# 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 matrix data, the quartile is computed for each column vector with Quartiles[{{x1,y1,},{x2,y2,},}] equivalent to {Quartiles[{x1,x2,}],Quartiles[{y1,y2,}]}. »
• For array 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 (default) {{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) {{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 {{0,0},{1,0}}.
• The data can have the following additional forms and interpretations:
•  Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channel's values or grayscale intensity value » Audio amplitude values of all channels »
• 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(2)

Quartiles for a list of exact numbers:

Quartiles of a parametric distribution:

## Scope(18)

### 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:

### 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 2023).

#### Text

Wolfram Research (2007), Quartiles, Wolfram Language function, https://reference.wolfram.com/language/ref/Quartiles.html (updated 2023).

#### CMS

Wolfram Language. 2007. "Quartiles." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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_2023_quartiles, author="Wolfram Research", title="{Quartiles}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Quartiles.html}", note=[Accessed: 27-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_quartiles, organization={Wolfram Research}, title={Quartiles}, year={2023}, url={https://reference.wolfram.com/language/ref/Quartiles.html}, note=[Accessed: 27-September-2023 ]}