# Skewness

Skewness[list]

gives the coefficient of skewness for the elements in list.

Skewness[dist]

gives the coefficient of skewness for the distribution dist.

# Details

• Skewness measures the asymmetry in list or of dist.
• A positive skewness indicates a distribution with a long right tail. A negative skewness indicates a distribution with a long left tail.
• Skewness handles both numerical and symbolic data.
• Skewness[{{x1,y1,},{x2,y2,},}] gives {Skewness[{x1,x2,}],Skewness[{y1,y2,}],}.
• Skewness[] is equivalent to CentralMoment[,3]/CentralMoment[,2]3/2.

# Examples

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## Basic Examples(2)

Skewness for a list of values:

Skewness for a parametric distribution:

## Scope(14)

### Data(10)

Exact input yields exact output:

Approximate input yields approximate output:

Skewness for a matrix gives column-wise skewness:

Works with large arrays:

SparseArray data can be used just like dense arrays:

Find the skewness of WeightedData:

Find the skewness of EventData:

Find the skewness of TemporalData:

Find the skewness of TimeSeries:

The skewness depends only on the values:

Find the skewness of data involving quantities:

### Distributions and Processes(4)

Find the skewness for univariate distributions:

Multivariate distributions:

Skewness for derived distributions:

Data distribution:

Skewness for distributions with quantities:

Skewness function for a random process:

## Applications(8)

Zero skewness indicates that the distribution is symmetric:

Distributions with longer tails to the right have positive skewness:

Distributions with longer tails to the left have negative skewness:

The limiting distribution for BinomialDistribution as is normal:

The limiting value of skewness is 0:

By the central limit theorem, skewness of normalized sums of random variables will converge to 0:

Define a Pearson distribution with zero mean and unit variance, parameterized by skewness and kurtosis:

Obtain parameter inequalities for Pearson types 1, 4, and 6:

The region plot for Pearson types depending on the values of skewness and kurtosis:

Generate a random sample from a ParetoDistribution:

Determine the type of PearsonDistribution with moments matching the sample moments:

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

Average number of steps:

Analyze the skewness as an indication of a tail in the daily step distribution:

The histogram of the frequency of daily counts confirms that the distribution has a longer left tail:

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

Skewness close to 0 indicates distribution symmetric around the mean:

## Properties & Relations(2)

Skewness for data can be computed from CentralMoment:

Skewness for a distribution can be computed from CentralMoment:

## Neat Examples(1)

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

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

#### Text

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

#### CMS

Wolfram Language. 2007. "Skewness." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Skewness.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_skewness, author="Wolfram Research", title="{Skewness}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Skewness.html}", note=[Accessed: 28-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_skewness, organization={Wolfram Research}, title={Skewness}, year={2007}, url={https://reference.wolfram.com/language/ref/Skewness.html}, note=[Accessed: 28-March-2023 ]}