How Unique is Sheldon Cooper? Monte Carlo Simulation and IQ Scores
Ian Gordon
2015-06-19
Introduction
Fans of the television show The Big Bang Theory know that Sheldon Cooper is, well, unique. For one thing, he claims to have an IQ of 187. After years spent watching Sheldon Cooper, I began to wonder just how unique his IQ actually is. How many Sheldon Coopers might there be in the world?
Discussion
IQ scores follow a normal (or Gaussian) distribution, with a mean of 100 and a standard deviation of 15, meaning that most people are concentrated around the middle of the bell curve. We know that about 99.7% of the population lie within 3 standard deviations of the mean and scores beyond that are very rare. How rare are these higher scores?
Using Monte Carlo simulation, we can approximate the number of people who have an IQ that is greater than or equal to a specified IQ score. We can then compare this result with the result based on the expected probability to judge the accuracy of the simulation.
Monte Carlo Simulation
To simulate the distribution of IQ scores, we draw random samples from a normal distribution and simply count the number that are greater than or equal to the specified score. For a large enough sample, this should give us a reasonable approximation.
How many people can we expted to have an IQ of 187 or higher? We can run the following R code to give the expected number based on the probability. Run the simulation and see how your results compare!
pnorm(187, mean=100, sd=15, lower=F) * 1000000000
[1] 3.315746
Try It!!!
To run the simulation, follow this link and enter the values for IQ, sample size, and the number of runs.
As we are dealing with random numbers, you can increase the number of runs to try to improve the accuracy of the estimate.
Beware, though. The larger the samples and the more runs, the longer the simulation will take. Try it out. If you get a result of zero, run again or increase the sample size and see what happens.