Tuesday, July 27, 2010

Sussex county tax scofflaws plotted and extended to $12 million

Sussex county in Delaware published their top 100 tax delinquents. The .pdf is available from the News Journal. pdf's are hard to use so I put the data into a Google Doc so that anyone can run calculations on it. Kudos to the News Journal and Sussex county for actually providing the data to the public.

First the graphs. The plot below shows amount owed by the tax delinquent plotted as a function of the rank for the top 100 tax delinquents in Sussex county according to their records.

The decay of the amount owed vs. the rank suggests plotting the data on a log-log scale (click for larger).

The plot of the data at higher ranks is a straight line. This dependence is characteristic of a power law function for amount owed vs. rank. This is typical for this kind of data. This plot is similar to a Zipf plot of ln(frequency) vs. ln(rank order), where we have used the amount owed instead of frequency.

Notice that that the first few items have higher values than a simple power law, the $80,000 of rank 1 and even the $30,000 of rank 7. Sometimes the tails of these types of distributions are called the "long tail" Most of the total can be in this tail, yet it is spread among many categories. I attempted three fits of the data. The first in blue includes the early parts of the data that are not really a power law, so this fit is not very good. The next two fits are for the data above rank 10, magenta, and then above rank 62, green, where the data is better suited to a power law fit.

The article reveals the total amount of tax that is delinquent for the top 100 and for the total for Sussex county:
"The list unveiled Tuesday includes developers, estates, homeowners and farmers who collectively owe about $1.6 million. The county collects taxes on its behalf and for all Sussex school districts. About $12 million is owed in total."
We can use the information from the article to figure out how many more tax delinquents there are in Sussex country by using the fit to the power law portion of the curve and extending it out to higher ranks until the total amount is the $12 million total quoted above. The chart below shows two fits, fit 1 for the portion of the the curve greater than rank 10 and fit 2 for the portion of the curve greater that rank 62, which are then extended to higher ranks.

The two curves are very similar in the tail portion and fit one yields 1829 tax delinquents and fit two yields 1717 tax delinquents to add up to $12 million of overdue taxes, with the lower ranks owing about $4000. This number is based on the assumptions above and is dependent on the curve fit and the extrapolation of the power law fit to these higher values. It is really best done with more orders of magnitude than with the data we have here. If Sussex county publishes their top 500 delinquents we might be able to check this work.

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