# Skewness Skewness[data]

gives the coefficient of skewness estimate for the elements in data.

Skewness[dist]

gives the coefficient of skewness for the distribution dist.

# Details  • Skewness measures the asymmetry in data or of dist.
• Skewness[] is equivalent to CentralMoment[,3]/CentralMoment[,2]3/2.
• A positive skewness indicates a distribution with a long right tail. A negative skewness indicates a distribution with a long left tail.
• • Skewness[{{x1,y1,},{x2,y2,},}] gives {Skewness[{x1,x2,}],Skewness[{y1,y2,}],}.
• Skewness handles both numerical and symbolic data.
• 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 weighted mean, 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 »
• For a random process proc, the skewness function can be computed for slice distribution at time t, SliceDistribution[proc,t], as α[t]=Skewness[SliceDistribution[proc,t]]. »
• # Examples

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## Basic Examples(2)

Skewness for a list of values:

Skewness for symbolic data:

Skewness for a parametric distribution:

## Scope(18)

### Basic Uses(7)

Exact input yields exact output:

Approximate input yields approximate output:

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:

### Array Data(5)

Skewness for a matrix gives columnwise skewness:

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find the skewness of a QuantityArray:

### Image and Audio Data(2)

Channelwise skewness value of an RGB image:

Skewness intensity value of a grayscale image:

On audio objects, Skewness works channelwise:

### 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:

## Possible Issues(1)

Skewness may be undefined for data:

Skewness may be undefined for a distribution:

## Neat Examples(1)

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