There are a number of studies that claim to have shown that men, on average, have significantly more (opposite-sex) sexual partners than women do. The numbers vary wildly, but the most conservative estimate in the book, coming from the National Center for Health Statistics in 2007, is that men have 75% more partners (on average), than women. However, there's a very cool and simple mathematical trick which will show that this result is literally impossible.
The key insight here is that the total number of sexual partners had by all men must be equal to the total number of sexual partners had by all women. You can see why this is true with this picture from the book:
The dots are people, and a line between two dots indicates that that those two dots got intimate at some point. You can see that if you count up the lines coming out of male dots, you get 2 + 0 + 1 + 3 + 1 = 7 total, and counting up the ones from women gives you 2 + 1 + 1 + 2 + 1 = 7 again, even though the actual distribution is different. In order for the totals to be different, there would have to be a line that came out of one dot but didn't go to another, which impossible given the way the problem is set up (i.e. male/female couples only).
So, now we know the total number of sexual partners is equal for both genders. Let's write the total for males as Y and the total for females as X (their chromosomes, if that helps you remember it). What we just showed with the picture is that Y = X. Next, let's call the total number of males M, and the total number of females F. Take the equation Y = X and multiply by 1/MF on both sides (you'll see why in a second). Now we have this equation:
This next equation is equivalent, I've just grouped the terms differently:
Now, this is the key: look at what we've put in the parentheses. What's Y/M? That's the total number of sexual partners for men divided by the total number of men. In other words, it's the average number of sexual partners for a given man. Similarly, X/F is the average number of partners for a woman.
Okay, so now what? We can multiply by F on both sides of the equation, which gives us this equation:
In other words, the average man has F/M times as many sexual partners as the average woman. So, let's look back at that original survey data - the most conservative estimate was that the average man has 75% more partners than the average woman, which means that F/M = 1.75. However, based on Census Bureau data, F/M = 1.035. In other words, the average man actually has only 3.5% more partners than the average woman. Whoops.
So what does that tell us? Well, the math can't lie, but people sure can. In fact, according to the textbook, "the principal author of the National Center for Health Statistics study reported that she knew the results had to be wrong, but that was the data collected, and her job was to report it".
The textbook gives another interesting application of the same principle:
A few years ago, the Boston Globe ran a story on a survey of the study habits of students on Boston area campuses. Their survey showed that on average, minority students tended to study with non-minority students more than the other way around. They went on at great length to explain why this “remarkable phenomenon” might be true. But it’s not remarkable at all—using our graph theory formulation, we can see that all it says is that there are fewer minority students than non-minority students, which is, of course what “minority” means.
Nick... awesome!
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