Another thing about variance...

That last post rambled on and somewhat lost it's coherence (me? incoherent? never!), but I did want to point out just one more thing about the effects of differing variances.

Last post was all about how having a greater variance in math ability will lead to gender imbalance as you get further from the mean (which we are assuming is equal, as the evidence would suggest that it is), however all the examples we were dealing with were about going above the mean, that is going towards people who are better and better at math.

But for a distribution that is centered on its mean, (which, once again, the normal is) the exact same thing will happen as you go to the left side of the distribution, that is towards people who are worse and worse at math. You will find the ratio of boys bad at math to girls bad at math skewing more and more towards the boys in exactly the same way you do among those hyper-good at math.

Is this observed? It is! What fun. Why do we never hear people complain about the severe gender imbalances in remedial math courses? Equality for women! We need more dumb girls!

(Relatedly, I always think of this whenever I hear too much complaining about gender imbalances in the "good" jobs and how we never hear about how awful it is that men dominate the "bad" jobs like, just off the top of my head: garbage-man, miner, janitor, felon, prisoner, etc., etc. In fact, did you know that there is a similar gap to the "gender wage gap" among the likelihood to die on the job? Men are significantly more likely to die on the job. So, far from employers being biased against women and showing it by conspiring to keep their wages down, they are actually biased against men and are showing it by literally killing them. Sneaky employers!)

Math is hard, so is science journalism...

So the NYTimes reports on a new study that shows no difference in average scores between boys and girls. They open, as one would expect, with the standard laugh at how this just proves once again how stupid, wrong and sexist old Larry Summers (youngest tenured professor in Harvard history, the idiot) was for suggesting that intrinsic aptitude might be at all responsible for the vastly greater percentage of men than women at top math and math-related faculties.

But, of course, Larry wasn't saying that men and women don't have equal average aptitude. His point was that all the studies of the issue show that men and women have equal averages but very unequal variances. The variance is the measure of how "spread-out" a distribution is. For a distribution centered about its mean, like a normal distribution, having a greater variance means that you will have higher and higher percentages the further you get from the mean.

This means when you're looking way, way in the tails (tails is a term used to refer to the far reaches, both left and right, of a distribution) if boys have a greater variance than girls, you will see a higher and higher ratio of boys to girls the further you get in the tails.

To put it in simple terms, if boys and girls have the same average but boys have a greater variance, you will expect to see 50-50 boys/girls at the average. As you start to go above the average, you will start to see that ratio skew towards the boys, say 52/50 at 1 SD (Standard Deviation, a measure of how far you are from the mean, in variance terms) above average (yes, I could do the math to get the actual differences with an implied variance, no I am not going to. Maybe later, if I get bored. BTW, I've never been that bored in my life so don't hold your breath.).

Now when you're talking SDs, the percentage of people you're talking about gets small, fast. About 15% of a population will be above 1SD, that drops to 2% more than 2 SDs above the mean (roughly, here, roughly), and only about 0.1% of a population will lie above 3SDs. (This one is near and dear to my heart for IQ related reasons...)

At 3SDs, or even further, say 4SDs (now you're talking like 1-in-a-million, or, say, like a Harvard math PHD's math ability level) the skew of boys-to-girls will get even more pronounced, if the boys have a higher variance. It might even get all the way to, say 85-to-15, which is the observed ratio.

So how did Larry and all the other studies get it wrong and this one get it right? Simple, this study agrees with Larry and all the others, but tries to play the difference down by suggesting that the variance they observed would account for a 75-25 ratio but only sexism could get it to the observed 85-15. Maybe it's sexism or maybe they're underestimating the variance measurement or maybe they're underestimating how far in the tails math-faculty really are, or maybe there are other attributes whose variance is different between men and women that reinforces the ratio or maybe lots of things.

Point is the nice lady at the NY Times either didn't understand what she was reporting or just decided to lie about what it said because it made a better story to laugh at dumb old Larry again rather than point out yet another study supporting his hypothesis.

Regardless, I'm sure the world will be a better place when we force extremely high-end math faculties to be evenly distributed between men and women. There's no way this will just result in either a) men being forced out as the departments shrink 'til only the number of men that equals the (much smaller) number of interested and capable women are allowed to do it or b) the lowering of standards until there is just no such thing as a capable and qualified math faculty.

After all, that's not at all what happened to college sports when this logic was applied to it... oh wait...



UPDATE: What do you know, I actually got that bored. Also I wanted to put up some pretty pictures to go with this post, so I did.


Here they are, maybe I'll try to work them into the text, though the fact that it was written picture-less somewhat complicates that, so I might not.


At any rate.

Here is a graph showing two normal probability distribution functions. They are both normal, with mean zero. The standard deviation of the "boys" function I have chosen to be 10% higher than the "girls" function. This gives a variance ratio (boys/girls) of 1.21, which is within the parameters seen in the latest study, though I just picked it arbitrarily as an example.


The x-axis is "girl" SDs. As you can see, the girl's graph is taller and skinnier than the boy's graph. This is because, as I said above, the higher variance "spreads out" the graph:

But, as I said, the real fun begins when you get into the tails. That's where the relatively small difference in variance starts to cause some real fun.

Here are two images of the left tail, that is the "bad-at-math" side of the distribution. The first shows the tail from 4 to 2 SDs below the mean and is, in effect, a "zoom-in" of the left part of the above graph. The second shows the same thing, but only 4 to 3 SDs below the mean, this is a zoom-in of the left half of the first zoom-in, if you will.

The thing to note is that while it looked above like the difference was vanishingly small in the tails, this is due mostly to the fact that in the tails you're dealing with such small numbers. When you zoom in, you see that the variance is in fact causing the difference between boys and girls to grow:

Finally, here is a graph showing how many boys versus girls you would expect out of 100 for the entire left half of the probability distribution. As you will be able to see, as you get further into the "worse at math" side of the spectrum, a random group of 100 made up of kids at that level or below gets more and more lopsidely male. At the mean it's 50-50, as expected since they have the same "average" ability, at 4 girl-SDs below the mean (which is the 1-in-10,000 bad at math kid, so the really, really bad at math kid) you're looking at 19 girls to 81 boys on average. Not because of sexism, but because there are more really dumb boys than girls in this model of math-ability.

(Note that, because the normal is symmetric about the mean, the "good-at-math" side of the dist. would display exactly the opposite tendency, that is out of a group of 100 random boys and girls, you would expect more and more boys as you get into "better" math territory and at 4 girl-SDs above the mean you would expect 81 boys to 19 girls. Note that this is the 1-in-10,000 "good-at-math" folk, or, in other words, the population that you are drawing from when selecting professors for the most elite universities.)

Look at me being all socially redeeming and all...

So via this post at Ross Douthat's blog at The Atlantic, I came across this post at the NYTimes Paper Cuts blog (which is, as the subtitle kindly points out, a blog about books). The post is about the insanely high prices you can find for certain not-that-old, not-that-special-seeming used paperbacks on services that allow you to sell used books, Amazon being a primary culprit according to the post.

Because most people are economically illiterate and Times readers are not as different from ordinary people as their egos would suggest, the common thread in the comments immediately seized upon "greed" as the reason for these high observed prices.

So I offered the following comment, in response to the first commenter who had suggested "greed" as the reason for the high prices (and who was, as my theory of economic illiteracy would suggest, the very first commenter):

quisqualis - As a general rule, greed is a tremendously unsatisfactory explanation of any price set in a market, as a moment’s thought will reveal.

While it may well be true that the seller is the very picture of greed, in a free market no one is compelled to buy their product and, in the case at hand, certainly no one is compelled to buy paperback books.

So while that greedy, greedy seller might well set the price at absurd heights and then cackle in delight when thinking of all the marvelous lucre they will get — they will not actually get anything unless someone decides to pay that price.

The person who decides to pay the price may well feel that it is too high, but in actually paying the price they clearly feel that the benefit they will get from the purchase exceeds even that high price, otherwise they would not buy it.

This is the crux of free-market transactions. Both parties must feel that the transaction adds to their well-being in some way or they would not pay it.

Personally, I do not like coffee and feel that at even a nickle a cup it is way, way overpriced, so I am shocked to see people shelling out 100 times that every day at Starbucks. But to the Starbucks shopper, the coffee (and perhaps the experience or whatever else makes up a part of the transaction from their standpoint) is worth more to them than the $5. From Starbuck’s point of view, the $5 is worth more than the coffee. So the coffee is sold for what to me seems an unreasonable price and both parties are better off.

Even if they are both greedy. In fact, BECAUSE, they are both greedy. If they didn’t see any benefit to be had from making the transaction, it would not be made.

Thus greed, while no doubt present in alarming amounts among most everyone, is not a good explanation for prices that we feel are outrageous.

— Posted by blighter

I thought that my habit of occassionally dropping comments on the odd blog here or there was totally unredeeming. After all, as the old saw has it: "Arguing on the Internet is like running in the special Olympics: even if you win, it's still retarded."

But low and behold, look what a later commenter dropped on the blog:

Appreciated ‘Blighter’s’ economic analysis (greed/ well-being) I had never thought of it quite in those terms but it sounds right. However one occasionally sees irrational prices that dealers are asking, sometimes 100 times the market value. I guess the explanation here is mania, delusion and temporary insanity. However one cannot help feel that it is initially provoked by greed. How do you explain the dealer who wants $9000 for a signed edition of 1000 of Galsworthy’s Plays, available at less than $100 all over the web? It has been for sale for 4 years to my knowledge.
— Posted by Laker

If I have helped even one person finally realize that free market transactions require both parties to be better off to become actual transactions, I have done the world a service. Even if they cling to their stupid love of "greed" as an explanation for other prices they feel are absurd.

So I think this counts as my good deed for the day week month.

good, good

A brief note to the creators, whoever you are, of the most annoying currently airing ad: the Wendy's ad for their warm chicken salad. The ad's main copy is as follows:

"If warm chicken tastes good... and cold, crunchy veggies taste good... wouldn't a warm chicken, cold, crunchy salad taste good good?"

The answer, in case anyone is so completely illiterate as to not know, is "Of course not, you blithering idiot." Adjectives in English are not increased in intensity by repetition. In fact, this is exactly the kind of linguistic habit that parents try to break small children of.

Thanks so much for simultaneously striking a blow against good grammar while polluting my favorite television shows with more vapid adverts for empty calories. The fact that you're advertising a salad does not make up for the assault on our language.

I sometimes wish I believed in hell so I could have the satisfaction of wishing it on the sorts of people who create these kinds of things. Alas, alas.

Leopard 1, Crocodile 0

Mammal pride, y'all!

Public Notice

Please be advised that the recent film "The Dark Knight" was very good and just plain fantastic for a super-hero film. If you are going to express contrary opinions, please consider the effect this will have on any listener's opinion of your intelligence and taste and preferably only express them in the solitude of your closet, at night, alone.

Please do not express them at work, in the cube next to mine.

Good grief, these are different chuckleheads than the cultural illiterates I heard bad-mouthing "There Will Be Blood" but, honestly, what kind of people do I work with here?

Back for the first time...

Due to popular (and salty) demand, this blog will no longer be on hiatus for the immediate future.

We apologize for any inconvenience this might render.

Sincerely,
The Management.