# Bounds on skew and kurtosis of IQ

The question of whether IQ is Normally distributed, or instead follows e.g. a Pearson type IV distribution, has been debated since at least the 1910s. The quotient- and deviation-based definitions give rise to very different eras in that debate, of course. (However, the distribution of an integer-valued IQ cannot be exactly Normal, even on a deviation-based definition.) A Normal distribution is uniquely characterised by its mean $$\mu$$ and standard deviation $$\sigma$$. Its next two moments are the skew $$\gamma_1=0$$ and excess kurtosis $$\kappa_\text{excess}=0$$. To disambiguate, I've defined

$$\gamma_1=\mathbb{E}\bigg(\tfrac{X-\mu}{\sigma}\bigg)^3,\,\kappa_\text{excess}:=\mathbb{E}\bigg(\tfrac{X-\mu}{\sigma}\bigg)^4-3.$$

By contrast, a Pearson type IV distribution requires all four moments to be specified.

While we can't literally prove $$\gamma_1=\kappa_\text{excess}=0$$ empirically, we can constrain such quantities. Have any empirical studies provided either upper or lower bounds on these moments of the IQ distribution (or something analogous such as another quantification estimating psychometric $$g$$), on either the quotient or deviation definition? In the interests of keeping this question appropriate to the site, I don't care what method of defining or measuring IQ was assumed in a particular study, so there's no need to take a stance on that.

There are studies where higher order moments are analyzed. Just off the top of my head, see (Johnson, Carothers, Deary, 2008). The actual point of this study was to examine the Greater Male Variability Hypothesis (which the data was found to be strongly consistent with), however they also analyzed the distributions of ability more generally. They analyze the Scottish Mental Survey data, which tested essentially all children of Scotland of a given age. They find that the distribution is definitely unsymmetrical with more people below the mode. Here is the relevant part of the abstract:

... Clear analysis of the actual distribution of general intelligence based on large and appropriately population-representative samples is rare, however. Using two population-wide surveys of general intelligence in 11-year-olds in Scotland, we showed that there were substantial departures from normality in the distribution, with less variability in the higher range than in the lower. Despite mean IQ-scale scores of 100, modal scores were about 105... This is consistent with a model of the population distribution of general intelligence as a mixture of two essentially normal distributions, one reflecting normal variation in general intelligence and one reflecting normal variation in effects of genetic and environmental conditions involving mental retardation.

See the study for further discussion around kurtosis and skewness. They also reference other studies you may find valuable.

• In summary (viz Table 1), skew was -0.13 for males and 0004 for females without intelligence-disrupting conditions, and 0.334 and 0.250 with them, in SMS32, with the SMS47 counterparts being -0.001, -0.004, 0156, -0.020, and these eight values' kurtosis counterparts were -0.476, -0.399, -0.290, -0.400, -0.550, -0.500, -0.688, -0.600. Thank you, +1. I won't select this answer just yet; I'll try to go through other studies as you mentioned, plus I wouldn't want to discourage other answers. – J.G. Nov 16 '18 at 9:21
• @J.G. Yes, but it's important to note that in that table they analyze disrupted and non-disrupted separately. The magnitude of the skew would be greater (i.e. more negative) if they didn't separate the analysis. They model the disruption versus non-disruption as two normal distributions. – Eff Nov 16 '18 at 9:23
• Yes, I noticed that. I'm hoping at least one of the mentioned studies looked at a population without such splitting. They probably will; there's clearly already been a lot of work on this subject. – J.G. Nov 16 '18 at 9:30
• @J.G. Yeah, it should be available somewhere. You could even e-mail one of the authors about the values when all data is analyzed together. – Eff Nov 16 '18 at 9:32

One answer is that, since $$g$$ does not really exist as a unidimensional biological entity (rather, it is a near infinite-dimensional soup of inherited DNA and accumulated life experiences), the question of its univariate distribution is moot.

• @PeterWesftall You're conflating $g$ depending on many causal factors with it being multivariate. – J.G. Nov 14 '18 at 13:39
• Rephrasing, "intelligence is not one dimensional, so it makes no sense to question the nature of its univariate distribution." But assuming, for the sake of argument, that there is a one-dimensional $g$ that is "intelligence," it is not observable, and its only salient quality is that it ranks every individual in the universe perfectly. As such, any monotone transformation of $g$ is equivalently called "intelligence." From that standpoint, the question of $g$'s distribution is still moot - $g$ can have any continuous distribution whatsoever, including, of course, N(0,1). – Peter Westfall Nov 15 '18 at 21:07
• You're still confusing dependence on something multivariate with being multivariate. You're also confusing the word "moot" with nontrivial. The whole point of my question was whether empirical studies had bounded $\gamma_1,\,\kappa_\text{excess}$. If you know of any, please mention them. – J.G. Nov 15 '18 at 21:24
• As far as empirical studies go, if you anchor manifest variables to a latent $g$, then that $g$ will inherit characteristics of the manifest measures. So then the question is not about how "intelligence" is distributed, but instead about how some particular manifest measure of "intelligence" is distributed. Which again renders the question moot, because it depends on the particulars of the given manifest measurement. So from that standpoint as well, the distribution could be anything whatsoever. – Peter Westfall Nov 16 '18 at 14:13
• @PeterWesftall Which might be why I asked about one specific quantification, namely IQ, rather than $g$ directly. – J.G. Nov 16 '18 at 14:14