Funny First Digit Distribution

Today I learnt one of the most interesting things I have ever known. Below is a graph of the first digit of many different physical constants against the number of constants with this first digit.

Of course this graph doesn’t look right at all. It seems to show that the probability of any universal constant beginning with the digit 1 is higher and this probability decreases as the first digit rises with very few beginning with 9. The truth is this exactly right, and doesn’t just apply to physical constants. If you look at the population, surface area, or border length of all countries you’ll find that about 30% of these values begin with 1. This is always true provided the data set covers quite a couple orders of magnitude and is not completely randomly generated. This 30% is just an average and for specific set sizes such as the numbers 1 to 9999 the probability of beginning with 1 drops to 11%. But when 1 to 19999 is examined the chance of beginning with 1 rises to 56%. The probability fluctuates between these two extremes and averages at 30% for an unknown set size. An interesting application is that when humans make up numbers it is part of the human psyche to try and distribute the numbers evenly. If these made up numbers are supposed to represent coefficients in scientific papers then we can see that they are clearly as forged as real data would generally follow this first digit law also known as Benford’s law. This interesting quirk or probability can be sued to spot these fraudulent physics papers and other cases where humans attempt to forge data.

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