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.