Thursday, October 13, 2005

WSJ: The Bell Curve: A quick primer WSJ: The Bell Curve: A quick primer

A quick statistical primer may be in order here in reference to Charles Murray's recent WSJ article reintroducing "The Bell Curve" that he published eleven years ago.

A lot of phenomena in life and nature fall in a statistical distribution known as the "normal" distribution. Given its shape, this distribution is often called the "Bell Curve". One of those phenomena is IQ.

The normal distribution is typically described by two measures, its "mean" and its "standard deviation". The mean is its center point. As the bell curve is symetric, the mean is also the median. The standard deviation is its spread-outedness. The greater the standard deviation, the flatter the bell curve. The first standard deviation is at approximately the 16th percentile on the low end and the 84th percentile on the high end. Thus, approximately 68% of a normal distribution falls within its standard deviation, and 32% outside, 16% on either end.

IQ tests are typically "normed" or centered at 100. This means that the scores are adjusted so that a score of 100 is the statistical mean. Similarly, they are normed so that 15 IQ points are one standard deviation. Thus, the resulting normed distributions of almost all IQ tests look identical - bell curves with a mean score of 100 points and standard deviation of 15 points.

But what has to be remembered is that these means and standard deviations are averages for the entire population sampled. The distributions of subpopulations within that population will also almost invariably be normal - but sometimes with different means and/or standard deviations.

Thus, if we assume equal male and female populations, we might have a mean male spatial reasoning score of 105 and mean female score of 95, resulting in an averaged mean of 100. Similarly, the male standard deviation may be 18, and the female standard deviation 12, averaged to the normed 15.

Let us though assume equal IQ means between the sexes. Still, there is a difference based on standard deviations. At one standard deviation (84%) above the mean, the male score is 118 and the female score is 112. However, at one standard deviation below the mean, the male score is 82 while the female score is 88. At two standard deviations (98%) above the mean, the male score is 136, while the female score is 124. Of course that also means that at two standard deviations below the mean, the male score is 64 and the female score is 76. Going one more standard deviation (99.7%), the male score is 154 and the female score is 136.

One important thing to note is that, just as there are more smart males than females, there are correspondingly more dumb males than females.

Let me note here that I just picked the 18 (15+3) and 12 (15-3) standard deviations for males and females out of thin air. In reality, the differences is most likely different - Murray didn't say. But with those values, at the 136 IQ level, we have 2% (2 std deviations) of males while we have 0.3% (3 std deviations) of females. The ratio is approximately 7:1. While my figures were admittedly invented out of thin air, males do outscore females at approximately this ratio, 7:1, at the top 1% level on the math SATs. This is a very comparable result, and is best explained by precisely this phenomenum - a larger male standard deviation.

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