In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of quantitative data. Robust measures of scale are used to complement or replace conventional estimates of scale such as the sample variance or sample standard deviation. As with other robust statistics, a robust measure of scale is minimally affected by a small fraction of outliers, at the cost of lower statistical efficiency when outliers are not present.
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The most familiar robust measures of scale are the interquartile range (IQR) and the median absolute deviation (MAD). The IQR is the difference between the 75th percentile and the 25th percentile of a sample. The interdecile range is a robust measure of scale that is closely related to the IQR. The MAD is the median of the absolute values of the differences between the data values and the overall median of the data set.
Rousseeuw and Croux^{[1]} propose alternatives to the MAD, motivated by two weaknesses of it:
They propose two alternatives statistics based on pairwise differences: S_{n} and Q_{n}, defined as:
These can be computed in O(n log n) time and O(n) space.
Neither of these requires location estimation, as they are based only on differences between values. They are both more efficient than the MAD under a Gaussian distribution: S_{n} is 58% efficient, while Q_{n} is 82% efficient.
For a large normal sample, 2.2219Q_{n} is approximately unbiased for the population standard deviation. For small or moderate samples, the expected value of Q_{n} under a normal distribution depends markedly on the sample size, so finite sample correction factors obtained from a table or from simulations are used to calibrate the scale of Q_{n}.
Like S_{n} and Q_{n}, the biweight midvariance aims to be robust without sacrificing too much efficiency. It is defined as
where I is the indicator function, Q is the sample median of the X_{i}, and
The estimator is robust since data points are downweighted as their distance from the median increases, with points more than 9 MAD units from the median having no influence at all.
In some cases, robust estimators of scale are used to estimate the population variance or population standard deviation. For example, the IQR is sometimes defined as the difference between the 75th and 25th percentiles divided by 1.349, so that it becomes unbiased for the population variance if the data follow a normal distribution.
In other situations, it makes more sense to think of a robust measure of scale as an estimator of its own expected value, interpreted as an alternative to the population variance or standard deviation as a measure of scale. 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.
Mizera & Müller (2004) propose a robust depthbased estimator for location and scale simultaneously.^{[2]}
