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.

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