1.3.EDA Techniques

1.3.5.Quantitative Techniques

## 1.3.5.6. | ## 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:

- How spread out are the data values near the center?
- How spread out are the tails?

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

*Definitions of Variability*For univariate data, there are several common numerical measures ofthe spread:

- variance - the variance is defined as
- \( s^{2} = \sum_{i=1}^{N}(Y_{i} - \bar{Y})^{2}/(N - 1) \)

where \(\bar{Y}\) is the mean of the data.

The variance is roughly the arithmetic average of the squared distance from the mean. Squaring the distance from the mean has the effect of giving greater weight to values that are further from the mean. For example, a point 2 units from the mean adds 4 to the above sum while a point 10 units from the mean adds 100 to the sum. Although the variance is intended to be an overall measure of spread, it can be greatly affected by the tail behavior.

- standard deviation - the standard deviation is the square root of the variance. That is,
- \( s = \sqrt{\sum_{i=1}^{N}(Y_{i} - \bar{Y})^{2}/(N - 1)} \)

The standard deviation restores the units of the spread to the original data units (the variance squares the units).

- range - the range is the largest value minus the smallest value in a data set. Note that this measure is based only on the lowest and highest extreme values in the sample. The spread near the center of the data is not captured at all.
- average absolute deviation - the average absolute deviation (AAD) is defined as
- \( AAD = \sum_{i=1}^{N}(|Y_{i} - \bar{Y}|)/N \)

where \(\bar{Y}\) is the mean of the data and

is the absolute value of*|Y|*. This measure does not square the distance from the mean, so it is less affected by extreme observations than are the variance and standard deviation.*Y* - median absolute deviation - the median absolute deviation (MAD) is defined as
- \( MAD = median (|Y_{i} - \tilde{Y}|) \)

where \(\tilde{Y}\) is the median of the data and

is the absolute value of*|Y|*. This is a variation of the average absolute deviation that is even less affected by extremes in the tail because the data in the tails have less influence on the calculation of the median than they do on the mean.*Y* - interquartile range - this is the value of the 75th percentile minus the value of the 25th percentile. This measure of scale attempts to measure the variability of points near the center.

*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 Distribution*The 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 Distribution*The 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 Distribution*The 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 Distribution*The fourth histogram is a sample from aTukey lambda distribution withshape parameter

*λ*= 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.

- 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.
- 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 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.