Pareto interpolation

Pareto interpolation is a nonlinear method of interpolation to find the median of a set of data. It is used in economics when analysing income figures. It assumes that the data fits a curve known as the Pareto distribution.

The median is given by

<math>{\rm median}=\kappa\,2^{1/\theta},</math>

where parameters κ and θ are given by:

<math>

K = \left( \frac{P_b - P_a} { \frac{1}{a^{\theta}} - \frac{1}{b^{\theta}}} \right) ^{ \frac{1} {\theta}} </math>

and

<math>

\theta \; = \; \frac{\log(1-P_a) - \log(1-P_b)} {\log(b) - \log(a)} </math>

where

a = lower limit of the category containing the median

b = upper limit of the category containing the median

P_a = proportion of the distribution that lies below the lower limit

P_b = proportion of the distribution that lies below the upper limit