Robust measures of scale

Qn estimatorrobust estimator of dispersionrobust measure of scale
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.wikipedia
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Statistical dispersion

dispersionvariabilityspread
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.
These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale. Robust measures of scale are those unaffected by a small number of outliers, and include the IQR and MAD.

Interquartile range

inter-quartile rangebelowinterquartile
The most common such statistics are the interquartile range (IQR) and the median absolute deviation (MAD). One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.
It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

Robust statistics

robustbreakdown pointrobustness
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.
The plots below show the bootstrap distributions of the standard deviation, median absolute deviation (MAD) and [[Robust measures of scale#Robust measures of scale based on absolute pairwise differences|Qn estimator]] of scale.

Median absolute deviation

MAD
The most common such statistics are the interquartile range (IQR) and the median absolute deviation (MAD). Another familiar robust measure of scale is the median absolute deviation (MAD), the median of the absolute values of the differences between the data values and the overall median of the data set; for a Gaussian distribution, MAD is related to \sigma as (the derivation can be found here).
Robust measures of scale

L-estimator

L-estimation
One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.
Robust L-estimators used to measure dispersion, such as the IQR, provide robust measures of scale.

Standard deviation

standard deviationssample standard deviationsigma
These are contrasted with conventional measures of scale, such as sample variance or sample standard deviation, which are non-robust, meaning greatly influenced by outliers. For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.
Robust standard deviation

Trimmed estimator

trimmedtrimming
One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.
When estimating a scale parameter, using a trimmed estimator as a robust measures of scale, such as to estimate the population variance or population standard deviation, one generally must multiply by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Interdecile range

Other trimmed ranges, such as the interdecile range (10% trimmed range) can also be used.
Robust measures of scale

Statistics

statisticalstatistical analysisstatistician
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.

Level of measurement

quantitativescaleinterval scale
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.

Data

statistical datascientific datadatum
In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data.

Outlier

outliersconservative estimateirregularities
These are contrasted with conventional measures of scale, such as sample variance or sample standard deviation, which are non-robust, meaning greatly influenced by outliers.

Estimator

estimatorsestimateestimates
These robust statistics are particularly used as estimators of a scale parameter, and have the advantages of both robustness and superior efficiency on contaminated data, at the cost of inferior efficiency on clean data from distributions such as the normal distribution. Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.

Scale parameter

scalerate parameterestimation
These robust statistics are particularly used as estimators of a scale parameter, and have the advantages of both robustness and superior efficiency on contaminated data, at the cost of inferior efficiency on clean data from distributions such as the normal distribution. For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Percentile

percentiles50th percentile85th percentile speed
One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.

Range (statistics)

rangerangingsample range
One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.

Median

averagesample medianmedian-unbiased estimator
Another familiar robust measure of scale is the median absolute deviation (MAD), the median of the absolute values of the differences between the data values and the overall median of the data set; for a Gaussian distribution, MAD is related to \sigma as (the derivation can be found here).

Estimation theory

parameter estimationestimationestimated
Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.

Expected value

expectationexpectedmean
Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.

Variance

sample variancepopulation variancevariability
These are contrasted with conventional measures of scale, such as sample variance or sample standard deviation, which are non-robust, meaning greatly influenced by outliers. For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Scale factor

scalescaling factorscaled
For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Bias of an estimator

unbiasedunbiased estimatorbias
For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Consistent estimator

consistentconsistencyinconsistent
For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

Cauchy distribution

LorentzianCauchyLorentzian profile
For example, the MAD of a sample from a standard Cauchy distribution is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.

Efficiency (statistics)

efficientefficiencyinefficient
These robust estimators typically have inferior statistical efficiency compared to conventional estimators for data drawn from a distribution without outliers (such as a normal distribution), but have superior efficiency for data drawn from a mixture distribution or from a heavy-tailed distribution, for which non-robust measures such as the standard deviation should not be used.