Quantile

Quantile[list,q]

gives the q ^(th) quantile of list.

Quantile[list,{q1,q2,…}]

gives a list of quantiles q1, q2, ….

Quantile[list,q,{{a,b},{c,d}}]

uses the quantile definition specified by parameters a, b, c, d.

Quantile[dist,q]

gives a quantile of the distribution dist.

Details

  • Quantile is also known as value at risk (VaR) or fractile.
  • Quantile[list,q] gives Sort[list,Less][[Max[Ceiling[qLength[list]],1]]].
  • Quantile[{{x1,y1,…},{x2,y2,…},…},q] gives {Quantile[{x1,x2,…},q],Quantile[{y1,y2,…},q]}.
  • For a list of length n, Quantile[list,q,{{a,b},{c,d}}] depends on x=a+(n+b)q. If x is an integer, the result is s[[x]], 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 {{0,0},{1,0}}.
  • 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 qn
    {{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[list,q] always gives a result equal to an element of list.
  • The same is true whenever d is 0.
  • When d is 1, Quantile is piecewise linear as a function of q.
  • Median[list] is equivalent to Quantile[list,1/2,{{1/2,0},{0,1}}].
  • About 10 different choices of parameters are in use in statistical work.
  • Quantile works with SparseArray objects.
  • Quantile[dist,q] is equivalent to InverseCDF[dist,q].

Examples

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Basic Examples  (6)

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^(th) quantile for a normal distribution:

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Quantile function for a continuous univariate distribution:

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Quantile function for a discrete univariate distribution:

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Scope  (24)

Applications  (7)

Properties & Relations  (9)

Possible Issues  (2)

Neat Examples  (1)

See Also

Median  Quartiles  Ordering  Variance  MedianDeviation  InterquartileRange  Sort  ListInterpolation  Nearest  InverseCDF  InverseSurvivalFunction  OrderDistribution

Tutorials

Introduced in 2003
(5.0)
| Updated in 2007
(6.0)