Pareto interpolation is a method of

estimating Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...

the

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...

and other properties of a population that follows a

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...

. It is used in

economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...

when analysing the distribution of incomes in a population, when one must base estimates on a relatively small random sample taken from the population. The family of Pareto distributions is parameterized by * a positive number κ that is the smallest value that a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

with a Pareto distribution can take. As applied to distribution of incomes, κ is the lowest income of any person in the population; and * a positive number θ the "Pareto index"; as this increases, the tail of the distribution gets thinner. As applied to distribution of incomes, this means that the larger the value of the Pareto index θ the smaller the proportion of incomes many times as big as the smallest incomes. Pareto interpolation can be used when the available information includes the proportion of the sample that falls below each of two specified numbers ''a'' < ''b''. For example, it may be observed that 45% of individuals in the sample have incomes below ''a'' = $35,000 per year, and 55% have incomes below ''b'' = $40,000 per year. Let :''P''_''a'' = proportion of the sample that lies below ''a''; :''P''_''b'' = proportion of the sample that lies below ''b''. Then the estimates of κ and θ are :

\widehat =
\left( 
\frac

\right)^

and :

\widehat \; = \;
\frac
.

The estimate of the median would then be :

\mbox=\widehat\cdot 2^,\,

since the actual population median is :

\mbox=\kappa\,2^.\,

References

U.S. Census Bureau, Memorandum on statistical techniques used in 2001 income survey (PDF)
See Equation 10 on p. 24.

Gives a derivation of the equations for Pareto interpolation. {{authority control Estimation methods Income inequality metrics Theory of probability distributions Parametric statistics Vilfredo Pareto