PowerShell and Benford's Law

Was reading through a statistics blog (R) the other day when I read a posting on Benford's law. The definition according to the blog is:

"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."

The probabilities are distributed as demonstration here.

This seemed counter-intuitive and I wanted to validate it myself. Let's look at the leading digit of all the txt files in one of my directories. Enter PowerShell.....

# Explore Benford's Law 

$array=@() 

foreach ($item in (Get-ChildItem -Path p:\ -Filter *.txt -Recurse)) 

{  

$array+= $item.length.toString()[0] 

} 



$array `

| Group-Object -NoElement `

| Sort-Object count -Descending `

| Format-Table @{label=”#”;expression={$_.Name}},  

@{label=”Count”;expression={"{0:%##}" -f $($_.Count/$array.Count)}},  

@{label=”Histogram”;expression={“▄” * $_.Count}} -autosize

I consider this a validation, but lets try one another example, this time looking at leading digits on the workingset of the processes on my desktop:

$array=@()       

foreach($a in (Get-Process))       

{        

$array+= $a.WorkingSet.toString()[0]       

}       

  

$array `

| Group-Object -NoElement `

| Sort-Object count -Descending `

| Format-Table @{label=”#”;expression={$_.Name}},        

@{label=”Count”;expression={"{0:%##}" -f $($_.Count/$array.Count)}},        

@{label=”Histogram”;expression={“▄” * $_.Count}} -autosize

Again, this seems to hold true. Now that I have examples of Benford's law, I feel compelled to try and understand it. Wish me luck!