Quantile

Quantile[list,q]
gives the ^(th) 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}}meanbased estimate (Weibull method)
    {{1, -1}, {0, 1}}modebased estimate
    {{1/3, 1/3}, {0, 1}}medianbased 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:

In[1]:=
Click for copyable input
Out[1]=

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]=

Quantile function for a continuous univariate distribution:

In[1]:=
Click for copyable input
Out[1]=

Quantile function for a discrete univariate distribution:

In[1]:=
Click for copyable input
In[2]:=
Click for copyable input
Out[2]=
Introduced in 2003
(5.0)
| Updated in 2007
(6.0)