In middle school I participated in a competition called Mathcounts. In this contest, each state held a competition to determine four people who go on to represent the state at Nationals.
I grew up in Maryland, and in 7th and 8th grade I just barely made Maryland’s team, getting fourth place both times. If I remember right, my score on the state competition was around 41, out of a total of 46.
National Mathcounts was tons of fun. In my 7th grade year the competition was held in Orlando, and they took us to a special section of Disney World featuring experimental rides that weren’t open to the public. I also remember the closing banquet, where the dessert consisted of a chocolate compass, protractor, and calculator. But perhaps the most memorable part was the pin trading: my coach handed me and my teammates 56 pins each representing Maryland, and we spent the day before the competition trading our pins for those of other states (and DC and territories).
One thing that bothered me, though, was that if I had been in California, I wouldn’t have been good enough to make it to the national competition. If you grew up in California, you needed to score a perfect 46 — and then go through tie-breakers with all the other people who got 46! It didn’t seem fair that it was way harder to make it to nationals from California than it was from Maryland, where in turn it was way harder than in Wyoming.
No surprise, then, how frequently the California team won the national competition: 8 times out of the 36 competitions since the inaugural one in 1984, with Texas (7 times) close behind. They had huge populations, which meant more talent to draw from!
Miss USA… or is it Miss America? I always forget…
Ah. Great, thanks Wikipedia.
Miss USA has a similar structure to Mathcounts. Each state (and D.C.) runs a competition to select a contestant to represent that state at the national competition. Miss America has the same format. So do some of the others listed, but Miss USA and Miss America are the most well-established, having been around since 1952 and 1920, respectively.
Here’s an interesting thing to think about: in the 69 years of competition since 1952, how many times do you think the representative from California has won Miss USA? If it helps, California has on average had about 11% of the United States’ population during this time period. Think about this for a bit, then read on.
There are two “extreme” answers you might give. One assumes that California is as likely to win Miss USA as any other state. In that case you’d expect about 1.35 winners. The other assumes that a state’s probability of winning is proportional to its population. In that case, you would expect around 7 or 8 winners from California.
So, if I told you that California has only won once, what would that tell you about the Miss USA pageant? What if I told you that California has won eight times?
I think it tells you how much luck there is in the pageant. To see this, you could imagine the following two extreme worlds:
- There is no luck at all. If you re-ran Miss USA and the state-level pageants again, the same people would win every time. In this case, at least if you assume that the skills and/or traits that make someone likely to win Miss USA are distributed evenly across the U.S. (more on this later), California should win a fraction of the time proportional to its population (11%).
- Miss USA is entirely luck: it may as well use a random number generator to choose a winner. In that case, since each state gets one representative, California should win 2% of the time. Note that this would also happen if the state-level competitions are entirely luck-based, because then we wouldn’t expect the representative from California to be any more skilled on average than the representative from Wyoming.
So, what’s the answer? As it turns out, California has won Miss USA 6 times since 1952 (about 9% of the time). By contrast, in the same time period, California has won Miss America just twice (3%). Huh!
But let’s do something more robust than just looking at California. Let’s plot the number of Miss USA winners each state has against the state’s population, and likewise for Miss America.
The relevant property of these two lines of best fit is their slope. In the “entirely luck” scenario, the slope would be 0. In the “no luck” scenario, it would be 0.29 (that’s the population of the U.S. in millions divided by the number of contests held since 1952). The closer the slope is to 0.29, the more skill is involved; the closer it is to 0, the more luck.
The slope of the Miss USA line is around 0.21; for Miss America, it is 0.14. Thus: Miss USA involves a lot less luck than Miss America. This somewhat surprised me: I expected about the same level of luck in both contests! I tested this finding for statistical significance, and it is fairly significant (under the usual assumptions) — though I can’t confidently rule out the null hypothesis that there’s the same amount of luck in both contests.
We can go further if we want: taking the ratio of each slope to the “no luck” slope of 0.29, we might say that Miss USA is 25% luck. Miss America is 52% luck. That’s pretty cool — I wish I’d pre-registered a hypothesis about these percentages!
(Depending on how strong your prior is that the amount of luck involved should be about the same, you might want to squeeze these percentages toward each other.)
But wait — to what extent are these percentages reasonable? To arrive at them, I’m making a pretty big assumption: that the skills and traits that make someone likely to do well at beauty pageants are equally distributed across the United States. The corresponding assumption is not true, for instance, for Mathcounts: I neglected to mention that next in line after California and Texas was Massachusetts, with 5 wins out of 36 — vastly out of proportion to its population. That’s not terribly surprising: Massachusetts has good schools, not to mention kids of MIT and Harvard professors. Could beauty pageants have a similar thing going on, where some states — for whatever reason — are much stronger than others?
One way to test this is to see whether the residuals off of the lines of best fit in the two charts above are correlated. In non-technical terms, if the same states tend to be above the line in the two charts, that would be a sign that our assumption was wrong. As an example, suppose that Oklahoma is really good at beauty pageants. Then you’d expect it to punch above its weight (be above the line) in both of the charts.
Instead, what we see is an impressive lack of correlation, indicating that our assumption was probably pretty reasonable.
I tried to posit a theory about why Miss America involves so much more luck than Miss USA, but couldn’t come up with a plausible one.1 The contest formats are pretty similar, so I’m guessing it has more to do with how contestants are judged. Unfortunately I don’t know enough about beauty pageants to come up with a good hypothesis here. If you have one, leave a comment!
1. One theory that came to mind: over the 69-year window we’ve been looking at, Miss USA winners have matched the racial makeup of the U.S. as a whole. This is because all winners before the 1980s were white, whereas in recent years racial minorities have been overrepresented among winners. On the other hand, Miss America winners have been disproportionately white over the same period, because (as with Miss USA) they were entirely white before then 1980s, while recently the winners have been representative of the U.S. racial makeup. As a result, regression dilution may be affecting Miss America’s slope, pushing it down. However, if this hypothesis were correct, we would expect states with more non-white residents to have been winning more frequently in recent years; this is not supported by the data.↩