Even as a young child I thought it was a stupid sentiment, since it was plainly evident that men were better than women at sports. Not much has really changed since I reached that conclusion, except now I reflexively cringe whenever I think it, having internalized the values of a fully modernized and politically correct system of sexual (and racial, and socioeconomic, etc.) oppression.

That is, I've been well trained to pay lip service to the idea that we are all equal, and to resist those who claim otherwise, meanwhile ignoring the obvious inequalities people suffer all around me. Of course we aren't all equal! Most everyone is different from most everyone else in fairly significant ways, and on occasion even rank generalizations have some validity.

For example, it's not news that men are much better (about twenty times) than women at producing testosterone (similarly for other androgens), and as a result men are—on average, and individually at the elite level—much stronger, faster, and have more endurance over the distances and durations we generally care about. ("Ultra" endurance competitions probably aren't competitive enough to say with any degree of confidence whether one sex has an advantage over the other.)

It's pointless for elite men and women to compete against each other in sports where strength, speed, and endurance matter, which is ostensibly why they don't. Still, despite the natural advantages men have over women in sports, stories abound of exceptional women challenging the fragile male ego, only to be chewed up and spit out by the patriarchy for their temerity.

[In 1902], [Madge] Syers entered the [figure skating] World Championships and won a silver medal. However, many thought she should have won gold, and there are reports that Ulrich Salchow, who was presented with gold, gave her his medal. The International Skating Union immediately changed the rules to ban women from competing against men after this upset.

When [Jackie Mitchell's] minor league team played an exhibition against the New York Yankees, Mitchell struck out Lou Gehrig and Babe Ruth. After this, Mitchell's contract was voided (baseball being "too strenuous" for women) and the commissioner instituted a ban against women in professional baseball that lasted until 1992.

^{#2}When [Jackie Mitchell's] minor league team played an exhibition against the New York Yankees, Mitchell struck out Lou Gehrig and Babe Ruth. After this, Mitchell's contract was voided (baseball being "too strenuous" for women) and the commissioner instituted a ban against women in professional baseball that lasted until 1992.

^{#6}I suppose those vignettes, as examples of female accomplishment, are more historical curiosities than anything else. In the first half of the 20th century, nutritional and athletic science were still in their infancy, and sports weren't the highly competitive, globe spanning, cash grabbing industries they are today. Most professional athletes weren't particularly good or well trained, so it's unsurprising the occasional elite woman could compare. Sadly, it's equally unsurprising the treatment such women got after embarrassing a man or two.

Nowadays, the theory goes, we recognize the necessity of sexually segregating sports, since male musculature is so advantageous. We hope such segregation is voluntary, as it is (now) in baseball, where women like Eri Yoshida are allowed to compete against men if they want to. Figure skating has yet to follow in baseball's footsteps.

To this day, Madge Sayers would be unable to repeat her 1902 performance. The International Skating Union explicitly forbids "ladies" from competing against "men",

^{Rule 538.2 (p. 105)}though Evan Lysacek's gold medal winning routines from the 2010 Winter Olympics, for example, are well within the physical capabilities of women. (It's worth noting that since that controversy, the point value of the essentially male-only quadruple jumps has been raised.)

Beyond that, the ISU's rules are also explicitly formulated so as to stymie casual comparison between men and women. The women's free skate is 30 seconds shorter than the men's,

^{Rule 501.2 (p. 81)}giving men more time to score points, and women's scores are arbitrarily multiplied by a smaller factor than men's after grading.

^{Rule 353.1.m.i (p. 20)}

But it comes as no surprise, really, that sports where women generally don't even want to compete against men, like baseball, are more magnanimous about it all. Eri Yoshida is an inspiration, but it's unlikely she or any woman could ever be good enough for the majors. She'll never throw a 100mph fastball, or even a 90mph one. Eri's cute, not a threat, and so she is accepted. Our colors show truer in sports where men have little or no advantage over women (or where positions exist where that's true.)

Women certainly have the physical capability to be elite short- to medium-range place kickers, for example, a position that mostly doesn't exist in the NFL, but easily could, and is fairly common in the NCAA. If you have the stomach for it, read Rick Reilly's harrowing report of the verbal and physical abuse (including molestation and rape) that Katie Hnida endured as a placekicker on the University of Colorado at Boulder football team. Recoil in abject disgust as Hnida recounts her experiences as a woman challenging gender norms in our superficially liberated society. And then forget, because it seems like the easiest thing to do.

Now let's turn away from the hellacious and criminal to the merely vile and discriminatory. Consider the case of Margaret Murdock, a legend in rifle shooting, a sport where it doesn't matter in the least what's between your legs, at least, not to the gun. In 1976, she was denied a gold medal at the Summer Olympics in Montreal. It's hard to get a clear picture of what exactly transpired, as accounts differ, but it seems likely unusual and unprecedented measures were taken to ensure Margaret didn't win.

An excerpt of a 2011 article from the Topeka Capital Journal:

Murdock and American teammate Lanny Bassham tied for the gold, but Bassham was awarded the medal on tie-breaking rules.

"I thought I had won and Lanny said he thought he had one more point," Murdock said. "They were going to check our targets and we waited about three hours and finally one of the guys came out and I asked if they had done the targets and he said, 'Oh, yeah, you won.'

"About a half-hour later they came out and changed it and gave Lanny the other point."

Murdock believes being female may have had something to do with the ruling.

"I had been screwed on a couple of other occasions big-time, so it doesn't surprise you," she said.

"I thought I had won and Lanny said he thought he had one more point," Murdock said. "They were going to check our targets and we waited about three hours and finally one of the guys came out and I asked if they had done the targets and he said, 'Oh, yeah, you won.'

"About a half-hour later they came out and changed it and gave Lanny the other point."

Murdock believes being female may have had something to do with the ruling.

"I had been screwed on a couple of other occasions big-time, so it doesn't surprise you," she said.

A 1988 article from the Pittsburgh Post-Gazette offers more details:

It all came to a head in the '76 Games. Murdock first nicked teammate Lanny Bassaham [sic] by a point for the gold medal in the small-bore three-position rifle event, but officials called for a controversial rescoring. It was announced Murdock and Bassaham [sic] tied.

After denying Murdock and Bassaham's [sic] request for a shootoff and refusing the International Olympic Committee's advice that two golds be awarded, international shooting authorities gave Bassaham [sic] the gold and Murdock the silver.

"That would have been traumatic for them if I had won the gold," Murdock said. "They never did congratulate me, which is kind of unusual."

After denying Murdock and Bassaham's [sic] request for a shootoff and refusing the International Olympic Committee's advice that two golds be awarded, international shooting authorities gave Bassaham [sic] the gold and Murdock the silver.

"That would have been traumatic for them if I had won the gold," Murdock said. "They never did congratulate me, which is kind of unusual."

And finally, a 1976 article from Sports Illustrated:

She would show extraordinary patience and acceptance in the following hours, because things were far from settled. All the targets had to be examined by three judges. A shot will score a 10 as long as it touches the line around the bull's-eye, and both Bassham and Murdock had 9s they felt would be raised to 10 under close scrutiny. After two hours the official scores were posted: 1,162 for Murdock, 1,161 for Bassham, 1,160 for Seibold.

Murdock was elated. "I've never seen judges change things once scores have been posted," she said.

Then it was announced that a clerical error had been discovered—a judge had simply put a 9 where he should have written a 10—and Bassham was tied with Murdock. The tie-breaking method is to compare scores for the last 10 shots. Murdock had 96 points, Bassham 98, so the victory was his. Murdock appealed for a second review of her targets.

...

When the review Murdock had requested failed to break the tie, she and Bassham asked the International Olympic Committee whether two gold medals could be given. Fine, said the IOC, if it's agreeable to the shooting union. It was not.

...

Everything was done by the book. There was no discrimination against Margaret because she was a woman. Not here.

Murdock was elated. "I've never seen judges change things once scores have been posted," she said.

Then it was announced that a clerical error had been discovered—a judge had simply put a 9 where he should have written a 10—and Bassham was tied with Murdock. The tie-breaking method is to compare scores for the last 10 shots. Murdock had 96 points, Bassham 98, so the victory was his. Murdock appealed for a second review of her targets.

...

When the review Murdock had requested failed to break the tie, she and Bassham asked the International Olympic Committee whether two gold medals could be given. Fine, said the IOC, if it's agreeable to the shooting union. It was not.

...

Everything was done by the book. There was no discrimination against Margaret because she was a woman. Not here.

Who were the judges? Were they all male? Was there any oversight? Did the judges know to whom each target belonged? How long did judging normally take? How unusual was it for the official scores to vary from the initial scores? How unusual was it for scores to be further modified after the official scores had been posted? Were the targets in this competition under an unusual amount of scrutiny? Were all targets reexamined in the hunt for points, or just those of men? How often did "clerical errors" occur? Had the International Shooting Sport Federation ever allowed ties in the past? (That is, were the tie-breaking rules always applied consistently?)

These are all questions that are very difficult to answer decades after the fact, but should have been asked, and answered, at the time. SI's Kenny Moore was clearly the wrong man for the job. He found it more important to subtly shame Murdock by reporting that she had "put on weight in the last couple of years," had "the beginnings of a double chin," and was studying to be a nurse, a nonthreatening and traditionally female job. He described none of the other competitors physically, other than to say one of them had a beard. Thank you, Ken.

Margaret Murdock isn't the only woman to be screwed over by the ISSF. You wouldn't know it by looking at their ISSF History Milestones timeline, though:

*Let's see, 1988, 1993 … nope, no 1992.*

They proudly proclaim each year in which they "added" women's events (which prior to 1992 meant an event that had been previously open was formally segregated), but curiously didn't think it worth mentioning when they did things like controversially forbid women from competing, which is exactly what happened after the 1992 Summer Olympics in Barcelona.

At those Games, skeet shooting remained one of the last "open" (that is, not sexually segregated) shooting events, and it was won, with an Olympic record, by China's Zhang Shan. A woman.

*She's a man eater.*

Luckily for her, it's impossible to fudge the numbers in skeet shooting. You either hit the clay pigeon or you don't. There's no secretive judging and rejudging of targets. No doubt the ISSF would have liked to cheat Zhang Shan out of her medal, just as they did for Margaret Murdock, but they couldn't find a way. So they did the next best thing.

The ISSF immediately abolished all open shooting events and formally barred women from men's competitions. Not only couldn't Zhang Shan defend her title in 1996 in Atlanta, there wasn't even an equivalent event for her to participate in. Women's skeet shooting wasn't added to the Olympic program until 2000.

*International Society for Sexist Fu—err, Funambulists*

The International Shooting Sport Federation also continues the proud tradition of intentionally confounding easy comparison between men and women, by arbitrarily having women take fewer shots, or by having them shoot from shorter distances. Take a look at the events in London this year:

- 50 m rifle three positions – men take 120+10 shots, women take 60+10 shots.
- 50 m rifle prone – no women's event.
- 10 m air rifle – men take 60+10 shots, women take 40+10 shots.
- Pistol – men shoot from 50 meters, women shoot from 25 meters.
- 25 m rapid fire pistol – no women's event.
- 10 m air pistol – men take 60+10 shots, women take 40+10 shots.
- Trap – men take 125+25 shots, women take 75+25 shots.
- Double trap – no women's event.
- Skeet – men take 125+25 shots, women take 75+25 shots.

That's all of them. There isn't a single event where men and women are directly comparable. It's very important to the ISSF that a woman never again be seen to outperform a man. But what if you're not a misogynist and you just want to know who the best skeet shooter in the world is, even if it's a woman? No competition can tell us plainly, so our only option is to do some math.

We're going to take it step by step, but if you're totally unfamiliar with probability and calculus, the following analysis may be challenging.

Let's compare the performances of Vincent Hancock (Qualifying, Finals) and Kimberly Rhode (Qualifying, Finals), respectively the male and female gold medalists in skeet shooting in London:

Rd. 1 | % | Rd. 2 | % | Rd. 3 | % | Rd. 4 | % | Rd. 5 | % | Rd. 6 | % | |

Kim | 25/25 | 100 | 25/25 | 100 | 24/25 | 98.7 | 25/25 | 99.0 | -/- | 99.0 | -/- | 99.0 |

Vince | 25/25 | 100 | 24/25 | 98.0 | 25/25 | 98.7 | 24/25 | 98.0 | 25/25 | 98.4 | 25/25 | 98.7 |

*Percentages are cumulative.*

Who was the better shooter in London? This is a difficult question. Kim had the superior performance, that much is clear. She hit a higher percentage of her targets. Not only that, in a round by round comparison, at no point was Vince leading her. But Vince took more shots, and most of us understand intuitively that his consistency over a larger sample means

*something*, even if it's hard to say exactly what.

For example, if Vince had hit 98,999/100,000 (98.999%), many of us would feel uncomfortable saying he was a worse skeet shooter than Kim, even though that would again be an inferior performance to Kim's 99% when just comparing percentages. But it doesn't sound so crazy for a 95% shooter to go on a hot streak and hit 99/100 on occasion. Things like that happen all the time, in fact. It's essentially impossible, however, for a 95% shooter to hit 98,999/100,000. (A statistician would explain this formally by saying the variance of the mean decreases as the number of shots increases.)

In reality it's overwhelmingly likely (>99.99%) that a "random" shooter with a 98,999/100,000 performance is actually more skilled than a "random" shooter with a 99/100 performance. Most 99/100 performances are just pretty good (but not great) shooters getting lucky. A shooter that puts up numbers like 98,999/100,000, on the other hand, is inarguably great.

Vince's actual 148/150 (98.667%) lies somewhere in the middle of our consistency continuum. How can we further quantify Kim's and Vince's performances to account for this? Let's approach this from the point of view of a probabilist who hasn't done hypothesis testing before.

Assume the following:

- In London, Kim and Vince each had an unknown but fixed percentage chance of hitting a target based on personal skill. That's two separate probabilities, one for Kim, and one for Vince. We don't know what they are, we just suppose they exist in theory, and that they are fixed numbers.
- We don't know anything else about those probabilities except they fall somewhere between 0% and 100%, and no value in that range is more likely than any other. (This doesn't seem entirely sensible on the surface, since we have reason to believe Vince and Kim are both very skilled, that is, close to 100%, but we will see later it doesn't matter.)
- Kim's performance didn't affect Vince's performance, and vice versa.

More formally, let p

_{K}= P(Kim hits), p

_{V}= P(Vince hits), and assume they are independent and distributed uniformly, strictly between 0 and 1.

We want to know P(p

_{K}> p

_{V}| Data), that is, the probability that Kim is actually more skilled than Vince given the observed data (their performances in London.) An important theorem in probability, Bayes' Theorem, gives us a handy expression for evaluating a conditional probability we don't know in terms of a few probabilities we hopefully do. Applying it to our problem, we get:

I suppose it's worth mentioning that, in math, theorem means "a thing we know to be true," not "a thing we're wondering about and might be false." If you believe 1+1=2, and you trust that mathematicians aren't collectively idiots or liars, you must also believe Bayes' Theorem. (You can also examine the proof of the theorem for yourself, and believe it by understanding it.)

What Bayes' Theorem tells us in this case is that the probability that Kim is more skilled than Vince given the London data is equal to the probability that we would witness the London data given that Kim is more skilled than Vince, times the probability that Kim is more skilled than Vince (given no conditions), divided by the probability that we would witness the London data (given no conditions.) Fortunately, all three probabilities from the right side of the equation can be computed using integrals over (portions of) the unit square.

First, the easy one: P(p

_{K}> p

_{V}) is the area of the triangle formed by all pairs (p

_{K}, p

_{V}) such that p

_{K}> p

_{V}. This can be done with an integral, but there is no need, as it is clearly ½.

Next, P(Data). This one is trickier. The London data is composed of two separate pieces of information. First, that Kim hit 99 of 100 targets. Second, that Vince hit 148 of 150 targets. What are the chances of each of those things happening? Well, each shot Kim or Vince takes is a Bernoulli Trial with success rate p

_{K}or p

_{V}, respectively.

Thus the probability that Kim would get exactly 99/100 is:

And the probability that Vince would get exactly 148/150 is:

Since they are independent, just multiply them together to get the probability they would happen in conjunction. More generally, we may define a function:

Which is just P(Data | p

_{K}, p

_{V}), that is, the conditional probability that we would witness the London data given a choice of some fixed p

_{K}and some fixed p

_{V}.

Also, let g(p

_{K}, p

_{V}) be the probability density function for our choice of p

_{K}and p

_{V}. p

_{K}and p

_{V}are assumed to be uniformly distributed between 0 and 1, that is, p

_{K}and p

_{V}are each equally likely to have any value between 0 and 1 as any other, so g is 1 inside the unit square and 0 elsewhere. (You may perform an additional check to see this makes sense by noting g's integral over its entire space is 1.)

What good are f and g? They can help us evaluate P(Data). Extending an idea from discrete probability, if you have some parameter that exhausts your probability space, the probability of observing an outcome is equal to the conditional probability of observing that outcome given some value for that parameter, times the probability the parameter actually is that value, integrated over the entire range of possible values for that parameter. That is:

_{K}> p

_{V}) is similar. From the definition of conditional probability, P(Data | p

_{K}> p

_{V}) = P(Data ∩ p

_{K}> p

_{V}) / P(p

_{K}> p

_{V}), so you do the same integral, except over the triangle where p

_{K}> p

_{V}rather than over the entire unit square, and then you divide by P(p

_{K}> p

_{V}), or ½:

Putting it all together (and conveniently canceling a 2 and a ½), you get that P(p

_{K}> p

_{V}| Data) is equal to the integral of f over the triangle where p

_{K}> p

_{V}, divided by the integral of f over the entire unit square. Plugging that into Mathematica (you didn't think I was going to do it by hand, did you?), we see that the answer is 1,152,511/2,187,465 or 52.6871%.

(And after the fact we note that the assumption of uniform distribution for p

_{K}and p

_{V}over their entire possible range was fine, as the observed data would be

*so*astronomically unlikely given low values for p

_{K}or p

_{V}that the portions of the integrals where p

_{K}or p

_{V}are low don't make a significant contribution to the final calculation.)

So, Kim didn't just put up the best numbers in London. Our analysis shows there's also a 52.6871% chance—it’s more likely than not—that Kim's superior performance in London was a result of superior skill. Congratulations, Kim, on being the best skeet shooter at the 2012 London Olympics, and we presume, in the entire world! (And may I one day not have to resort to a math degree to figure that out.)

As for the other shooting events ... well, I could repeat this analysis for trap shooting, in which female gold medalist Jessica Rossi of Italy hit 99/100 (99%) of her targets, and male gold medalist Giovanni Cernogoraz of Croatia hit only 146/150 (97.33%) of his, but I'm an American. I obviously don't care about a bunch of foreigners.

Jessica Rossi was amazingly cool... And it doesn't matter whether you are a male or female when you are that cool... You are going to hit the target, period.

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