IQ Range Probability

Calculator

IQ range

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Probability:
Frequency:
Total in the world[1]:
Total ever lived[1]:

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[1] Population estimates as of 2023.

Formula

The IQ "scale" is not a direct measure, but a rank, which only has meaning relative to other scores. It is defined to have a normal distribution with mean value $\mu = 100$ and (typically) a standard deviation of $\sigma = 15$: \begin{align}f(x = IQ) &= \frac{1}{\sigma\sqrt{2\pi}} \exp\left[\frac{-(x - \mu)^2}{2\sigma^2}\right] \\\\ &= \frac{1}{15\sqrt{2\pi}} \exp\left[\frac{-(IQ - 100)^2}{2\cdot 15^2}\right]. \end{align} Surprisingly, it is possible to convert any continuous distribution of test scores into a normal distribution that preserves the order. The probability that an IQ is less than $IQ_{max}$ is given by:

\begin{align} P(IQ < IQ_{max}) &= \int_{-\infty}^{IQ_{max}} \frac{1}{\sigma\sqrt{2\pi}} \exp\left[\frac{-(IQ - \mu)^2}{2\sigma^2}\right] \,\mathrm dx \\\\ &= \frac12\left[1 + \operatorname{erf}\left(\frac{IQ_{max} - \mu}{\sigma\sqrt{2}}\right)\right] \\\\ &= \frac12\left[1 + \operatorname{erf}\left(\frac{IQ_{max} - 100}{15\sqrt{2}}\right)\right]. \end{align}

So the probability that $IQ_{min} < IQ < IQ_{max}$ is: $$\frac12\left[\operatorname{erf}\left(\frac{IQ_{max} - 100}{15\sqrt{2}}\right) - \operatorname{erf}\left(\frac{IQ_{min} - 100}{15\sqrt{2}}\right)\right].$$