BUILT-IN MATHEMATICA SYMBOL

# Quantile

Quantile[list, q]
gives the quantile of list.

Quantile[list, {q1, q2, ...}]
gives a list of quantiles , , ....

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.

## DetailsDetails

• Quantile is also known as value at risk (VaR) or fractile.
• Quantile[list, q] gives Sort[list, Less][[Ceiling[qLength[list]]]].
• Quantile[{{x1, y1, ...}, {x2, y2, ...}, ...}, q] gives .
• For a list of length n, Quantile[list, q, {{a, b}, {c, d}}] 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[list, q] 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.
• 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].

## ExamplesExamplesopen allclose all

### Basic Examples (6)Basic Examples (6)

Find the halfway value (median) of a list:

 Out[1]=

Find the quarter-way value (lower quartile) of a list:

 Out[1]=

Lower and upper quartiles:

 Out[1]=

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]=