# WinsorizedVariance

WinsorizedVariance[list,f]

gives the variance of the elements in list after replacing the fraction f of the smallest and largest elements by the remaining extreme values.

WinsorizedVariance[list,{f1,f2}]

gives the variance when the fraction f1 of the smallest elements and the fraction f2 of the largest elements are replaced by the remaining extreme values.

WinsorizedVariance[list]

gives the 5% winsorized variance WinsorizedVariance[list,0.05].

WinsorizedVariance[dist,]

gives the winsorized variance of a univariate distribution dist.

# Details • WinsorizedVariance gives a robust estimate of the variance, as more extreme values are replaced by less extreme ones.
• The winsorizing fraction is determined by the parameters f1 and f2, which indicate the fraction f1 of the smallest elements and the fraction f2 of the largest elements to be replaced by the remaining extreme values.
• WinsorizedVariance[list,{f1,f2}] gives the variance of Clip[list,{z1,z2}] where z1 equals RankedMin[list,1+ ], z2 equals RankedMax[list,1+ ], and n equals the length of list.
• • WinsorizedVariance[{{x1,y1,},{x2,y2,},},f] gives {WinsorizedVariance[{x1,x2,},f],WinsorizedVariance[{y1,y2,},f],}.
• WinsorizedVariance[dist,{f1,f2}] gives Variance[CensoredDistribution[Quantile[dist,{f1,1-f2}],dist]] for a univariate distribution dist.

# Examples

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## Basic Examples(3)

Winsorized variance after removing extreme values:

Winsorized variance after removing the smallest extreme values:

Winsorized variance of a symbolic distribution:

## Scope(8)

### Data(7)

Exact input yields exact output:

Approximate input yields approximate output:

Winsorized variance of a matrix gives column-wise variances:

Winsorized variance of a large array:

SparseArray data can be used just like dense arrays:

Winsorized variance of a TimeSeries:

The winsorized variance depends only on the values:

Winsorized variance works with data involving quantities:

### Distributions(1)

Winsorized variance of a univariate distribution:

## Applications(2)

Obtain a robust estimate of location when outliers are present:

Extreme values have a large influence on the variance:

Find a winsorized variance for the heights of children in a class:

The 5% winsorized mean:

Plot the winsorized variance as a function of the fraction parameter:

Plot the square root of the winsorized variance with respect to the winsorized mean:

## Properties & Relations(5)

A 0% WinsorizedVariance is equivalent to Variance:

WinsorizedVariance approaches 0 as f approaches 1/2:

WinsorizedVariance of a distribution is the variance of its CensoredDistribution:

Variance of the CensoredDistribution with appropriate bounds:

WinsorizedVariance of a sample gives an estimate of the variance of a censored distribution:

Variance of the CensoredDistribution with appropriate bounds:

TrimmedVariance drops the data beyond a certain quantile level, then computes the sample mean:

WinsorizedVariance clips the data beyond a certain quantile level, then computes the sample mean:

Plot the sorted data against the sample with elements removed and the clipped sample:

## Possible Issues(1)

WinsorizedVariance requires numeric values: 