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
- Pa = proportion of the distribution that lies below the lower limit
- Pb = proportion of the distribution that lies below the upper limit