Visualising sex differences, Emil and Tufte

 

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Edward R. Tufte’s “The Visual Display of Quantitative Information” is a well-known delight which, once read, will make you see informative displays in a new and clearer light. It was in Tufte’s book that I came across his championing of “the typographical delight of the statistician W.J. Youden whose illustration of the normal curve I reproduce above.

I had Tufte’s insights in mind when working on the original depiction of sex differences in the distribution of intelligence.

https://drjamesthompson.blogspot.co.uk/2013/09/are-girls-too-normal-sex-differences-in.htm

The Gaussian distribution, or standard normal curve, or Bell Curve, never ceases to amaze. The shape is plain to see, but the area under the normal curve is hard to estimate by  eye, hence the need for visualizers with area calculations attached.

What prompts these reflections? No sooner did I post about the visualisation of sex differences than a sharp eyed Danish commentator shot back with a good suggestion, which I have incorporated immediately.

http://emilkirkegaard.dk/understanding_statistics/?app=tail_effects

To recap: Suppose someone was to do a study on adults, not children, and test the intelligence of these adults while also having done brain scans so that we can calculate the link between the brains and mental abilities. If such a study were to show a 4 IQ point male advantage, and the sd of male intelligence is, say 15 and the female sd closer to 14, what would the distributions of male and female intelligence look like at the higher ends of the ability scale?

If we use Emil’s visualizer and put in mean=104 (sd 15) for the men in blue, and mean=100 (sd 14) for the women in red, and set the high mark cut-off as IQ 130 (corresponding to the top 2.28% of the overall population) then 4.15% of men make the cut and only 1.60% of women: the sex ratio will be 2.58 to 1. That means that 72% of bright people will be men.

Or, as has just been suggested to me from Denmark, we can use the visualizer and put in mean=102 (sd 15) for the men in blue and mean=98 (sd 14) for the women and once again set the cut-of at IQ 130. Then 3.13% of men make the cut and only 0.65% of women: the sex ratio will be 4.8 to 1. That means that 83% of bright people will be men.

In both cases there is a 4 IQ point difference, but by choosing to express this difference straddling the mean rather than above the mean, the sex ratio increases. I chose the first way because I had been triggered by the higher than average results from the Scottish National Survey. I am grateful to Emil for pointing out that I should try it closer to the mean, a more representative situation.

Using the old approach of Males 104, Females 100, moving up to IQ 140 (the top 0.38% of the overall population) then 0.82% of men make the cut and only 0.21% women: the sex ratio is 3.8 to 1. That means that 80% of these even brighter people will be men.

Using the new approach of Males 102 Females 98 then at IQ140 0.56% of men make the cut and 0.13% of women: the sex ratio is 4.18. That means that 81% of bright people will be men.

Using the old approach, moving up to IQ 145 (the top 0.13% of the population) then 0.31% of men make the cut and only 0.06% of women: the sex ratio is 4.8 to 1. That means that 82.7% of these very bright people (the three sigmas) will be men.

Using the new approach, moving up to IQ 145 (the top 0.13% of the population) then 0.21% of men make the cut and only 0.04% of women: the sex ratio is 5.3 to 1. That means that 84% of these very bright people (the three sigmas) will be men.

In the refined company of my loyal readers, you may well say that IQ 145 is no great shakes: there will be 13 three sigmas in a thousand at this level of intellect. Too common. What if we take the 1 in a thousand criterion, equivalent to an IQ of about 155 (3.7 sigma). At that refined level the sex ratio will be 7.9 to 1. Call it an 8 to 1 chance that this very bright person will be a man.

Using the Males 102 Females 98 version 0.021% of men and 0002% of women make the cut, and the sex ratio is 8.8 to 1.

So, there are slight differences in the sex ratio depending on: whether there is a mean difference of intelligence between men and women of 4 IQ points; whether the variance of male intelligence is greater than that of females; and to some extent about where one places the discrepant male/female means on the standard normal curve.

It is Bank Holiday in England, which means clouds, cold winds and no sunshine. Banks have a lot to answer for.

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