I became aware of Benford's Law when it was explained by Ole as part of his brilliant "Watch Out for Number One!" speech. According to the law, there is an inherent distribution of the first digits when considering any set of numbers. So if you say look at the file sizes of 1000 random files, the probability that the file size will start with the digit "1" will be about 30%. In general, the probability that the number will start with digit "d" where "d" ranges from 1 to 9 is log10 (1 + 1/d).
Since my curiosity was piqued, I decided to give it a try. I started by looking at the size of all the files in my computer. I was not expecting any success as the files in my machine range from system DLLs to spreadsheets. However when the numbers were crunched, lo and behold! Benford's law was rearing its head. Same with word count of all the Python files in my computer. Finally I looked at 10,000 random mediants to approximate Pi. The mediants matched the law to a tee!
There are several interesting practical applications for this law. For e.g. credit card companies and Income Tax agencies can use this law to detect fraudulent transactions or income tax returns.