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Statistics

  

Quantile

  

compute quantiles

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

Quantile(A, p, ds_options)

Quantile(X, p, rv_options)

Parameters

A

-

data set or Matrix data set

X

-

algebraic; random variable or distribution

p

-

algebraic; probability

ds_options

-

(optional) equation(s) of the form option=value where option is one of ignore, method, or weights; specify options for computing the quantile of a data set

rv_options

-

(optional) equation of the form numeric=value; specifies options for computing the quantile of a random variable

Description

• 

The Quantile function computes the quantile corresponding to the given probability p for the specified random variable or data set.

• 

For a real valued random variable X with distribution function Fx, and any p between 0 and 1, the pth quantile of X is defined as inf{y|Fyp}. For continuous random variables this is equivalent to the inverse distribution function.

• 

For more details on sample quantiles see option method below.

• 

The first parameter can be a data set (e.g., a Vector), a Matrix data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

• 

The second parameter p is the probability.

Computation

• 

All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

• 

By default, all computations involving random variables are performed symbolically (see option numeric below).

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Data Set Options

  

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

• 

ignore=truefalse -- This option controls how missing data is handled by the Quantile command. Missing items are represented by undefined or Float(undefined). If ignore=false and A contains missing data, the missing data elements will be considered greater than all present data points. If ignore=true all missing items in A will be ignored. The default value is false.

• 

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight 1.

• 

method=integer[1..8] -- Method for calculating the quantiles. Let n denote the number of non-missing elements in A and for i=1..n let Bi denotes the ith order statistic of A. The first two methods for calculating quantiles are defined as follows.

1. 

Bj, where j=floornp+1;

2. 

Bj, where j=floornp+12;

  

The remaining quantiles are calculated in the form Bj+Bj+1Bjr, where j=floorq, r=fracq, and q is one of the quantities given below.

3. 

q=np;

4. 

q=np+12;

5. 

q=n+1p;

6. 

q=1+n1p;

7. 

q=13+n+13p;  (default method)

8. 

q=38+n+14p.

Random Variable Options

  

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

• 

numeric=truefalse -- By default, the quantile is computed using exact arithmetic. To compute the quantile numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

Compute the quantile of the Weibull distribution with parameters a and b.

QuantileWeibulla,b,13

aln321b

(1)

Use numeric parameters.

QuantileWeibull3,5,13

3ln321/5

(2)

QuantileWeibull3,5,0.3333333333

2.50444761563527

(3)

QuantileWeibull3,5,13,numeric

2.50444761563527

(4)

Generate a random sample of size 100000 drawn from the above distribution and compute the sample quantile.

ASampleWeibull3,5,105:

QuantileA,13

2.50274848246744

(5)

Compute the standard error of the sample quantile for the normal distribution with parameters 5 and 2.

XNormal5,2

XNormal5,2

(6)

BSampleX,106:

QuantileX,13,numeric,StandardError[106]Quantile,X,13,numeric

4.13854540122573,0.00259298577070808

(7)

QuantileB,13

4.13682046222954

(8)

Create two normal random variables and compute the quantiles of their sum.

XRandomVariableNormal5,2:

YRandomVariableNormal2,5:

QuantileX+Y,13

1587292+58RootOf3erf_Z+1292

(9)

QuantileX+Y,13,numeric

4.68046250585916

(10)

Verify this using simulation.

CSampleX+Y,106:

QuantileC,13

4.67576726413908

(11)

Compute the quantile of a weighted data set.

Vseqi,i=57..77,undefined:

W2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5:

QuantileV,13,weights=W

65.5485434888542

(12)

QuantileV,13,weights=W,ignore=true

65.5538510125591

(13)

Consider the following Matrix data set.

MMatrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M31130114694415271273683907884642878964844995128007

(14)

We compute the 37 quantile of each of the columns.

QuantileM,37

3.961.4761904761901.07756857142857105

(15)

References

  

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

  

Hyndman, R.J., and Fan, Y. "Sample Quantiles in Statistical Packages." American Statistician, Vol. 50. (1996): 361-365.

Compatibility

• 

The A parameter was updated in Maple 16.

See Also

Statistics

Statistics[Computation]

Statistics[CumulativeDistributionFunction]

Statistics[Decile]

Statistics[DescriptiveStatistics]

Statistics[Distributions]

Statistics[ExpectedValue]

Statistics[Percentile]

Statistics[Quartile]

Statistics[RandomVariables]

Statistics[StandardError]