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This is another “fun with math” post meant to impart simple math knowledge that could make the world a better place.
If you aren’t a statistician/math aficionado/empirical economist, you’ve probably never thought about what a “percentage point” is or how it’s different from a “percent”. Frankly, I hadn’t either until one of my advisers in graduate school asked if I was reporting results in percent or percentage points. The realization that there was a difference was eye-opening. Basically, a “percentage point” is always out of 100, whereas a “percent” is always relative to some baseline rate. We can also think of percentage point as telling us something on an absolute scale and a percent as telling us something on a relative scale.
Let me give you an example of why it matters. What sounds scarier, if I tell you that your probability of getting into a fatal car crash is 50 percent higher when driving over the speed limit or if I tell you that your probability of getting into a fatal car crash is 0.5 percentage points higher when driving over the speed limit? Chances are, the first one sounds a lot worse. But the 50% number is relative to some baseline crash rate, which is probably very very low (maybe 1 in a million if we’re talking about a day’s worth of driving). So multiplying that by 1.5 still leaves you pretty safe. By contrast, raising your risk of a fatal car crash by 0.5 percentage points brings it from 0.0001% to 0.5001% - more than a 5000% increase! (By the way, “%” usually refers to “percent”. If you want to talk about percentage points, you should just write it out.)
Why should you care about this difference? Because it’s often helpful to know differences in percentage points rather than percent, especially when it comes to rare events. For example, the risks of birth defects rise dramatically in percent/relative terms with the mother’s age, but the percentage points/absolute changes are actually pretty small. According to this page, 20 year old women have about a 0.19% chance of having a baby with some chromosomal anomalies, whereas 40-year-old women have a 1.52% chance. If you calculate the percent increase in risk, it’s huge, almost 800%. But the percentage point change is clearly much smaller, just 1.33.
Moral of the story - as a rule of thumb, if you want to scare or impress someone, use percent. If someone is trying to scare or impress YOU, ask them what the percentage point/raw difference is.
Search for “What successful people do” and you’ll find dozens of articles and books divulging the secrets. Success could be measured by learning a language, starting a business, having good relationships with people, being happy, etc. Typically, the writers will find some “successful” people and scrutinize their strategies. But what few people realize is that looking at what successful people do is not enough to figure out whether a particular strategy is associated with a higher probability of success. Why? Because you need to (at least) know the rate at which “normal” people employ that strategy!
Let me give you a simple example. Suppose that you read in a book that 80 percent of “successful” people get up at or before 6am every day. 80 percent - that’s a lot! But suppose then I tell you that 80 percent of “normal” people get up at or before 6am every day. All of a sudden, the strategy of getting up at 6am doesn’t look so impressive – it’s just something that all people do. If I tell you that 90 percent of normal people get up at 6am, then it looks like you’re better off sleeping in!
Conversely, just because a strategy is rare among “successful” people doesn’t mean that it’s not a good strategy for success. Let’s go back to the 6am example and say that only 10 percent of successful people get up at or before 6am every day. Is it beneficial to sleep in then? Not if I tell you that only 2 percent of “normal” people get up that early – then successful people are 5 times more likely to get up early than normal ones! So it’s vital to know the rate of a behavior in the “general population” if you want to understand its correlation with “success”.
Of course, I have to end this post with a “correlation-is-not-causation” cautionary tale, as every good scientist should. Suppose I tell you that 90 percent of successful people get up later than 6am (as in the previous example), but only 10 percent of “normal” people do (so success = 9 times more likely to get up later than 6am!). Does this mean sleeping in will bring you good fortune? Absolutely not. It’s very possible (and maybe even likely) that success allows you to “rest on your laurels” and relax, so it’s being successful that causes sleeping in, not the other way around.
Here’s a very practical (and somewhat simplified) math problem that I was faced with today.
You’re deciding whether or not to take the bus or the subway. The stations are right next to each other, they cost the same amount of money, and they’re equally uncomfortable to ride. Suppose that the subway runs every five minutes, but you don’t know how often the bus runs or when the last bus came. Once you get on the bus, it only takes you 20 minutes to get to the destination. It takes 40 minutes by subway. You’re running late, so you want to minimize your travel time. If you’re really into economic details, assume you’re risk neutral and there’s no discounting.
Question 1: How do you figure out how long it’s been since the last bus came?
Question 2: What other piece of information do you need to figure this out exactly? (There are at least two acceptable answers here, but there may be more)
Question 3: Assume that piece of information is 30. Do you take the bus or the subway?
Question 4: Wasn’t that fun?
If you see me standing at a bus stop, this is a good example of what I’m thinking about.
See the first comment for the answer!