Law Of Anomalous Numbers

Life is full of coincidences. Yesterday I wrote a post and extolled the virtues of a certain Leighton Vaughan Williams, while almost ten years ago, I wrote about Benford numbers, and included an article written by the same Professor Williams on the subject. 

Casually included towards the end of that article, were these lines:

The same principle applies to trailing (i.e. last) digits. It's a great way, therefore, of checking the veracity of receipts. If, for example, there is an unusual number of trailing digit '7's, there's a decent chance that the figures are cooked. Tax authorities are alert to this.

In the past, I have published the Benford numbers from my Betfair trading spreadsheet, which is now into its 14th year, having been started on January 1st 2006. They're nothing more than a curiosity, but I like numbers so it's a fun annual exercise. 

Having re-read the article referenced above, I was a little sceptical that the claim that the law applies to trailing digits was correct, and sure enough it isn't. If tax authorities are analysing trailing digits, they are wasting their time.

Below are my 13 years of Benford data, including for the first (and last) time, the results for trailing digits:

Other than my 7s and 9s being 0.8% lower and higher than expected respectively, the numbers are pretty close to expectation, but for the trailing digits, the evidence is clear that every digit has pretty much the same chance of appearing. Benford's law does NOT appear to apply to trailing digits.

Benford's Law does apply to the second digit, although within a narrower range, and again my results are fairly close to those expected, but from third digit on the differences are even less. 

The law is useful for detecting bogus numbers, speaking of which, an astonishing claim was made overnight regarding my post on horse-racing and beating the insider traders.

Aluckyaday Tweeted:

The effort required to profit from horse racing at softbooks is very minimal. My selections take me about 15 minutes a day, and have over the last 3 years returned a 27.5% ROI. If you are able to bet with softbooks then racing offers more profit opportunities than any other sport

Goodness me. And there I was thinking an ROI of 2.75% over one year was good. 

Curious though as to what planet these soft-books are located on, that allow someone to win more than peanuts over three years at anywhere near this rate of return?