Pop quiz: what’s the difference between the two x-y graphs below?
The answer is that they’re same graph — but the first picture above, we’ve zoomed in on one side of the bell curve.
In the book How Not To Be Wrong: A Guide To Mathematical Thinking, author Jordan Ellenberg points out that if you focus on a small part of a curve, it looks rather a lot like a line. Human like lines because they’re easy to understand. Plot some data and draw a line, and you have visual evidence that more of this produces more of that, or that more of this results in less of that. If the x-axis represents a span of time, you can even predict what the data will look like years from now by extending the line into the future.
The trouble is, real life tends to be more curved and less tidy than our beloved trendlines would have us believe. Two variables that were happily holding hands and running uphill together on our chart can suddenly break up and go their separate ways. So the trendline flattens out or reverses direction completely. That’s often how real life is, and if we don’t want to be wrong (as the title of the book says), we shouldn’t assume that a trendline heading in a particular direction will continue in that direction forever.
For example, you’ve probably seen headlines announcing that by the year 2040, everyone in America will be obese. Yup, every single person. Those predictions are based on charts that look something like this:
Researchers take a 25-year trendline that runs from 1990 to 2015 and simply extend it another 25 years. That is, of course, a wee bit ridiculous. Some decent-sized fraction of the population is highly resistant to becoming fat. Even if the Standard American Diet doesn’t undergo a positive shift (which I think it already is), the likely scenario is that when the actual data is plotted in 2040, it will look something like this:
Trendlines can flatten out. It happens all the time. It’s what we see when the law of diminishing returns kicks in. If I’ve been lifting weights once per month and switch to twice per month, I’ll likely become stronger. Kick it up to once per week and I’ll probably become stronger again. But at some point, more frequent workouts won’t make me stronger. There’s a limit to how quickly a body can produce new, stronger muscle cells.
Trendlines can also rise and then fall, producing a bell curve. More of this means more of that up to a point, but then even more of this leads to less of that. Interestingly (at least for those of us who enjoy reading about economics), Ellenberg introduces the concept by writing about taxes and the revenues they produce.
I’ve had debates with big-government-lovin’ acquaintances who are convinced there’s no national problem we couldn’t solve if we just had the political courage to seriously jack up taxes – especially on those darned rich people. In their minds, the relationship between tax rates and tax revenues looks like this:
Higher tax rates always produce higher revenues and therefore more government goodies for all – no matter how high the rates go.
But that’s not how it works in real life. Ellenberg doesn’t argue in favor of any particular tax rate, but he points out what any sane economist knows: at some point, higher tax rates produce less revenue, not more, because (among other reasons):
- People have less disposable income to buy goods and services, which means fewer people will be employed to provide them.
- More people decide to participate in the underground economy to avoid high taxes.
- Fewer people can save the capital to start a new business.
- People who already have the capital to start a new business decide they won’t bother if they’re taking all the risks but Uncle Sam is going to take most of the reward.
- People in the highest income brackets often quit working for the year once any additional income they earn is taxed at a rate they find unacceptable.
So the relationship between tax rates and tax revenues is a bell curve:
Again, Ellenberg doesn’t advocate for a particular tax rate. He simply points out that in any given set of economic circumstances, there’s going to be a rate far greater than 0% and far less than 100% that produces the most revenue. If we only focus on the left side of the curve, we’ll end up believing that higher rates always produce more revenue. If we only focus on the right side of the curve, we’ll end up believing that lower rates always produce more revenue. Both beliefs are wrong – and of course, the title of the book is How Not To Be Wrong.
One way to avoid being wrong is to understand that in real life, the relationship between this-and-that often takes the form of a bell curve. Starting from a low level, more of something may be good … but that doesn’t necessarily mean a LOT more is better. Starting at a high level, less of something might also be good … but that doesn’t mean zero is better.
We see the bell-curve relationship all the time in the health sciences. How much vitamin D should you get? The answer would surely look something like this:
Starting from a low level, more is better. But that doesn’t necessarily mean a LOT more is better. At some point, a LOT more is toxic.
How much protein should you eat? We know too little is bad for your health. You’ll lose muscle mass and your immune system will weaken. But too much can cause diarrhea and dehydration. So the relationship between protein and health is a bell curve.
But my bell curve probably doesn’t peak at the same point yours does. And a muscular athlete who engages in heavy workouts would need protein than I do. So the relationship between protein and health may look something like this:
Not every relationship in health is a bell curve, of course. If we’re talking about rat poison and health, the relationship probably looks like this:
Less is better, period. Sometimes that’s the reality. Chareva’s mother is highly allergic to walnuts. So for her at least, the walnut-to-health relationship would look very much like the rat poison-to-health relationship above.
But getting back to the How Not To Be Wrong theme, here’s a graphic representation of a belief I once held, but now consider wrong:
Metabolic health is the highest at close to zero carbs, then goes down from there. I believed that because I went from a high-carb diet to a low-carb diet, and my health dramatically improved. So for awhile, I figured if low is good, close to zero must be even better. It may be true for some people, but it’s clearly not true for everyone.
On the low-carb cruise, I was pleased to hear Dr. Justin Marchiagiani respond to a question about ketogenic diets by saying he doesn’t recommend them for everyone. He works with each patient to find his or her ideal carbohydrate intake, which will depend on a number of factors. For some, it’s 50 grams per day or less; others feel better and lose weight more easily at 100 grams per day or more. He said he feels at his best at between 75 and 100 grams per day. That’s about where I am now as well.
There is no level of carb intake that’s ideal for everyone. So the relationship between carbs and metabolic health looks something like this:
If we only focus on the right side of the curve, we’ll end up believing that more carbs always means worse metabolic health, so the ideal level of carb intake is close to zero.
But that ignores the left side of the curve. And that’s an easy way to be wrong.