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Quantile

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Quantile[list, q]
gives the qth 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:
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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 qth quantile for a normal distribution:
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