## Encyclopedia > Pareto interpolation

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# 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

${\rm median}=\kappa\,2^{1/\theta},$

where parameters κ and θ are given by:

$K = \left( \frac{P_b - P_a} { \frac{1}{a^{\theta}} - \frac{1}{b^{\theta}}} \right) ^{ \frac{1} {\theta}}$

and

$\theta \; = \; \frac{\log(1-P_a) - \log(1-P_b)} {\log(b) - \log(a)}$

where

a = lower limit of the category containing the median

b = upper limit of the category containing the median

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

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

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