benford.php

<?php
$limit = 1000000;
$counts = [
    '1' => 0,
    '2' => 0,
    '3' => 0,
    '4' => 0,
    '5' => 0,
    '6' => 0,
    '7' => 0,
    '8' => 0,
    '9' => 0
];

for ($current = 0; $current < $limit; $current++) {
    //$number = (string) rand(1, getrandmax());
    $number = (string) random_int(1, PHP_INT_MAX);
    $counts[$number[0]]++;
}

foreach ($counts as $number => $count) {
    echo $number . ': ' . ($count/$limit)*100 . '%' . PHP_EOL;
}

And you can start to see what's going on here - you can't bin them by the starting digit and expect to see a uniform distribution.

So, for example, if you're generating numbers between 1 and 10, you're counting 2-9 in there own buckets (which get about 10% probability you get from chance). But the outcomes for 1s and 10s are combined in the "1" bucket going on.

Similarly between 1 and 50. There are a lot of numbers in that range that start with the digits 1, 2, 3, and 4 (1 think 11 of them for each). But for 5, 6, 7, 8, 9 there are fewer. 5 is just 5 and 50, 6 is only 6, etc...

Yup, that's the issue: https://twitter.com/tjlytle/status/1202990988240850947

Max in size cuts out most of the set of numbers that start with 9 (near the max)

If you set the min and max to 11--99, you get a normal distribution:

1: 11.1096%
2: 11.1252%
3: 11.058%
4: 11.1275%
5: 11.0732%
6: 11.1191%
7: 11.1153%
8: 11.1355%
9: 11.1366%

Based on what I read, it looks like this implementation is wrong (those are the percentage of occurences), but should look like:

<?php
$limit = 1000000;
$counts = [
    '1' => 0,
    '2' => 0,
    '3' => 0,
    '4' => 0,
    '5' => 0,
    '6' => 0,
    '7' => 0,
    '8' => 0,
    '9' => 0
];

for ($current = 0; $current < $limit; $current++) {
    //$number = (string) rand(1, getrandmax());
    $number = (string) random_int(1, PHP_INT_MAX);
    $counts[$number[0]]++;
}

foreach ($counts as $number => $count) {
    echo $number . ': ' . benfordLaw($number) .PHP_EOL;
}

function benfordLaw($leading){
  return (log(1+(1/$leading))/log(10));
}

Output

1: 0.30102999566398
2: 0.17609125905568
3: 0.1249387366083
4: 0.096910013008056
5: 0.079181246047625
6: 0.066946789630613
7: 0.057991946977687
8: 0.051152522447381
9: 0.045757490560675

See implementation and expected outputs in other languages: https://rosettacode.org/wiki/Benford%27s_law#JavaScript