Fun hate fact about the bell curve

Today’s hate fact is hyperexponential decay.

The normal survival function (the number of cases in a normal distribution that are more than x, one minus the cumulative distribution) approaches zero as x increases hyperexponentially, which is to say, very fast. This often makes it possible, under many common circumstances, to draw conclusions about individual cases, to infer a particular person’s character and or ability from his race or sex, and to infer a particular individual’s race or sex from his ability or character, to draw conclusions about particular identifiable people from average racial characteristics.

Normal distribution, probability density of cases having value x:

normal distribution, probability of cases having value x

normal distribution

Normal Survival function, proportion of cases x or more:

cumulative normal distribution

normal survival function

This is the probability that a particular case will exceed x, for example the proportion of people in the population that exceed x, where x is some characteristic that is normally distributed, such as height or intelligence or criminality.

I have used the less common definition, integral from x to infinity, (normal survival function) rather than the more common definition, integral from minus infinity to x (cumulative normal distribution), since minus infinity is counter intuitive, and the fact that there are very few really smart people, and very few really violent people, is intuitive, so we want the function to approach zero, the number of people smarter than x, rather than unity, the proportion of people dumber than x.

Survival function at the extremes:

cumulative normal distribution at the extremes

normal survival function at the extremes

For x more than one or two standard deviations above the mean, the following approximation to the normal survival function is good enough:

approximation for cumulative normal distribution at extremes

approximation for the normal survival function

Obviously any category that is likely to be of interest (such as the category of people competent enough to do such and such, or the category of people wicked enough to do so and so) is going to deviate from the mean by one or two standard deviations, so we can almost always use this approximation.

And now for today’s hate fact:

Notice that the normal survival function falls off hyperexponentially, in other words, very abruptly, falls off faster than exponentially.

This is what makes it possible to deduce facts about people’s characteristics from their race in particular individual cases, and their race from their characteristics in particular individual cases.

It follows from hyperexponential decay that if you select a subgroup from the population that meets some high standard, for example the entry requirements of a university course, or fails some low standard, for example performs an act that is both stupid and criminal, then the vast majority of those selected will only just meet the standard.

Hyperexponential decay means that if you have finite population of cases, you are going to hit zero mighty fast.

So if you have one standard for white students, and another standard for black students, there is a good chance that every single black student in a class will be inferior to any white student in that class, since the vast majority of blacks will be close to the their minimum, which is lower than the white minimum. Similarly for women in computer science classes.

Even though there is a lot of overlap in the population as a whole, in the selected category, very little overlap, so chances are that in any small group of the category, such as students at a class, every person who got in on his merits will be better than any single person who got in on affirmative action.

Thus for example: a class of fifty students, six of them are black. None of the white essays are plagiarized, all of the black essays are plagiarized.

If there are a thousand blacks in the university, there is going to be some overlap. If there are six blacks in the class, probably not.  Particularly as the likes of Michelle Obama are not in the class, because they probably got a luxury all expenses paid scholarship to grace Princeton with their presence.

Conversely, if you look at the work of a person who is a member of a group where they were selected for being good enough, and yet that work is is not good enough, you can be pretty sure he belongs to the category benefited by affirmative action. You can tell the skin color of first lady Michelle Obama from the fact that her Princeton University senior thesis is incoherent and full of has spelling and grammar errors. Obviously, you cannot conclude that someone is black from the fact that their essay is no good, but from the fact that it is senior thesis at an ivy league university, and nonetheless no good, you can tell that she is black.  Thanks to hyperexponential decay, even Princeton cannot find enough black women who can write decent essays.

The Ivy League are hard up for enough black men, hard up for enough females, and to make their quota of black female graduates, they have to really scrape the bottom of the barrel, first lady Michelle Obama being what they find at the bottom of the barrel.

If a crime is violent, you cannot necessarily know the perpetrator is black. If a crime is stupid, you cannot necessarily know the perpetrator is black. But if a crime is violent and stupid, you can be pretty sure the perpetrator was black, thanks to hyperexponential decay:

Two women attempt to pass a counterfeit $50 bill at McDonald’s. The cashier refuses it. One of them punches the cashier, a large male. He retreats. They jump the counter. He flees all the way to the corner. Then they attack him in the corner. He loses control, beats the crap out of both of them, damn near kills one of them. Guess the race.

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14 Responses to “Fun hate fact about the bell curve”

  1. Lemniscate says:

    Affirmative action is the opposite of what should be done in a rational world. It should be harder for members of less able groups to gain admission, because a passing score for a member of a less able group is more likely to be a fluke misrepresenting true ability than for a member of a more able group. One should use a Bayesian approach, building different prior distributions for each group. I think this is the least likely thing to happen in the world, though. Can you imagine an admissions brochure proudly proclaiming, “We only let in poor people, blacks and women with higher test scores than rich, white men!”

    See this excellent post: http://statsquatch.blogspot.com/2011/01/overprediction-of-minority-performance.html.

    Oh and the density function for the normal distribution does not represent the probability of a value being x (which is zero) but the ‘probability density’ of the distribution at that point.

    • jim says:

      fixed the probability density.

      One of the major causes of passing scores is fraud. There is pressure to “close the gap” Fraud closes the gap. Under represented minorities tend to be heavily overrepresented in misconduct cases. There is pressure to close that gap also. So turning a blind eye to fraud, and indeed actively encouraging it, is win win. Hence the financial collapse. The vast majority of dud loans were made to Hispanics, who either committed fraud, or their loan officer committed fraud on their behalf.

  2. Zach says:

    Nice! Are those your maths?

    Further, I didn’t see a thesis that was “full” of errors. Anybody got a link to the whole thing?

    • jim says:

      My maths, though I am sure lots of other people have made the same calculation.

      Her thesis was not “full” of errors, I got a little carried away, but it was rather unimpressive. Probably would have merited a B in high school, but rather underwhelming as a Princeton thesis.

  3. Steve Johnson says:

    jim,

    What you’ve described is not a cumulative distribution function (goes from negative infinity to x) but a survival function (goes from x to infinity).

    Agreed on the reasoning and very well said.

  4. PRCalDude says:

    Couldn’t the cashier have beaten the obnoxious white whale also? He showed a bit more restraint than was due, IMO.

  5. jim says:

    I did not see a white whale in the video. The two black women crossing the counter are sisters. A white women who had her fake fifty rejected might have made a scene, but would not have jumped the counter and chased the cashier into a corner.

  6. PRCalDude says:

    You didn’t see the obnoxious fat white lady yelling across the counter at the guy to stop? Maybe I’m seeing things.

    • jim says:

      Oh yes, saw her, but he was probably busy with the black women, who needed a beating even more than the white whale.

      • Zach says:

        There is something hilarious about the above comment. Not wrong, just funny.

        The video was taken down fyi.

  7. PRCalDude says:

    :51 in “STOOOOPPPP STOPPPPPP STOPPPPP111111”

  8. […] as affirmative action makes the differences between blacks and whites starkly visible to everyone at the same time as it makes it a criminal offense to notice, or even think about, those […]

  9. […] This, however, only works if you hire on the basis of quotas – the smartest x% of males, the smartest 1.5x% of females. But if x is a rather small number, it then becomes obvious that every person hired to fulfill the quota is dumber than every person hi… […]

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