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# Linearly Nonlinear

Take two!

If taking one vitamin pill per day is good for you, then taking two pills per day should be better, shouldn’t it? If two is better than one, then three should be better than two.

If we continue to follow this logic, we’ll soon be taking a whole bottle of vitamins every day. Why don’t we? Because of two limiting factors:

• Diminishing returns – each additional pill gives less benefit than the previous pill;
• Negative returns – beyond a certain point, each additional pill makes things worse.

It’s easy to figure this out for simple items like vitamin pills. But, in more complex decisions, we tend to have a linear bias. If there’s a linear relationship between two variables, we assume that the line continues forever.

Let’s take schools, for instance. In the United States, we’re obsessed with measuring student performance in schools and tracking it over time. We create performance tables to identify the schools that provide the best education and those that provide the worst.

You may notice that small schools are often at the top of the charts. You might conclude that school size and performance are linearly related. It might be wise to build more small schools and fewer large schools. Unfortunately, you’re suffering from linear bias.

To find the error, just look at the bottom of the performance charts. You’ll probably find many small schools there as well. Small schools dominate the top and bottom of the chart; large schools tend to fall into the middle range.

What’s going on? It’s the variability of small samples. If you flip a coin ten times, you might get eight tails. If you flip the same coin a thousand times, it’s very unlikely that you’ll get 800 tails. With larger samples, things tend to even out.

The same happens in schools. Larger schools have larger samples of students and their performance tends to even out. Performance in small schools is much more variable, both upward and downward. The relationship between school size and performance is a curve, not a straight line.

For the same reason, I was briefly (but gloriously) the most accurate shooter on my high school basketball team. After three games, I had taken only one shot, but I made it! In other words, I made 100% of my shots – the top of the performance chart. But what if I had missed that one shot? My accuracy would have fallen to 0%, the very bottom of the chart. With one flip of my wrist, I could have gone from best to worst. That’s the volatile variability of small samples.

A straight line is the simplest relationship one can find between two variables. I generally believe that simpler is better. But many relationships simply aren’t simple. They change in nonlinear ways. By trying to make them linear, we over-simplify and run the risk of significant mistakes. Here are two:

• If an hour of exercise is good for you, then two hours must be better. The assumption is that more exercise equals better health. It’s a linear relationship. But is it really? I have friends who exercise for hours a day. I worry for their health (and sanity).
• If cutting taxes by 10% is good for the economy, then cutting taxes by 20% must be better. We assume that lower taxes stimulate the economy in a linear fashion. But, at some point, we must get negative returns.

What’s the bottom line here? If someone argues that the relationship between two variables is a straight line, take it with a grain of salt. And if one grain of salt is good for you, then two grains should be better. And if two grains are better, then three grains … well, you get the picture.

(I adapted the school example from Jordan Ellenberg’s excellent new book, How Not To Be Wrong: The Power Of Mathematical Thinking).

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