# Quantile

Quantile[list,q]

gives the q 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 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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