# How Not to Be Wrong – Jordan Ellenberg | Free Book Summary

## How Not to Be Wrong – Jordan Ellenberg

How not to be deceived by mathematical traps.

A mathematician is always asking, “What assumptions are you making? And are they justified?”

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## Math is the study of avoiding errors

Math is a science that helps us solve common problems and enable us to be right about our decisions. Math is mostly just common sense. We intuitively use logic and reason to solve “math” problems.

However, we often fail at this logic—for example, in survivorship bias. When analyzing data, survivorship bias only focuses on the people or things that succeeded while ignoring the ones that failed.

## We confuse probability with risk

We use probability to assess how risky something is, such as an investment or an action we want to take. But if we just use probability to assess risk, we fail to take into account how bad the potential negative outcomes will be if they do occur.

For example, would you rather get

\$50,000

or have a 50/50 chance of losing \$100,000 or getting \$200,000?

The expected value is the same, but the negative results in the second scenario would be really bad. The risk is much higher.

## Take the findings of scientific research with a grain of salt

Headlines like “New study shows milk is related to Alzheimer’s” “This study reveals how much you really do at work.” is problematic for these reasons:

• Sometimes insignificant results can pass statistical tests.
• Unsuccessful studies are rarely published. Example: 20 studies test chocolate for causing constipation. If 19 studies fail and only the one that finds a correlation is published, it can change our perception.
• Researchers fabricate results. If they only need one extra percentage point to comply with scientific standards, they may tweak the data because they believe what they found is true.

Dividing one number by another is mere computation. Figuring out what to divide is mathematics.

## Other mathematical traps

• Small datasets have a lot more noise in them. When looking at conclusions based on data sets, see how much noise or signal they contain.
• Percentages are good for positive quantities but should be avoided for negative numbers.
• Statistical observations. People are quick to see patterns that don’t exist.
• Correlation is not transitive. A father is related by blood to his son; the son is related by blood to the mother; but the mother and father are not related by blood.
• Uncorrelated does not imply unrelated.

## False linearity

People often assume that the rate of change of a quantity will remain the same over time.

If obesity increased 1% last year, they assume the same will happen in the following years. One research paper concluded that 100% of Americans will be obese by 2048, then applied false linearity to the same data and concluded that only 80% of black men will be obese at that time.

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