# 1.3.5.6. Measures of Scale (2023)

1.Exploratory Data Analysis
1.3.EDA Techniques
1.3.5.Quantitative Techniques

## Measures of Scale

Scale, Variability, or Spread A fundamental task in many statistical analyses is to characterizethe spread, or variability, of a data set. Measures of scaleare simply attempts to estimate this variability.

When assessing the variability of a data set, there are two keycomponents:

1. How spread out are the data values near the center?
2. How spread out are the tails?
Different numerical summaries will give different weight to thesetwo elements. The choice of scale estimator is often drivenby which of these components you want to emphasize.

The histogram is an effective graphicaltechnique for showing both of these components of the spread.

Definitions of VariabilityFor univariate data, there are several common numerical measures ofthe spread:In summary, the variance, standard deviation, average absolutedeviation, and median absolute deviation measureboth aspects of the variability; that is, the variability nearthe center and the variability in the tails. They differ inthat the average absolute deviation and median absolute deviationdo not give undue weight to the tail behavior. On the otherhand, the range only uses the two most extreme points and theinterquartile range only uses the middle portion of the data.Why Different Measures?The following example helps to clarify why these alternativedefintions of spread are useful and necessary.

This plot shows histograms for 10,000 random numbers generated froma normal, a double exponential, a Cauchy, and a Tukey-Lambdadistribution. Normal DistributionThe first histogram is a sample from anormal distribution. The standard deviation is 0.997, the median absolute deviation is0.681, and the range is 7.87.

The normal distribution is a symmetric distribution withwell-behaved tails and a single peak at the center of the distribution.By symmetric, we mean that the distribution can be folded aboutan axis so that the two sides coincide. That is, it behaves thesame to the left and right of some center point. In this case,the median absolute deviation is a bit less than the standarddeviation due to the downweighting of the tails. The rangeof a little less than 8 indicates the extreme values fall withinabout 4 standard deviations of the mean.If a histogram or normal probability plot indicatesthat your data are approximated well by a normal distribution, thenit is reasonable to use the standard deviation as the spreadestimator.

Double Exponential DistributionThe second histogram is a sample from adouble exponential distribution.The standard deviation is 1.417, the median absolutedeviation is 0.706, and the range is 17.556.

Comparing the double exponential and the normal histogramsshows that the double exponential has a stronger peak atthe center, decays more rapidly near the center, and has much longertails. Due to the longer tails, the standard deviation tendsto be inflated compared to the normal. On the other hand, themedian absolute deviation is only slightly larger than it is forthe normal data. The longer tails are clearly reflected in the valueof the range, which shows that the extremes fall about 6 standarddeviations from the mean compared to about 4 for the normal data.

Cauchy DistributionThe third histogram is a sample from aCauchy distribution. Thestandard deviation is 998.389, the median absolute deviation is1.16, and the range is 118,953.6.

The Cauchy distribution is a symmetric distribution with heavytails and a single peak at the center of the distribution.The Cauchy distribution has the interesting property thatcollecting more data does not provide a more accurate estimatefor the mean or standard deviation. That is, the samplingdistribution of the means and standard deviationare equivalent to the sampling distribution of the original data.That means that for the Cauchy distribution the standard deviationis useless as a measure of the spread. From the histogram, it isclear that just about all the data are between about -5 and 5.However, a few very extreme values cause both the standarddeviation and range to be extremely large. However, the medianabsolute deviation is only slightly larger than it is for thenormal distribution. In this case, the median absolute deviationis clearly the better measure of spread.

Although the Cauchy distribution is an extreme case, it doesillustrate the importance of heavy tails in measuring the spread. Extreme values in the tails can distort the standarddeviation. However, these extreme values do not distort the medianabsolute deviation since the median absolute deviation is based onranks. In general, for data with extreme values inthe tails, the median absolute deviation or interquartile rangecan provide a more stable estimate of spread than the standarddeviation.

Tukey-Lambda DistributionThe fourth histogram is a sample from aTukey lambda distribution withshape parameterλ = 1.2. The standard deviation is 0.49, the medianabsolute deviation is 0.427, and the range is 1.666.

The Tukey lambda distribution has a range limited to(-1/λ,1/λ).That is, it has truncated tails. In this case the standard deviationand median absolute deviation have closer values than for theother three examples which have significant tails.

Robustness

Tukey and Mostellerdefined two types of robustness where robustness is alack of susceptibility to the effects of nonnormality.

1. Robustness of validity means that the confidence intervals for a measure of the population spread (e.g., the standard deviation) have a 95 % chance of covering the true value (i.e., the population value) of that measure of spread regardless of the underlying distribution.
2. Robustness of efficiency refers to high effectiveness in the face of non-normal tails. That is, confidence intervals for the measure of spread tend to be almost as narrow as the best that could be done if we knew the true shape of the distribution.
The standard deviation is an example of an estimator that is the bestwe can do if the underlying distribution is normal. However, it lacksrobustness of validity. That is, confidence intervals basedon the standard deviation tend to lack precision if the underlyingdistribution is in fact not normal.

The median absolute deviation and the interquartile range are estimatesof scale that have robustness of validity. However, they are notparticularly strong for robustness of efficiency.

If histograms and probability plots indicate that your dataare in fact reasonably approximated by a normal distribution,then it makes sense to use the standard deviation as the estimateof scale. However, if your data are not normal, and in particularif there are long tails, then using an alternative measure suchas the median absolute deviation, average absolute deviation, orinterquartile range makes sense. The range is used in someapplications, such as quality control, for its simplicity.In addition, comparing the range to the standard deviation gives anindication of the spread of the data in the tails.

Since the range is determined by the two most extreme points inthe data set, we should be cautious about its use for large valuesof N.

Tukey and Mostellergive a scale estimator that has both robustness of validity androbustness of efficiency. However, it is more complicatedand we do not give the formula here.

Software Most general purpose statistical software programs can generate at least some of the measures of scale discusssed above.
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