Dispersion

Measurements of central tendency (mean, mode and median) locate the distribution within the range of possible values, measurements of dispersion describe the spread of values.

Variance

These are the most familiar measurements of dispersion.   Variance is the arithmetic mean (average) of the square of the difference between the value of an observation and the arithmetic mean of the value of all observations.  It is also referred to as  the second moment about the mean.  The formal definition of variance being:

For computation purposes, the formula can be used in the form shown below which allows the variance to be derived without first calculating the mean:

Standard Deviation

Standard deviation is the square root of the variance:

Normalized Standard Deviation

It is often useful to express the difference between the mean and a given value in units of standard deviation.

The normalized standard deviation is often referred to as z.  Probability tables for the normal distribution are usually based on z.

Mean Absolute Deviation (MAD)

A weakness of standard deviation as a measure of dispersion is its sensitivity to anomalous values which are a feature of real life data.  This is a result of the square of the difference between a value and the mean, this conveniently gets rid of negative values, but at the expense of increasing the significance of extreme ones.  An alternative is based the absolute value of the difference between a given value and the mean:

The downside is that the use of absolute values makes the analytical treatment of functions difficult, but this is a small price to pay for such an acronym.

In situations where the median is a more stable measure of central tendency, it is used in place of the mean.

The example below compares the standard deviation and the MAD for a small sample which contains an anomalous extreme value.  The measures of central tendency for the sample are:

Mean	1.7
Median	1.5

1.2	0.5	0.3	0.25
1.4	0.3	0.1	0.09
1.5	0.2	0.0	0.04
1.6	0.1	0.1	0.01
2.8	1.1	1.3	1.21
Totals	2.2	1.8	1.60

Mean Absolute Deviation	0.44
Median Absolute Deviation	0.36
Standard Deviation	0.57

The MAD statistics are less sensitive to extreme anomalous values, however, it is important to use the statistic which is best suited for a given analysis.

Page updated: 30-Mar-2006

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