
SN SCALEName:
Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency. The most common estimate of scale, the standard deviation, is the most efficient estimate of scale if the data come from a normal distribution. However, the standard deviation is not robust in the sense that changing even one value can dramatically change the computed value of the standard deviation (i.e., poor resistance). In addition, it does not have robustness of efficiency for nonnormal data. The median absolute deviation (MAD) and interquartile range are the two most commonly used robust alternatives to the standard deviation. The MAD in particular is a very robust scale estimator. However, the MAD has the following limitations:
Rousseeuw and Croux proposed the S_{n} estimate of scale as an alternative to the MAD. It shares desirable robustness properties with MAD (50% breakdown point, bounded influence function). In addition, it has significantly better normal efficiency (58%) and it does not depend on symmetry. The S_{n} scale estimate is defined as:
That is, for each i we compute the median of {x_{i}  x_{j} j = 1, ..., n}. The median of these n numbers is then the estimate of S_{n}. The constant c is determined to make S_{n} a consistent estimator. The value used is 1.1926 (this is the value needed to make S_{n} a consistent estimator for normal data). The S_{n} statistic measures typical distances between values in contrast to the MAD and the standard deviation which measure the distance from a central location. This is why the Sn is appropriate for asymmetic distributions as well symmetric distributions. The Rousseeuw and Croux article (see the Reference section below) discusses the properties of the S_{n} estimate in detail.
where <y> is the response variable; <par> is a parameter where the computed Sn scale statistic is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = SN SCALE Y1 SUBSET TAG > 2
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics", , AddisonWesley, pp. 203209.
MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 100 MULTIPLOT SCALE FACTOR 2 X1LABEL DISPLACEMENT 12 . LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 200 LET SIGMA = 1 LET Y2 = LOGNORMAL RANDOM NUMBERS FOR I = 1 1 200 . BOOTSTRAP SAMPLES 500 BOOTSTRAP SN SCALE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT X1LABEL . BOOTSTRAP SN SCALE PLOT Y2 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT . END OF MULTIPLOT JUSTIFICATION CENTER MOVE 50 96 TEXT SN SCALE BOOTSTRAP: NORMAL MOVE 50 46 TEXT SN SCALE BOOTSTRAP: LOGNORMAL
 
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Date created: 05/05/2003 