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 (if you want more digits, here’s a link to the first million digits):

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

Digit | ||

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 and 12 digits 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 , so we get ; . The standard deviation is

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