Wednesday, February 08, 2012

Benford's Law analysis of my own tax return

I've written in the past about Benford's Law as applied to elections and BBC executives' expenses and MPs' expenses, but as the British tax season has just ended I thought it might be fun to apply it in one area where it's commonly used: financial fraud.

At hand I had my own (non-fraudulent) tax return for 2010 and so I entered all the figures on it and looked at the distribution of initial numbers. The green line shows the idealized initial number rates and the blue the actual. Looks quite close:

But the only reliable way to check is a statistical test for goodness-of-fit between my numbers and the idealized distribution. I chose to use a Fisher's Exact Test as chi-squared would be inappropriate due to the low frequencies involved.

Here's a quick R session:
> tax <- matrix(c(18, 12.9, 8, 7.6, 3, 5.4, 5, 4.2, 0, 3.4, 1,
2.9, 1, 2.5, 3, 2.2, 4, 2.0), ncol=2, byrow=T)
> tax
      [,1] [,2]
 [1,]   18 12.9
 [2,]    8  7.6
 [3,]    3  5.4
 [4,]    5  4.2
 [5,]    0  3.4
 [6,]    1  2.9
 [7,]    1  2.5
 [8,]    3  2.2
 [9,]    4  2.0
> fisher.test(tax)

 Fisher's Exact Test for Count Data

data:  tax 
p-value = 0.6451
So, with a p-value that high there's no way to reject the null hypothesis that my tax return fits the expected Benford's Law distribution.

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