BiweightMidvariance
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BiweightMidvariance

Details

- BiweightMidvariance is a robust dispersion estimator.
- BiweightMidvariance is given by a weighted second-order central moment with Median as its center. Elements farther from the center have lower weights.
- The width scale of the weight function is controlled by a parameter c. Larger c indicates more data values are included in the computation of the statistic, and vice versa.
- For the list {x1,x2,…,xn}, the value of the biweight midvariance estimator is given by
, where
,
is Median[{x1,x2,…,xn}], and
is MedianDeviation[{x1,x2,…,xn}].
- BiweightMidvariance[list] is equivalent to BiweightMidvariance[list,9].
- BiweightMidvariance[{{x1,y1,…},{x2,y2,…},…}] gives {BiweightMidvariance[{x1,x2,…}],BiweightMidvariance[{y1,y2,…}],…}.
- BiweightMidvariance allows c to be any positive real number.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
BiweightMidvariance of a list:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-wd9

BiweightMidvariance of columns of a matrix:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-vu6sr

BiweightMidvariance of a list with scaling factor 8:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-whtpl

BiweightMidvariance of a list of dates:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ziof1v

Scope (9)Survey of the scope of standard use cases
Exact input yields exact output:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ug7y2

Approximate input yields approximate output:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-cg1nsz


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-dvod55

Biweight midvariance with different scaling parameters:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-fgu628


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-bdt2rg

Biweight midvariance for a matrix gives columnwise estimate:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-jywoa6

Biweight midvariance of a large array:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-enve04


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-if5yx4

Find a biweight midvariance of a TimeSeries:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-tg8p6z

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ffhpdi

The biweight midvariance depends only on the values:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-fy9fte

Biweight midvariance works with data involving quantities:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-jopin9


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-e8c21s

Compute biweight midvariance of dates:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-b1smxx

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-pa4nmn


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-uok1il


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-o9ersi

Compute biweight midvariance of times:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-et9bla


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ztsexm

List of times with different time zone specifications:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-mrqghz


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ow7hca

Applications (5)Sample problems that can be solved with this function
Obtain a robust estimate of dispersion when extreme values are present:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-b99oaf

Sample variance is heavily influenced by extreme values:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-3fz1a

Identify periods of high volatility in stock data:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-nj16d1

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-qnjs45

Smooth the data using the square root of a five-year moving biweight midvariance:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-kfgcti

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-bef0x

Compute biweight midvariance for slices of a collection of paths of a random process:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-8se1zg

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-52xxug

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-iakfqb

Plot biweight midvariances over these paths:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-tvmkqe

Find the biweight midvariance of the heights for the children in a class:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-cevfij

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-fllmtw


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-celepo

Plot the square root of the biweight midvariance with respect to the median:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-g98mgx

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-mm15h9

Consider data from standard normal distribution with outliers modeled by another normal distribution with large spread:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-hby4vw

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-0arthm

Test the data against standard normal distribution:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-7wya5

Compute the biweight midvariance:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-pszm77

Remove outliers by picking data points that are within three times the square root of biweight midvariance from the sample median:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-bf1y7q

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ehmy0c

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-xr5fir

Test the new data against standard normal distribution:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-i4ekyf

Properties & Relations (3)Properties of the function, and connections to other functions
Compute the biweight midvariance of a sample:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-dcvsx3

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ctetjg

Values outside of the interval have no effect on the statistic. Here
is the sample median and
is the median absolute deviation.
is a scaling parameter with default value equal to 9:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-fi8qr

The shape of the weight function w(x) being used in computing biweight midvariance:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-d006dh

Multiply the smallest and the largest values in the sample by 2 and compute the biweight midvariance again:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-jyiz5y

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-eup6o0


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-dgbfj

BiweightMidvariance and Variance are dispersion estimators of data:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-by15b

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-cp1as4

Resample the data to generate bootstrap estimates:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-sep9r
Compute the standard deviation/mean ratio of the bootstrap estimates for each estimator; smaller value indicates more accurate dispersion measure:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-c7rozz


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-4rrmo

BiweightMidvariance converges to the second central moment for large values of the parameter c:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-dy2d21

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-r55ck1


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-ed6bml

Possible Issues (1)Common pitfalls and unexpected behavior
Biweight midvariance may be undefined for vectors of an even number of elements with a small scaling parameter:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-sz8yw2

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-im2iz

For vectors of odd length and relatively small c, biweight midvariance might assume very large values:

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-3ny0o1

https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-snkb8d


https://wolfram.com/xid/0jvqhof73zrnjn1ve7oe-3ya2s4

Wolfram Research (2017), BiweightMidvariance, Wolfram Language function, https://reference.wolfram.com/language/ref/BiweightMidvariance.html (updated 2024).
Text
Wolfram Research (2017), BiweightMidvariance, Wolfram Language function, https://reference.wolfram.com/language/ref/BiweightMidvariance.html (updated 2024).
Wolfram Research (2017), BiweightMidvariance, Wolfram Language function, https://reference.wolfram.com/language/ref/BiweightMidvariance.html (updated 2024).
CMS
Wolfram Language. 2017. "BiweightMidvariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/BiweightMidvariance.html.
Wolfram Language. 2017. "BiweightMidvariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/BiweightMidvariance.html.
APA
Wolfram Language. (2017). BiweightMidvariance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BiweightMidvariance.html
Wolfram Language. (2017). BiweightMidvariance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BiweightMidvariance.html
BibTeX
@misc{reference.wolfram_2025_biweightmidvariance, author="Wolfram Research", title="{BiweightMidvariance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/BiweightMidvariance.html}", note=[Accessed: 29-April-2025
]}
BibLaTeX
@online{reference.wolfram_2025_biweightmidvariance, organization={Wolfram Research}, title={BiweightMidvariance}, year={2024}, url={https://reference.wolfram.com/language/ref/BiweightMidvariance.html}, note=[Accessed: 29-April-2025
]}