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: 01-June-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: 01-June-2023 ]}