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**Benford’s law**, also called the

*first-digit law*, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time.

Wikipedia, retrieved 08/25/2011

R simulation:

```
library(MASS)
benford <- function(m, n){
list <- c()
# compute all m^n, for n= 1, 2, ..., i, ..., n
for(i in 1:n){
list[i] <- m^i
}
# a function to extract the first digit from a number
bben <- function(k){
as.numeric(head(strsplit(as.character(k),'')[[1]],n=1))
}
# extract the first digit from all numbers computed
first.digit <- sapply(list, bben)
# plot frequency of first digits
truehist(first.digit, nbins=10, main=m)
}
par(mfrow=c(2,2))
benford(2,1000)
benford(3,640) # if n is greater, it returns "inf" (on my pc)
benford(4,500)
benford(5,440)
```

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