## Robust Measures of Scale

The sample standard deviation is a commonly used estimator of
the population scale. However, it is sensitive to outliers
and may not remain bounded
when a single data point is replaced by an arbitrary number.
With robust scale estimators, the estimates remain bounded even
when a portion of the data points are replaced by arbitrary numbers.
A simple robust scale estimator is the interquartile range,
which is the difference between the upper and lower quartiles.
For a normal population, the standard deviation can
be estimated by dividing the interquartile range by 1.34898.

Gini's mean difference is also a robust estimator of
the standard deviation .It is computed as

If the observations are from a normal distribution,
then is an unbiased estimator of
the standard deviation .

A very robust scale estimator is
the median absolute deviation (*MAD*) about the median (Hampel 1974).

where the inner median, *med*_{j} (*y*_{j}), is the median
of the *n* observations and the outer median, *med*_{i},
is the median of the *n* absolute values of the deviations about
the median.
For a normal distribution, 1.4826 *MAD* can be used to estimate the
standard deviation .

The *MAD* statistic has low efficiency for normal distributions
and it may not be appropriate for symmetric distributions.
Rousseeuw and Croux (1993) proposed two new statistics
as alternatives to the *MAD* statistic, *S*_{n} and
*Q*_{n}.

where the outer median, *med*_{i}, is the median of the
*n* medians of

To reduce small-sample bias, *c*_{sn}*S*_{n} is used to
estimate the standard deviation , where
*c*_{sn} is a correction factor (Croux and Rousseeuw 1992).

The second statistic is computed as

where , *h* = [*n*/2]+1 and
[*n*/2] is the integer part of *n*/2.
That is, *Q*_{n} is 2.2219 times the *k*th order statistic
of the distances between data points.
The bias-corrected statistic *c*_{qn}*Q*_{n} is used to
estimate the standard deviation , where *c*_{qn}is the correction factor.

A **Robust Measures of Scale** table includes
the interquartile range, Gini's mean difference,
*MAD*, *S*_{n}, and *Q*_{n},
with their corresponding estimates of ,as shown in Figure 38.14.

**Figure 38.14:** Robust Measures of Scale and Tests for Normality

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.