# 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

[itex]{\rm median}=\kappa\,2^{1/\theta},[itex]

where parameters κ and θ are given by:

[itex]

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

and

[itex]

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

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 limitzh:帕累托插值

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