The fall-out from Boris Johnson's comments on IQ and the distribution of wealth brings out the fact that many misunderstandings about social inequality are also confusions about mathematics.
Chris Dillow and an editorial in the Guardian both point out a howler in Johnson's speech. When he said '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', he seems to have been unaware that he was using IQ scores standardised so that 16% of the population will always fall below 85 and 2% above 130. The actual scores add no further information once the percentage of the population is given, and vice versa. He literally said nothing, twice.

But I suspect that Johnson's error is even more profound than failing to recognise that he was quoting standardised scores. What did he think was the relevance of the IQ scores he cited with so little understanding? The argument is an old favourite of the right since the 18th century: look, here's the scale of inequality in natural abilities, so what do you expect wealth inequality to look like? But that argument would fail even if, by some miracle, people's position in the distribution of IQ matched their position in the wealth distribution.

If you have six million in wealth and I have to struggle by on a mere three million, you are twice as wealthy as me. By the same token, it would seem that if you have an IQ of 150 and I have an IQ of 75, you are twice as intelligent as me. Not so: IQ test scores can't be divided into each other. They are measured on an 'interval' scale, meaning a scale which allows for addition and subtraction, but not division or multiplication. This isn't just a matter of convention: the use of an interval scale reflects the fact that the notion of zero intelligence is not meaningful. (Consider what would be involved in attributing zero intelligence to any entity to which it could conceivably be meaningful to attribute intelligence at all.) In contrast the notion of zero wealth is all too meaningful, and wealth is measured using a 'ratio' scale which allows for multiplication and division.

So we can never say any individual is twice, or 1.2 or 0.75 as intelligent as another*. Note: we can't say this in principle, whatever measurement we use. This point is conceptual and is independent of any views about the validity or otherwise of IQ measurements. The most we can say, if we accept that IQ tests measure something we're willing to call 'intelligence', is that the difference in intelligence between two individuals or sets of individuals is a multiple (by some positive real number < = or > 1) of the difference between any two other individuals or sets. And this makes inequality in intelligence something very different to inequality in income and wealth.

People often appeal to the idea of meritocracy to explain or justify economic inequalities- the idea apparently being that unequal rewards (income, wealth) are or should be proportional to some combination of unequal effort and talent. If we take IQ as the measure of talent, we've just seen why this idea is conceptually incoherent. There could be any number of relationships between IQ and wealth, but proportionality isn't one of them. Perhaps in some meritocratic utopia, positions on the interval scale of intelligence would map on to positions on the ratio scale of wealth. This could involve no more than a correspondence in rankings, so if X is smarter than Y and Y is smarter than Z, the income ranking is always XYZ for all Xs Ys and Zs, but the difference in income could take any positive value for any pair. Or it could involve a tighter relationship where the differences in the two measures are linked by some function: say if the difference in IQ scores was multiplied by a constant, or squared or whatever to get to the difference in wealth. But the function linking IQ and wealth will not make wealth proportional to IQ because that would require both variables to be measured on an ratio scale. (I take it that any composite of effort and IQ would also need to be measured on an interval scale, so these comments don't just apply to intelligence as a basis for meritocratic distribution.)

This conceptual problem in meritocratic accounts of inequality is actually rather promising. What it suggests is that even if we accepted meritocracy as a plausible ethical ideal there are any number of distributions that might meet the requirement for a relationship betetween talent and reward, and meritocracy gives us no further basis for choosing between them, allowing other considerations such as egalitarianism or allocative efficiency to come fully into play. We might accept that position in the wealth or income distribution should correspond to position in the IQ distribution, but that differences in income and wealth should be mimimised, or maximised, or left completely arbitrary. We might, closer to the animating spirit of meritocracy, call for differences in income to be proportional to differences in IQ, but that proportionality could be based on multiplication by a constant of .0001 or 10,000,000. We might decide that the appropriate reward for talent was to distribute feathers of continuously graded shades of pink to be worn at civic celebrations. None of these options seem to me to be inconsistent with the basic meritocratic position.

So Johnson's notion that current levels of inequality are just the expression of natural differences in talent isn't just wrong, it really doesn't make sense, as those natural differences could be expressed by any number of functions linking IQ and wealth. But some functions are ruled out as explanations of the wealth distribution. IQ and wealth could in principle be linked by a function which reproduces the shape of the IQ distribution, such as multiplying differences in IQ by a constant, but they are not in fact so linked. The reason is that IQ is normally distributed in the familar symmetrical bell-shaped curve, so that high and low scores balance out at population level **. But wealth and income are not normally distributed. The distribution is skewed so that the rich get a much bigger share of the wealth than they would under a bell-shaped distribution. (There is an excellent account using U.S. data here .) If the wealth distribution had the same shape as the IQ distribution, wealth would be much more equally shared. So I don't see how you can be a consistent meritocrat, believe that IQ measures intelligence and think that the current distribution of wealth is equitable.

All of which suggests that meritocrats who are happy with the status quo should drop IQ like a hot potato.

*The zero in the Celsius temperature scale is not an absolute zero- it's the freezing point of water, not the complete absence of heat- and it follows that a temperature of 40 degrees Celsius is not twice as hot as a temperature of 20 degrees. But zero on the Kelvin scale is an absolute, so ratios of magnitudes are meaningful. If we convert Celsius temperatures to Kelvins, then 40 degrees Celsius corresponds to a Kelvin temperature about 7% higher than 20 degrees Celsius. (I appreciate that it doesn't necessarily feel like that.) The point is that there is nothing which corresponds to the notion of absolute zero when it comes to intelligence. There is no Kelvin scale for IQ.

** This is not just a matter of measurement convention. Normal distributions have been found empirically to be, well, the norm for measures of intelligence and aptitude .