# Benford’s law, or the First-digit law

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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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