# QuartileSkewness

QuartileSkewness[data]

gives the coefficient of quartile skewness for the elements in list.

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

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

QuartileSkewness[dist]

gives the coefficient of quartile skewness for the distribution dist.

# Details

• QuartileSkewness[data] is given by , where is given by Quartiles[data].
• A positive value of quartile skewness indicates the median is closer to the lower quartile than the upper quartile .
• A negative value of quartile skewness indicates the median is closer to the upper quartile .
• QuartileSkewness[data,{{a,b},{c,d}}] uses computed as Quartiles[data, {{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:
•  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 channels values or grayscale intensity value » Audio amplitude values of all channels »

# Examples

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## Basic Examples(2)

Quartile skewness for a list of exact numbers:

Quartile skewness 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 quartile skewness for WeightedData:

Find the quartile skewness for EventData:

Find the quartile skewness for TemporalData:

Find the quartile skewness of TimeSeries:

The quartile skewness depends only on the values:

Find the quartile skewness for data involving quantities:

### Array Data(5)

QuartileSkewness for a matrix gives columnwise ranges:

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

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find quartile skewness of a QuantityArray:

### Image and Audio Data(2)

Channel-wise quartile skewness value of an RGB image:

Quartile skewness intensity value of a grayscale image:

Quartile skewness amplitude of all amplitude values of all channels:

### Distributions and Processes(3)

Find the quartile skewness for a parametric distribution:

Quartile skewness for a derived distribution:

Data distribution:

Quartile skewness for a time slice of a random process:

## Applications(6)

Zero QuartileSkewness indicates the median is equally distant from the remaining quartiles:

Positive QuartileSkewness indicates that the median is closer to the lower quartile:

Negative QuartileSkewness indicates that the median is closer to the upper quartile:

Obtain a robust estimate of asymmetry when extreme values are present:

Measures based on the Mean are heavily influenced by extreme values:

This time series contains the number of steps taken daily by a person during a period of five months:

Median number of steps:

Analyze whether the step distribution is skewed toward the lower or the upper quartile:

The histogram of the frequency of daily counts shows that the median is closer to the upper quartile:

Find the quartile skewness for the heights of children in a class:

Negative quartile skewness indicates that the median is closer to the lower quartile:

## Properties & Relations(3)

QuartileSkewness is a function of linearly interpolated Quantile values:

QuartileSkewness is a function of quartiles:

QuartileSkewness is a function of the median, first quartile and a dispersion measure:

## Possible Issues(1)

QuartileSkewness requires numeric values:

## Neat Examples(1)

The distribution of QuartileSkewness estimates for 50, 100 and 300 samples:

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

#### Text

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

#### CMS

Wolfram Language. 2007. "QuartileSkewness." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/QuartileSkewness.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_quartileskewness, author="Wolfram Research", title="{QuartileSkewness}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/QuartileSkewness.html}", note=[Accessed: 01-March-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_quartileskewness, organization={Wolfram Research}, title={QuartileSkewness}, year={2023}, url={https://reference.wolfram.com/language/ref/QuartileSkewness.html}, note=[Accessed: 01-March-2024 ]}