Benford’s law, or the First-digit law

August 25, 2011

(This article was first published on Statistic on aiR, and kindly contributed to R-bloggers)

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:


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){


# 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)




benford(3,640) # if n is greater, it returns "inf" (on my pc)



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