<p>There are studies where higher order moments are analyzed. Just off the top of my head, see (<a href="https://doi.org/10.1111/j.1745-6924.2008.00096.x" rel="nofollow noreferrer">Johnson, Carothers, Deary, 2008</a>). 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:</p>

<blockquote>
  <p>... 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, <strong>we showed that there were substantial
  departures from normality in the distribution,
  with less variability in the higher range than in the lower</strong>.
  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.</p>
</blockquote>

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

CC BY-SA 4.0
2018-11-16T09:21:26.523Z
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.
CC BY-SA 4.0
2018-11-16T09:23:29.697Z
@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
CC BY-SA 4.0
2018-11-16T09:30:38.740Z
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.
CC BY-SA 4.0
2018-11-16T09:32:22.897Z
@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
CC BY-SA 4.0
2018-11-16T08:55:45.057Z
2018-11-16T09:20:57.663Z
2018-11-16T09:20:57.663Z
Eff
<p>One answer is that, since <span class="math-container">$g$</span> 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.</p>

CC BY-SA 4.0
2018-11-14T13:39:39.300Z
@PeterWesftall You're conflating $g$ depending on many causal factors with it being multivariate.
J.G.
CC BY-SA 4.0
2018-11-15T21:07:10.653Z
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).
BigBendRegion
CC BY-SA 4.0
2018-11-15T21:24:59.187Z
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.
CC BY-SA 4.0
2018-11-16T14:13:21.543Z
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.
BigBendRegion
CC BY-SA 4.0
2018-11-16T14:14:19.620Z
@PeterWesftall Which might be why I asked about one specific quantification, namely IQ, rather than $g$ directly.
J.G.
CC BY-SA 4.0
2018-11-17T16:31:54.070Z
Ok, got it. But since you state "I don't care what method of defining or measuring IQ was assumed," the answer remains that the distribution could have any moments whatsoever - no constraints. The question of the particular distribution is necessarily tied to method (and design) of the measurement. After all, any monotone transformation of a particular measurement of IQ is also a measurement of IQ.
BigBendRegion
CC BY-SA 4.0
2018-11-17T18:20:00.317Z
But the facts remains the one that was assumed wasn't as open-ended as that in any study. It's a question about what was found, not what could have been found.
J.G.
CC BY-SA 4.0
2018-11-18T15:39:46.240Z
Ok, it's a lit search question. You might include "fat tails" or "heavy tails" in your Googling, as those terms are synonymous with "kurtosis." Some people think kurtosis refers to "peakedness," and might miss some relevant papers.  For example, this pops up: https://arxiv.org/pdf/1510.04245.pdf . Although the article does not precisely refer to intelligence, it does refer to cognitive functions, and makes my broader point that "IQ" could be measured in various ways, having any distributions whatsoever.
BigBendRegion
CC BY-SA 4.0
2018-11-14T13:08:53.157Z
2018-11-14T13:08:53.157Z
BigBendRegion
<p>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 <span class="math-container">$\mu$</span> and standard deviation <span class="math-container">$\sigma$</span>. Its next two moments are the skew <span class="math-container">$\gamma_1=0$</span> and excess kurtosis <span class="math-container">$\kappa_\text{excess}=0$</span>. To disambiguate, I've defined</p>

<p><span class="math-container">$$\gamma_1=\mathbb{E}\bigg(\tfrac{X-\mu}{\sigma}\bigg)^3,\,\kappa_\text{excess}:=\mathbb{E}\bigg(\tfrac{X-\mu}{\sigma}\bigg)^4-3.$$</span></p>

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

<p>While we can't literally prove <span class="math-container">$\gamma_1=\kappa_\text{excess}=0$</span> 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 <span class="math-container">$g$</span>), 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.</p>

<measurement><intelligence><statistics><iq>
Bounds on skew and kurtosis of IQ
J.G.
# Bounds on skew and kurtosis of IQ

(Asked by: username_0 on 2018-11-10)

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.

**Tags:** measurement, intelligence, statistics, iq


---

## 2 answers

### Answer by username_1 on 2018-11-16. Score: 2

There are studies where higher order moments are analyzed. Just off the top of my head, see ([Johnson, Carothers, Deary, 2008](https://doi.org/10.1111/j.1745-6924.2008.00096.x)). 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.

> **Comments:**
> **username_0**: 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.  
> **username_1**: @username_0 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.  
> **username_0**: 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.  
> **username_1**: @username_0 Yeah, it should be available somewhere. You could even e-mail one of the authors about the values when all data is analyzed together.  

---

### Answer by username_2 on 2018-11-14. Score: -1

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.

> **Comments:**
> **username_0**: @PeterWesftall You're conflating $g$ depending on many causal factors with it being multivariate.  
> **username_2**: 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).  
> **username_0**: 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.  
> **username_2**: 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.  
> **username_0**: @PeterWesftall Which might be why I asked about one specific quantification, namely IQ, rather than $g$ directly.  
> **username_2**: Ok, got it. But since you state "I don't care what method of defining or measuring IQ was assumed," the answer remains that the distribution could have any moments whatsoever - no constraints. The question of the particular distribution is necessarily tied to method (and design) of the measurement. After all, any monotone transformation of a particular measurement of IQ is also a measurement of IQ.  
> **username_0**: But the facts remains the one that was assumed wasn't as open-ended as that in any study. It's a question about what was found, not what could have been found.  
> **username_2**: Ok, it's a lit search question. You might include "fat tails" or "heavy tails" in your Googling, as those terms are synonymous with "kurtosis." Some people think kurtosis refers to "peakedness," and might miss some relevant papers.  For example, this pops up: https://arxiv.org/pdf/1510.04245.pdf . Although the article does not precisely refer to intelligence, it does refer to cognitive functions, and makes my broader point that "IQ" could be measured in various ways, having any distributions whatsoever.  

---

