statistics-0.13.3.0: A library of statistical types, data, and functions

Statistics.Quantile

Description

Functions for approximating quantiles, i.e. points taken at regular intervals from the cumulative distribution function of a random variable.

The number of quantiles is described below by the variable q, so with q=4, a 4-quantile (also known as a quartile) has 4 intervals, and contains 5 points. The parameter k describes the desired point, where 0 ≤ kq.

Synopsis

# Quantile estimation functions

Arguments

 :: Vector v Double => Int k, the desired quantile. -> Int q, the number of quantiles. -> v Double x, the sample data. -> Double

O(n log n). Estimate the kth q-quantile of a sample, using the weighted average method.

data ContParam Source #

Parameters a and b to the continuousBy function.

Constructors

 ContParam !Double !Double

Arguments

 :: Vector v Double => ContParam Parameters a and b. -> Int k, the desired quantile. -> Int q, the number of quantiles. -> v Double x, the sample data. -> Double

O(n log n). Estimate the kth q-quantile of a sample x, using the continuous sample method with the given parameters. This is the method used by most statistical software, such as R, Mathematica, SPSS, and S.

Arguments

 :: Vector v Double => ContParam Parameters a and b. -> Int q, the number of quantiles. -> v Double x, the sample data. -> Double

O(n log n). Estimate the range between q-quantiles 1 and q-1 of a sample x, using the continuous sample method with the given parameters.

For instance, the interquartile range (IQR) can be estimated as follows:

midspread medianUnbiased 4 (U.fromList [1,1,2,2,3])
==> 1.333333

# Parameters for the continuous sample method

California Department of Public Works definition, a=0, b=1. Gives a linear interpolation of the empirical CDF. This corresponds to method 4 in R and Mathematica.

Hazen's definition, a=0.5, b=0.5. This is claimed to be popular among hydrologists. This corresponds to method 5 in R and Mathematica.

Definition used by the S statistics application, with a=1, b=1. The interpolation points divide the sample range into n-1 intervals. This corresponds to method 7 in R and Mathematica.

Definition used by the SPSS statistics application, with a=0, b=0 (also known as Weibull's definition). This corresponds to method 6 in R and Mathematica.

Median unbiased definition, a=1/3, b=1/3. The resulting quantile estimates are approximately median unbiased regardless of the distribution of x. This corresponds to method 8 in R and Mathematica.

Normal unbiased definition, a=3/8, b=3/8. An approximately unbiased estimate if the empirical distribution approximates the normal distribution. This corresponds to method 9 in R and Mathematica.