Key Messages
Unlock The Hidden Math Of Everyday Life To Avoid Being Wrong
It’s amazing how much mathematics influences our lives, yet so many of us are intimidated by its complexities or think it is irrelevant.
In truth, the language of mathematics is simply a way to capture and communicate ideas we already understand intuitively.
With How Not to Be Wrong, readers can dig deep into understanding how math influences everything and how they can use it to make sure they’re not wrong.
For example, you’ll learn why research findings published in journals can be incorrect and a debut novelist’s second book usually isn’t as memorable as their first—all topics that require math acumen to truly understand.
Ultimately, you will come away from this book with an appreciation for the power of math on everything around us and an ability to reason through problems in an informed manner.
This understanding goes beyond just “public opinion”—it gives you the power to draw powerful conclusions that go beyond what anyone else has discovered!
Math Is The Science Of Common Sense – Using It To Avoid Wrong Answers
Mathematics is not something reserved for high-level mathematicians or engineers.
It’s the science of avoiding being wrong, and it’s based on our common sense.
Consider this example: during WWII, planes returned from Europe with bullet holes in their fuselages.
A mathematician suggested that rather than outfitting those returning planes with better armor in their fuselages, they should instead fortify their engines to minimize the number of planes that were shot down.
This may not seem like a math problem at first glance, but it actually illustrates an issue known as survivorship bias – the logic error of concentrating on the things that “survived” some process – which can only be solved by applying math correctly.
Moreover, mathematics is also derived from our intuition; understanding why adding seven stones to five stones is the same as adding five stones to seven stones requires no mathematical explanation – it simply makes intuitive sense.
Math helps us understand subjectivity and validate our common sense in a way that can be used to identify pitfalls before making decisions or formulating conclusions – essentially, its purpose is to help us stay right when presented with complicated problems or situations.
That’s why math is not limited to complex equations; it’s about using reason to not be wrong about things and make sure we’re always right.
Linearity, The Idea Of Simplifying Difficult Problems Into Easier Ones, Is Used Across A Variety Of Fields
Linearity is one of the most powerful tools we have for simplifying mathematical problems, and the concept has been used by mathematicians throughout history.
By assuming linearity, hard problems can be broken down into easier ones.
This can be done in a variety of ways, such as assuming that straight lines represent linearity in geometry and applying this concept to estimating the area of a circle.
In statistics, linear regression can be used to measure how two variables are related – such as salary level and voting preference- by plotting data points on a graph and approximating their trend using a straight line.
Thanks to linearity, challenging mathematical problems can be made much more manageable, allowing us to uncover relationships between different factors without having to account for each individual data point separately.
Chance And Probability: Navigating The Unpredictability Of Science Through Data Analysis
Drawing conclusions from observational data can be an uncertain process.
While scientists use data to build theories, random coincidence can also be responsible for it.
To illustrate this point, consider a popular experiment where a neuroscientist showed photos of people to a dead fish and measured its brain activity; surprisingly the fish responded accurately to the emotions in the pictures.
This example demonstrates how easy it is for research findings to arise simply by chance.
One way of avoiding such inconclusive results is by using probability theory.
For instance, if you’re testing a new drug to determine its efficacy, you could employ a null hypothesis significance test.
This involves setting assumptions around what will happen in the test before conducting it; usually that the new drug does nothing at all under normal circumstances.
Then, you evaluate your results and assess the likelihood that they occurred by chance – referred to as the p-value.
Usually, if it falls under 0.05 then your data can be considered statistically significant and with 95 percent certainty you know that your drug has some effect.*
In conclusion, drawing conclusion from observational data can be somewhat precarious but probability techniques are useful in making sure our findings are reliable.
The Power Of Probability: Know The Expected Value And Risk Before You Take A Chance
Probability theory can give us insight into the expected outcome of a given bet.
It can tell us the likely result when we buy a lottery ticket, make an investment, or purchase life insurance.
By calculating the expected value of each outcome, we can make more informed decisions.
However, while probability theory helps us know what to expect from a bet, it doesn’t always account for risk.
The expected value of taking part in a 50/50 bet might be the same as receiving $50,000 outright—but if you lose that bet, your losses could be much higher than doing nothing at all!
That’s why it’s important to consider the risks before making any kind of risky bets or investments.
If you don’t have enough money to cover the potential losses that come with a certain bet or investment, it might not make sense to take part in it.
Why Regression To The Mean Sends A Writer’S Second Book Spiralling: A Phenomenon Called The “Regression Effect”
The regression effect states that an unlikely event that happens is likely to happen less in the future, and more unlikely to occur for consecutive times.
This phenomenon can be observed in all areas of life, but it’s often not recognized.
Take the example of short people usually having short children and tall people usually having tall children.
While their subsequent offspring might still be tall or short, there is a chance that they will end up closer to the mean than their parents were, due to factors such as health, eating habits and sheer luck that may not line up again for the very tall or very short parents.
Similarly, research into bran regulating digestion was similarly misattributed – if a person reports faster digestion rates one day, then they are more likely to have slower digestion rates the next regardless of whether they consume bran or not due to the regression effect.
The same principle applies when analyzing a famous writer’s second book which isn’t as well-crafted as his first – critics often attribute this to exhaustion when really it’s just because of regression taking place.
The regression effect occurs everywhere and should definitely be taken into account whenever looking at data or why certain events happen!
Linear Regression Is A Powerful Tool, But It Can Lead To Misleading Results When Applied Incorrectly
Linear regression is an invaluable statistical tool to help us make sense of how variables are related.
However, linear regression can be misleading if used incorrectly – namely, if it’s assumed that a data set has linear trends when in fact those trends may not exist.
In order to use linear regression properly, all the points in the data set must be plotted on a graph and then find the line that comes closest to passing through them.
But linear regression will only work if the data points are in a generally linear shape.
Using this technique for nonlinear phenomena can produce inaccurate results.
This happened in 2008 with a paper published by Obesity claiming that all Americans would be overweight or obese by 2048, determined by using linear regression on a graph indicating obesity levels against time.
But because these levels tend to increase or decrease gradually over long periods of time rather than linearly, their predictions were incorrect – 109% of Americans would have been obese by 2060!
Simply put, while linear regression is an extremely powerful analytical tool – it is important to take into account whether or not your data has actually followed a linear pattern before jumping to conclusions based off of its application.
John Ioannidis’ Publication Shows That Most Published Research Findings Are Not Reliable
John Ioannidis’ influential 2005 paper, “Why Most Published Research Findings are False,” illustrated why many research findings can be incorrect.
One of the main reasons is due to misused data and probability calculations.
For example, Ioannidis discussed that even if only 10 genes might be related to schizophrenia, it’s possible that up to 5,000 genes could pass a 95 percent significance test by chance.
Another issue involves researchers overlooking or hiding the results of unsuccessful experiments.
This can lead to unjustified attention and focus on experiments that have produced successful outcomes, while ones which measure something different remain obscure.
Additionally, scientists may too often tweak their results to make them statistically significant if they’re marginally below certain cutoffs.
These examples show how easily an experiment can give false evidence if data is not handled correctly.
This overreliance on single studies with little oversight can create an unhelpful echo chamber for scientific knowledge and give rise to misjudgment in the field of science.
Measuring Public Opinion: Dig Deeper Than A Simple Majority
When trying to measure public opinion, it’s important to be wary of polls that appear to represent this opinion.
People may have conflicting opinions and can even contradict themselves.
For example, a January 2011 CBS News poll reported that 77 percent of respondents thought cutting spending was the best way to address the federal budget deficit, while only one month later a Pew Research poll showed that in 11 of 13 categories more people wanted to increase government spending than cut it.
Speaking of “the majority” can also be misleading since it relies on having only two options.
An October 2010 poll found that 52 percent of respondents opposed the U.S.
Affordable Care Act, with 41 percent supporting it – however when the numbers were broken down differently it revealed that almost half of those who had voted against the bill actually supported its main premise.
This occurred too in the 2000 U.S presidential election, where George Bush won 48.85 percent of the Florida vote compared to Al Gore’s 48.84 percent and Ralph Nader’s 1.6 percent – yet nearly every person who voted for Nader would’ve prefered Gore over Bush, so it is safe to say that a 51% majority would prefer Gore over Bush despite what was officially announced as the outcome.
These examples show how polls and elections which make statements about public opinion are often wrong or misleading- which is why we must always take their results with a grain of salt!
Wrap Up
In ‘How Not to Be Wrong’, the key message is that mathematics is derived from common sense and can teach us how to avoid being wrong.
Using mathematical ideas, we can uncover hidden math in our lives which can help us in various situations, from understanding the newspaper to knowing whether to buy life insurance.
The actionable advice given by this book is not to buy lottery tickets.
Lotteries make their money by convincing people to pay for a ticket with little likelihood of reward.
Calculating the expected value of the payoff shows that in most cases, you’ll end up losing more money than you win.
The book provides an enlightening summary that helps us understand where mathematical ideas come from and how they help us make better decisions in our lives.
It serves as a wonderful reminder not to take things at face value, but rather to trust ourselves and use proper math-based strategies whenever practical.
