Small population issue in statistics

From The Rockettes and Race: A Very White Christmas by Ginia Bellafante. It is kind of typical of the anger, hatred, prejudice and racism of the Social Justice postmodernists. It is also kind of typical of the innumeracy of journalists who work themselves up into a frenzy based on their own ignorance.

Bellafante attends the Christmas Spectacular Starring the Radio City Rockettes:

There were plenty of children in the audience though, even if it was a school day, and plenty of old people and middle-aged people. Many of them were wearing Santa hats. The crowd was overwhelmingly white.

I mention this fact because at the end of the second decade of the 21st century, the Rockettes, whose performances are taken in by almost one million people every holiday season, are themselves almost all white. The show I saw featured, as far as I could tell, only one African-American dancer in the lineup of close to 40. There were, in the end, more camels on stage than black women.

Among the 80 dancers who make up the Rockettes corps, 10 percent are women of color, a spokeswoman for the company told me; you are only seeing half the cast during any given show because there are so many performances to fill — on weekends, up to six a day. Regardless of that, any variance in skin tone is obscured by lighting and makeup that have the effect of creating a stultifying homogeneity, which is the point and amounts, ultimately, to an eerie celebration of whiteness. Ancillary cast members in the pageant — non-Rockettes — include a black man playing an elf and a black man playing a bellman.

Bellafante is the racist in this screed. She cares not for the merits of any individual. She is counting races as a measure of performance. But she doesn't count very well. There were apparently probably 4 African American dancers and she only counted one. Her inability to count then provides the basis to mount a moral dais to proclaim her disappointment "of the second decade of the 21st century."

How tone-deaf, arrogant, and pretentious can the Mandarin class be in the second decade of the 21st century? This level of tone-deafness, arrogance, and pretentiousness.

Bellafante has access to the facts. Among the 80 Rockettes, "10 percent are women of color," But Bellafante is still disappointed. Having conjured a racism narrative, she is going to stick with it.

Ten percent of the Rockettes are black. Thirteen percent of the population at large is black. So there is a three percentage variance in racial representation in the Rockettes. When dealing with small populations, it is by no means uncommon to have a few percentage points variance from large population norms. I would not be surprised to find that there are two or three percent greater or lesser representation of right handers or brown eyes or any other random trait in a small population of 80.

This variance of three percent would not seem a sufficient variance to lay a charge of old-style racism which is what Bellafante is implying. She is projecting onto the Rockettes the entirety of her own racism, stereotypes and prejudices. She clearly thinks that because the Rockettes are majority white, that they are racist and that they are watched by racists and cater to racists. That is what she wants to be true, even if the numbers don't bear her out. She does not appear to be sensitive to the fact that if the US is majority white, then the Rockettes, absent any confounding factors, are also likely to be majority white.

Ann Althouse puts it nicely. My emphasis added.

If I did attend the show because I wanted to write about it, I would be aware of my own difference from Rockettes enthusiasts, and I'd try to understand why the people who love it love it. I wouldn't present my emotional response as objective, and I hope I wouldn't distance myself from the crowd. There's something especially bad about putting yourself above other people — thinking you're the one with the good taste — and then racializing the difference.

Bellafante - everything that brings the press into disrepute; ideological, ignorant, innumerate and a bearer of racist animus.

But for all that, there is an interesting issue in here that is real but not really resolvable. It is that small population issue.

Up front, I acknowledge that the social justice critical theorists and the mainstream media have fudged definitions to such an extent that it is almost impossible to speak with clarity. Race is mixed with geography is mixed with ethnicity is mixed with culture. Stripping out Hispanic because it is a non-race category, based on the best information, census and otherwise, the US very roughly self-identifies as ~80% white, 15% black, 5% Asian and Native American. More precisely, I think it is about 77% white, 13% black, 6% Asian, and 4% Native American/other/mixed.

And those are just the straight racial numbers. Whenever you do a comparison, you have to compare apples to apples. You have to make sure you are incorporating the appropriate definitional attributes. Race might be one (if you are Bellafante). A good example is in the tech industry - You would expect the racial make-up of your employees to match that of the racial make-up of the population of people with a BS degree in computer sciences more than to the population at large. And indeed it does.

So the population with a demonstrated interest in professional dancing might be the better comparison base than the population at large (and even there, you get in to issues of type of dancing). Does the professional dancing population vary from the at-large population by race? No idea. The BLS data for dancers and choreographers is too small a number for them to break out. We'll just have to assume that the number of black dancers is the same as that for the whole population, 12.6%.

On this count, the Rockettes are within spitting distance of representation. 10.0% versus 12.6%. That is easily within the margin of error for small population comparisons.

On the other hand, and this is my point, that is a whole population perspective. It looks inherently reasonable, and indeed it is.

On the other hand, if you are a member of that particular minority, then that is the base you are working from. In that instance, the Rockettes aren't 2.6% too few black dancers. They are in fact nearly 21% under the number of black dancers that there should be (12.6-10/12.6). A 20% variance is not nothing. It is outrageous. From this perspective, Bellafante is justified in her outrage and her suspicion that the Rockettes are racially discriminating against black dancers.

Both perspectives are true. The Rockettes are nearly fully represented in terms of number of black dancers, AND there are strikingly too few black Rockettes. The paradox is, in effect, something like the conundrum of flipping a coin and getting five heads. What's the probability of flipping a sixth head? Most people feel it to be some tiny number (there is only a 1.5% of throwing six head in a row) and yet the probability that the next flip is a heads is 50%. What is the chance of flipping a fair coin six times and getting six heads? 1.5%. What is the chance that the next flip after the first five is going to be a heads? 50%. BOTH are true statements.

With our dancers, it is a small population issue. Small absolute variances are exaggerated in terms of proportion. And there is nothing you can really do about it. It is a small population issue that will always be there even if the system is perfectly colorblind. Random variance in any given year will mean that in the short within a small population, everyone is going to be under or over-represented. For any given snapshot in time, that variance, within a small range, provides no predictive value in terms of assessing whether there is prejudice or discrimination at play.

Occam's Razor points towards Bellafante being a typical journalist - ideological, ignorant, innumerate and a bearer of racist animus. But there remains a real possibility that she has a sophisticated understanding of statistics and that it is the 21% variance from target population that is exercising her. But I wouldn't count on it. In fact, I would guess the odds to be less than that of flipping a heads six times straight.