This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Quantile

 Quantilegives the quantile of list. Quantilegives a list of quantiles , , .... Quantileuses the quantile definition specified by parameters a, b, c, d. Quantilegives a quantile of the symbolic distribution dist.
• For a list of length n, Quantile depends on . If x is an integer, the result is , where s=Sort[list, Less]. Otherwise the result is s[[Floor[x]]]+(s[[Ceiling[x]]]-s[[Floor[x]]])(c+dFractionalPart[x]), with the indices taken to be 1 or n if they are out of range.
• The default choice of parameters is .
• Common choices of parameters include:
 {{0, 0}, {1, 0}} inverse empirical CDF (default) {{0, 0}, {0, 1}} linear interpolation (California method) {{1/2, 0}, {0, 0}} element numbered closest to {{1/2, 0}, {0, 1}} linear interpolation (hydrologist method) {{0, 1}, {0, 1}} mean-based estimate (Weibull method) {{1, -1}, {0, 1}} mode-based estimate {{1/3, 1/3}, {0, 1}} median-based estimate {{3/8, 1/4}, {0, 1}} normal distribution estimate
• Quantile always gives a result equal to an element of list.
• The same is true whenever d is .
• When d is , Quantile is piecewise linear as a function of q.
• About 10 different choices of parameters are in use in statistical work.
Find the halfway value (median) of a list:
Find the quarter-way value (lower quartile) of a list:
Lower and upper quartiles:
The q quantile for a normal distribution:
Quantile function for a continuous univariate distribution:
Quantile function for a discrete univariate distribution:
Find the halfway value (median) of a list:
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Find the quarter-way value (lower quartile) of a list:
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Lower and upper quartiles:
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The q quantile for a normal distribution:
 Out[1]=

Quantile function for a continuous univariate distribution:
 Out[1]=

Quantile function for a discrete univariate distribution:
 Out[2]=
 Scope   (15)
Quantile works with any real numeric quantities:
Find quantiles of elements in each column:
Find multiple quantiles of elements in each column:
Obtain results at any precision:
Compute results for a large vector or matrix:
Compute results using other parametrizations:
Obtain exact numeric results:
Obtain a machine-precision result:
Obtain a result at any precision for a continuous distribution:
Obtain a symbolic expression for the quantile:
Quantile for nonparametric distributions:
Compare with the value for the underlying parametric distribution:
Plot the quantile for a histogram distribution:
Quantile for a truncated distribution:
Quadratic transformation of an exponential distribution:
Censored distribution:
Compute results for a SparseArray:
 Applications   (3)
Plot the q quantile for a list:
Plot a linearly interpolated quantile:
Generate a random number from a distribution:
With default parameters Quantile always returns an element of the list:
Quartiles gives linearly interpolated Quantile values for a list:
InterquartileRange is the difference of linearly interpolated Quantile values for a list:
QuartileDeviation is half the difference of linearly interpolated Quantile values for a list:
QuartileSkewness uses linearly interpolated Quantile values as a skewness measure:
Quantile is equivalent to InverseCDF for distributions:
Symbolic closed forms do not exist for some distributions:
Numerical evaluation works:
Substitution of invalid values into symbolic outputs gives results that are not meaningful:
Passing it as an argument, it stays unevaluated:
Here is the 1/2 quantile with two varying parameters: