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Quartiles[data]

gives the quantile estimates of the elements in data.

Quartiles[data,{{a,b},{c,d}}]

uses the quantile definition specified by parameters a, b, c, d.

Quartiles[dist]

gives the quantiles of the distribution dist.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Array Data  
Image and Audio Data  
Date and Time  
Distributions and Processes  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
History
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Quartiles

Quartiles[data]

gives the quantile estimates of the elements in data.

Quartiles[data,{{a,b},{c,d}}]

uses the quantile definition specified by parameters a, b, c, d.

Quartiles[dist]

gives the quantiles of the distribution dist.

Details

  • is equivalent to the median. »
  • is equivalent to the average of the medians of the and smallest elements in data if is odd, and the median of the smallest elements if is even.
  • is defined like , but with the largest rather than smallest elements.
  • For MatrixQ data, the quartile is computed for each column vector with Quartiles[{{x1,y1,…},{x2,y2,…},…}] equivalent to {Quartiles[{x1,x2,…}],Quartiles[{y1,y2,…}]}. »
  • For ArrayQ data, quartiles are equivalent to ArrayReduce[Quartiles,data,1]. »
  • Quartiles[data] is equivalent to Quantile[data,{1,2,3}/4,{{1/2,0},{0,1}}]. »
  • Quartiles[data,{{a,b},{c,d}}] is equivalent to Quantile[data,{1,2,3}/4,{{a,b},{c,d}}].
  • Common choices of parameters {{a,b},{c,d}} include:
  • {{0,0},{1,0}}inverse empirical CDF
    {{0,0},{0,1}}linear interpolation (California method)
    {{1/2,0},{0,0}}element numbered closest to p n
    {{1/2,0},{0,1}}linear interpolation (hydrologist method; default)
    {{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
  • The default choice of parameters is {{1/2,0},{0,1}}. »
  • The data can have the following additional forms and interpretations:
  • Associationthe values (the keys are ignored) »
    SparseArrayas an array, equivalent to Normal[data] »
    QuantityArrayquantities as an array »
    WeightedDatabased on the underlying EmpiricalDistribution »
    EventDatabased on the underlying SurvivalDistribution »
    TimeSeries, TemporalData, …vector or array of values (the time stamps ignored) »
    Image,Image3DRGB channel's values or grayscale intensity value »
    Audioamplitude values of all channels »
    DateObject, TimeObjectlist of dates or list of times »
  • Quartiles[dist] gives the list corresponding to Quantile[dist]. »
  • For a random process proc, the quartiles function can be computed for slice distribution at time t, SliceDistribution[proc,t], as Quartiles[SliceDistribution[proc,t]]. »

Examples

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Basic Examples  (3)

Quartiles for a list of exact numbers:

Quartiles of a list of dates:

Quartiles of a parametric distribution:

Scope  (22)

Basic Uses  (8)

Exact input yields exact output:

Approximate input yields approximate output:

Compute results using other parametrizations:

Find the quartiles of WeightedData:

Find the quartiles of EventData:

Find the quartiles of TemporalData:

Find the quartiles of TimeSeries:

The quartiles depend only on the values:

Find the quartiles for data involving quantities:

Array Data  (5)

Quartiles for a matrix gives columnwise quartiles:

Quartiles for a tensor gives columnwise medians at the first level:

Works with large arrays:

When the input is an Association, Quartiles works on its values:

SparseArray data can be used just like dense arrays:

Find quartiles of a QuantityArray:

Image and Audio Data  (2)

Channelwise quartile values of an RGB image:

Quartile intensity values of a grayscale image:

Quartile amplitudes of all channels:

Date and Time  (4)

Compute quartiles of dates:

Compute the weighted quartiles of dates:

Compute the quartiles of dates given in different calendars:

The mean is given in one of the input calendars:

Compute the quartiles of times:

List of times with different time zone specifications:

Distributions and Processes  (3)

Find the quartiles for a parametric distribution:

Quartiles for a derived distribution:

Data distribution:

Quartile functions for a random process:

Applications  (4)

Quartiles divide a distribution in four equal probability sections:

Find a moving quartile envelope for a time series:

Data smoothed by moving median:

Moving envelope of first and third quartiles:

Find the quartiles for data representing the top oil-producing fields in 2001:

Compare with the minimum and maximum values for the data:

Plot the data with quartile lines:

Compute the quartiles for the heights of children in a class:

Properties & Relations  (6)

Quartiles are given by linearly interpolated Quantile values:

The default parameters for Quantile give a different result:

The second quartile of the data is the Median:

The quantile of 1/2 does not average the two middle elements for lists of even length:

InterquartileRange is the difference between the first and third quartiles:

QuartileDeviation is half the difference between the first and third quartiles:

QuartileSkewness is a skewness measure obtained from the quartiles:

BoxWhiskerChart shows the quartiles for data:

Possible Issues  (2)

Quartiles requires numeric values in data:

The symbolic closed form may exist for some distributions:

Quartiles of data computed via Quantile do not always agree with Quartiles:

Specify linear interpolation parameters in Quantile:

Wolfram Research (2007), Quartiles, Wolfram Language function, https://reference.wolfram.com/language/ref/Quartiles.html (updated 2024).

Text

Wolfram Research (2007), Quartiles, Wolfram Language function, https://reference.wolfram.com/language/ref/Quartiles.html (updated 2024).

CMS

Wolfram Language. 2007. "Quartiles." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Quartiles.html.

APA

Wolfram Language. (2007). Quartiles. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Quartiles.html

BibTeX

@misc{reference.wolfram_2025_quartiles, author="Wolfram Research", title="{Quartiles}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Quartiles.html}", note=[Accessed: 30-October-2025]}

BibLaTeX

@online{reference.wolfram_2025_quartiles, organization={Wolfram Research}, title={Quartiles}, year={2024}, url={https://reference.wolfram.com/language/ref/Quartiles.html}, note=[Accessed: 30-October-2025]}