Answer to Question #118631 in Statistics and Probability for Michael

Question

IQ scores of people around the World are normally distributed, with a mean of 100 and a standard deviation of 15. A genius is someone with an IQ greater than or equal to 140. What percent of the population is considered genius ?

Answer 1

Since IQ of people has a normal distribution with $\mu=100,\,\,\sigma=15$ , its probability density function (see e.g., https://en.wikipedia.org/wiki/Normal_distribution) is

$p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12(\frac{x-\mu}{\sigma})^2}$ , where $\mu=100,\,\,\sigma=15$ .

We compute the following integral:

$\int_{140}^{+\infty}p(x)dx=\frac{1}{15\sqrt{2\pi}}\int_{140}^{+\infty}e^{-\frac12(\frac{x-100}{15})^2}dx\approx0.0038$

Then, the percent of people with IQ greater than or equal to 140 is approximately 0.0038*100%=0.38%

Answer:0.38%

P.S. the integral was computed with a help of Anaconda (https://www.anaconda.com/products/individual).

A code for the computation is:

_{from scipy import integrate}

_{import numpy as np}

_import math

_{func = lambda x:(1/(15*math.sqrt(2)*math.sqrt(math.pi)))*math.exp(-1/2*((x-100)/15)*}

_{((x-100)/15))}

_{Pr = integrate.quad(func, 140, np.Infinity)}

_print(Pr)