# A few statistics on the first 25 digits of pi

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.10.

Here are a few statistical properties of the first 25 digits of ${\pi}$ (if you want more digits, here’s a link to the first million digits):

$\displaystyle \pi=3.141592653589793238462643\ldots \ \ \ \ \ (1)$

The frequency of each digit and the probability of getting each one are:

 Digit ${j}$ ${N_{j}}$ ${P_{j}}$ 0 0 0 1 2 0.08 2 3 0.12 3 5 0.2 4 3 0.12 5 3 0.12 6 3 0.12 7 1 0.04 8 2 0.08 9 3 0.12

The most probable digit is 3, the median is 4 (there are 10 digits ${<4}$ and 12 digits ${>4}$ so that’s as close as we can get to dividing the distribution equally) and the average is 4.72.

We can get the variance by calculating ${\left\langle N^{2}\right\rangle -\left\langle N\right\rangle ^{2}}$, so we get ${\left\langle N^{2}\right\rangle =\frac{710}{25}=28.4}$; ${\sigma^{2}=28.4-\left(4.72\right)^{2}=6.1216}$. The standard deviation is

$\displaystyle \sigma=2.474 \ \ \ \ \ (2)$

We’d need to use quite a few more digits to get a properly random collection of numbers.