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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 symbolic distribution dist.
  • 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=0.
  • When d=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 ten different choices of parameters are in use in statistical work.
EXAMPLESCLOSE ALL
Basic Examples   (4)
Find the half-way value (median) of a list:
Find the quarter-way value (lower quartile) of a list:
Lower and upper quartiles:
The q^(th) quantile for a normal distribution:
Find the half-way value (median) of a list:
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Find the quarter-way value (lower quartile) of a list:
In[1]:=
Click for copyable input
Out[1]=
 
Lower and upper quartiles:
In[1]:=
Click for copyable input
Out[1]=
 
The q^(th) quantile for a normal distribution:
In[1]:=
Click for copyable input
Out[1]=
Scope   (6)
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:
Applications   (3)
Plot the q^(th) quantile for a list:
Plot a linearly interpolated quantile:
Generate a random number from a distribution:
Properties & Relations   (6)
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:
Neat Examples   (1)
Here is the 1/2 quantile with two varying parameters:
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