# Boris Johnson and the statistics of IQ

I’ve been putting off this post – I don’t really get a kick out of demolishing the statistical claims of politicians – but it intersects with league tables, an old area of interest, so here goes.

Boris Johnson, the mayor of London who enjoys some popularity by cultivating his lovable-eccentric persona, said last week:

Whatever you may think of the value of IQ tests, it is surely relevant to a conversation about equality that as many as 16% of our species have an IQ below 85, while about 2% have an IQ above 130.

You can read the whole speech here; the IQ bit is in page 7. Others have been alarmed by various aspects of this and the surrounding flurry of strange metaphors: society as breakfast cereal, society as centrifuge… but I will confine myself to the stats. “IQ” could mean any number of test procedures and questions. There is considerable debate about what it actually measures. There is also considerable debate about what we might want to measure, even if we could measure it. Bear that in mind, because Johnson sweeps those concerns aside.

What you do is to run your test on a bunch of normal people (small alarm bell should be ringing here about who is normal). Their results might come up with a certain mean and standard deviation, and you want your results to be comparable (read “indistinguishable”) to others, so you add or subtract something to bring the mean to 100, and multiply or divide to bring the SD to 15. Now you have an IQ score. So far, we have uncertainty arising from defining the construct, the validity of the measure, and the differences between. But with a nice normal distribution like that, it’s quite tempting to ignore those issues and move straight on to the number-crunching.

If the mean is 100 and the SD 15, then 16% of the distribution will be less than 85.1, and 2% of the distribution will be above 130.8. What Johnson is describing is the normal distribution’s shape. Why is it like that? Because you made it that way, remember, you squashed and shoved it into standardised shape. What else creates a normal distribution in nature? Adding unrelated things together (central limit theorem) and random noise. Saying that 16% are below 85 is no more meaningful that saying half the population is below the average. Everybody now knows to laugh at politicians and their apparatchiks who say things like that. I think that’s a good thing, because we are learning little by little not to have the wool puled over our eyes by “dilettantes and heartless manipulators”. Don’t fall for Johnson’s nonsense either.

A final note: why have the bigger portion at the low end and the smaller at the top? Clearly he is trying to appeal to a listener who thinks of themselves in the top 2%, and considers with distaste the rabble far below. Yet you could measure any old nonsense and talk with the same air of scientific veracity about quantiles of the normal distribution.

PS: I usually put a picture in, because blog readers like pictures. But on this occasion I fear it would only encourage him. You’ll note I don’t call him by his near-trademarked first name either.