics should be thought of as the science of common sense. Yes, it gets complicated at times and may seem like a foreign language that only some understand, but ultimately it’s not that difficult to use. If you learn to think like a mathematician, there’s a good chance that you could learn how not to be wrong.

In this book summary readers will discover:

  • How mathematics can solve everyday problems
  • Breaking down hard problems
  • Linear regression doesn’t always work
  • Probability theory
  • What’s wrong with most research findings

Key lesson one: How mathematics can solve everyday problems

Mathematics is key to solving common problems every day. We just don’t recognize it because it may not be condensed into a complicated formula. In fact, it may seem totally unrelated to what you even consider to be math.

Take, for example, American planes in World War II. Those that returned were riddled in bullet holes and in particular, the plane’s main body contained more holes than the engine. So, to better protect the planes, military advisors put forward the suggestion to improve the armour of the body of the plane. However, a mathematician suggested that a better option would be to improve the armour of the engine. This confused everyone as the engine was not the area found with the most bullet holes. The mathematician pointed out that that was why these planes had returned. Those planes that had been shot in the engine did not make it back at all. Therefore, it made sense to better protect the engines and possibly save more planes. The advisors exhibited survivorship bias, a common mathematical occurrence whereby they focused on the planes that survived and not the ones that did not. 

Even though the example seems to be logical, it is actually mathematical. Mathematics is derived from common sense. It may be sometimes hard to explain because it seems so straightforward but it is a source of what we know intuitively – without any complicated formulae.

Key lesson two: Breaking down hard problems

It’s one of the first things you learn when doing mathematics. If you are faced with a difficult problem, try breaking it down into smaller problems. By solving the small problems, you may solve the big problem or at least get closer to the solution. 

In mathematics, it is assumed that difficult problems have linearity or can be considered as straight lines. Curves, like those found in geometry, represent nonlinearity. If we zoom in on a circle close enough, a curve might actually look like a straight line if you are looking at a small enough piece of it. This can mean that the curve of the circle is almost like a series of straight lines at a slight angle. Continuing with this train of thought, if we wanted to measure the area of the circle, it could be done by placing a square within the circle so that the corners touch the circle. The area of the square is easy to calculate and for the spaces that remain within the circle, other polygons can be inserted to do the same. In this way, the area of the circle can be approximated using just straight lines. This is a simple example of linearity.

Linearity is commonly used in statistics in the form of linear regression. It’s used as a way of measuring how observations are related. It does this by plotting observations against each other on a graph and assuming linearity. It does not connect each data point on the graph but shows the overall trend of the data represented as a straight line. 

Key lesson three: Linear regression doesn’t always work

Linear regression is useful to show us how variables are related but you can’t use it all the time. The reality is that some data is just not linear and if you try to use linear regression on data like this, your results will be greatly skewed. Linear regression is only feasible when the data points are already forming a linear shape. 

A great way to understand this is if you consider a missile’s trajectory. It takes a curve-like path after it is fired. If you take a short segment of the trajectory, much like the circle discussed earlier, it appears to be linear. This means you could easily calculate where the missile will be in a few seconds later. However, we needed to calculate the missile’s location much later on, the path is no longer straight and if you assume it still is you will calculate the location incorrectly. 

This is important when considering data in our daily lives. Many findings are reported to the public without being analysed correctly. That’s why we need to be a bit more critical and carefully consider how the data is presented in a study to have confidence in the results.

Key lesson four: Probability theory

Most of the time, when scientists collect data, it is observational data. They use this data to formulate possible theories which they will investigate. But, observational data can be tricky. Sometimes, drawing conclusions from simple observations can be your downfall. 

To avoid this problem, scientists use something called probability theory. This entails them following a specific process. For example, if a new drug is being tested to cure a disease, researchers aim to test the null hypothesis. This is basically the opposite of what you want your result to be. In this case, the null hypothesis would be that the new drug does not have any effect at all. When you consider your data, you need to look at the deviation of the date. The p-value in statistics refers to the probability that your data came about by chance. If the probability is less than the p-value of 0.05, the data is considered to be statistically significant. This means that your data shows, with 95 per cent certainty, that the drug does have an effect as the results are significantly different.

Probability theory can be used in any situation that we are uncertain about. It can tell us the most likely outcome. It can even be used to let you know what a lottery ticket’s expected value really is. For example, a lottery of $6 million is available to win and 10 million people, including yourself, purchased a single ticket for $1, the expected value of a single ticket is 60 cents. This would mean that you should expect to lose 40 cents for the ticket you purchased for $1. Calculating expected value like this is also used when pricing stock options or considering life insurance. 

Expected value, however, does not reflect the risk associated with any given bet. A classic example is when the expected value is calculated to be the same on either side of the bet. Take for example a situation where you could either receive $50 000 or take part in a coin toss between losing $100 000 and gaining $200 000. The expected value is the same either way but if you lost the coin toss you would be out $100 000. This is why a risky investment should only be an option if you can cover any possible losses. You have to consider the risks carefully when taking a bet.

Key lesson five: What’s wrong with most research findings

It has already been discussed as to how data can be misinterpreted if not analyzed correctly, but why do seasoned scientists still make these mistakes? Professor John Ioannidis published a paper to discuss why it occurs so often.

His first point was that by chance, some observations can pass significance tests. Observations have to be tested thoroughly no matter how significant they seem. If we had to consider the probability of someone getting schizophrenia based on their genes, scientists would have to examine thousands of genes without even knowing how many would be associated with schizophrenia. If they considered 100 000 genes with the most commonly used significance of 95 per cent, 5 per cent of the genes or 5000 genes considered would pass the test by chance. This test is in no way specific enough in this scenario. What if there were only 15 genes associated with schizophrenia?

Ioannidis’s second point was that too many studies go unpublished because they don’t end up with successful results. This is quite common as journals tend to favour publishing positive results as opposed to experiments that did not work. As a result, some studies receive more attention and can cause a stir even if the results came about by pure chance. This may sound confusing but just consider 10 labs that test gummy bears to see if it has an effect on your skin. Out of the 10 labs, 9 of them don’t get a significant result and thus don’t bother publishing their work. The one lab that did get a significant result, does get published. So, now everyone believes, without a doubt, that gummy bear affects your skin without knowing about the results found at the 9 other labs.

Ioannidis’s last point brought to light that some scientists alter their results to obtain a statistical difference so that they can publish their work. This happens when the experiments yield results that reach close to 95 per cent certainty but do not hit the mark. If it is close enough, it is possible to tweak results so that it does get to 95 per cent. You would be surprised how often this actually occurs. Researchers tend to do it because they truly believe their hypotheses, especially since it was almost significant – but it is still wrong.

The key takeaway from How Not to Be Wrong is:

Mathematics plays a bigger role in our lives than we often realise. We use it almost daily without us even being aware of it. It is an important part of our lives and if we learn to understand how it is used outside of the classroom, we will be able to learn how not to be wrong.

How can I implement the lessons learned in How Not to Be Wrong:

As an exercise to gain awareness, start taking note of how often you use mathematics in your daily life. You may be surprised as to how important it really is – even if you are just calculating how much money you lost by buying that lottery ticket!

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