Sunday, 30 September 2012

Distributions, SAT scores, the Cult of Equality

I tell you one and one makes three
I’m the cult of equality

Based on the statistical shenanigans used in Lewontin's fallacy, there are two ways an argument of equality is put forward:

1)The differences between the groups are much smaller than the differences amongst the groups themselves.
2)There is more overlap between two groups than non-overlap.

The first one is less disingenuous than the second, so lets start with the second one.

To do so, we first require to lay down some groundwork about statistics.

With an average male height of 5'10'' it's much easier to find men 6' or 5'8'' inches tall, than 6'10'' or 4'10''.  Note that the average of both sets is 5'10''. It's an example of the common observations that more people are closer to the average than extremes when considering various human traits.

The graphic representation of this fact is the bell curve/normal distribution.

The mean(μ) being the average, and the standard deviation(σ) a measure of how spread out the distribution is. A higher mean means taller people, a higher standard deviation means more taller and shorter people than the mean.(higher SD - flatter curve, lower SD - more peaking curve)

Note that for the same difference, namely of 1 SD, there are very different proportions of people between the bounds of mean and one SD, one SD from the mean and two SD from the mean, and 2SD from the mean and 3SD from the mean, viz. 68.2%, 27.2% and 4.2% respectively.

Coming back to the argument, "there is more overlap than there is non-overlap", consider two populations whose means are separated by a difference of 2SD, but have the same number of individuals and standard deviation. They do overlap, the right half of one overlaps almost completely the left half of the other.

Using the numbers from above bell curve, there is about 52% of overlap in both the distributions.
So there is more overlap among the two groups than the their non-overlapping parts. However now consider how many of people in group 2 are above the average of the people in group 1.
While the average of group 1 is at 50th percentile, the corresponding person in the other group has to be almost at 98th percentile to make the cut.
The ratio of those making the cut from the different groups is thus almost 50:2 or 25:1.

If you increase the criteria to one SD above M1, then the ratios go to about 16:0.1 or 160:1.

This is not a proof of equality, but humongous inequality.

A difference expressed in terms of SD is therefore a big fucking deal.


Regarding SAT scores and how the boy-girl ratio has been falling(written here before):

The 2005 repetition found that there were 2.8 boys for each girl in the group which scored over seven hundred. Remember that the numbers were 13 boys to one girl in the early 1980s.  

The thing to realize about SAT distributions being the the jump in ratios when one goes farther at the ends of distributions. The ratio for one more SD increase, jumped from 25:1 to 160:1, i.e. a more than five-fold increase, even though the absolute numbers themselves are falling.(50% and 2.1% in the former, while 16% and 0.1% in the latter).

So while the ratio of 2.42 to 2.1 is close to 1, the ratio of 0.42 to 0.1 is more than 4.(not actual numbers). Secondly, it's easier to smother the difference when the numbers make up less than 1% of the distribution. Or even less than 0.1% (1 in 1000)
An easier test can go a long way at making the sex-differences disappear since the distributions are much closer together(the means are almost equal) and the small difference in the SD of the two distributions(men are more variable) only makes a big difference at the ends.

Irwing and Lynn (2005) found that men outnumber women 2-to-1 at the 125 IQ level, and 5.5-to-1 at 155.

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