Pareto interpolation is a method of estimating

Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....

 the median

Median

In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

 and other properties of a population that follows a Pareto distribution. It is used in economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

 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
  Random variable
  In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
  
  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


and


The estimate of the median would then be


since the actual population median is