Is IQ normally distributed (i.e., is it really a ‘bell curve’)?

Note: This is a section of a longer article. To go to the start, click here.

Yes, in our dataset IQ was approximately normally distributed. We decided to test this only on participants that came from Positly (our web platform for recruitment of study participants) because the rest of our sample that came from social media is highly non-representative. As we wrote previously, our social media sample is younger, mostly male and of above average IQ and therefore inappropriate for generalizing about the IQ distribution.

After we selected Positly participants (n = 1853), we plotted their IQs. Here is a histogram and density plot that shows the distribution of IQs in our Positly sample (blue vertical dots indicate the average IQ of the sample). Note that we have not transformed the shape of this distribution at all (i.e., we did not do any processing that would force it to be a bell curve). Since IQ is based on principle component analysis (i.e., taking the first principle component of the matrix of task z-scores scores for all participants, and then calculating the loading of each task on the principle component), IQ scores end up being a weighted average of task scores. Since weighted averages have a tendency to be normal distributed due to the central limit theorem when applied to statistically independent random variables, this raises a question of how much of the bell curve shape comes from the fact that it's a weighted average of different test results, and how much of it comes from the underlying nature of intelligence. One argument sometimes made in favor of IQ being inherently normally distributed is that if it is the result of many small, independent additive factors (e.g., in a person's life, or in our genes, or in the brain) then that would produce a bell curve naturally. To what extent this may or may not be true is beyond the scope of this report.   

The distribution looks pretty bell-curved, i.e. normally distributed. However, to test this formally, we conducted the Kolomogorov-Smirnov test, which is a statistical test that tests whether the distribution statistically significantly deviates from normal. The test was non-significant (D = 0.019, p = 0.53), meaning that the difference between a normal distribution and the actual IQ distribution we measured in our sample is not statistically significant.

What do the other studies say?

Studies generally agree that IQ is normally distributed (e.g. Godwin & Smith, 2012; Kaplan et al., 2000). 

Takeaways

• In our sample, IQ was normally distributed, which agrees with prior studies.

If you'd like to read the full report, of which this is a section, as one long PDF, you can download it here.

And if you'd like to understand where your intellectual strengths and weaknesses lie, try the cognitive assessment tool that we developed out of this research:

Launch the Cognitive Assessment!