## Measures Of Dispersion

In addition to knowing the "typical" value for a given sample of data, it is important to know the degree to which individual observations vary around this level. Are the data tightly clustered around the typical value, or are the data widely dispersed? If the data is tightly clustered about the typical level, then measures of central tendency provide a close approximation to individual values drawn from the sample. If the data are widely dispersed around typical values, then measures of central tendency offer only a poor approximation to individual values that might be drawn from the sample. As in the case of measures of central tendency, statisticians have constructed several useful measures of such dispersion. In general, measures of dispersion describe variation in the data in terms of the distance between selected observations or in terms of the average deviation among sample observations. Managers often focus on the range, variance and standard deviation, and coefficient of variation. Which among these is most appropriate for a given task depends on the nature of the underlying data and the need being addressed by the manager.

smallest observations

Range range The simplest and most commonly employed measure of dispersion is the sample range, or the scope from largest to difference between the largest and smallest sample observations. In the telecommunications services example, the sample range in net profit is defined by the \$7.6 million earned in the most profitable sample market to the \$2.9 million earned in the least profitable sample observation. Note the very high degree of dispersion in net profits over the sample. The highest level of firm profits earned is more than two and one-half times, or 150 percent, greater than the lowest profit level. The range in net profit margin, though substantial, is much lower because these data are implicitly size-adjusted. The 16.4 percent earned in the market with the highest net profit margin is only 34 percent greater than the 12.2 percent margin earned in the market with the lowest profit margin. Profit variation is much less when one explicitly controls for firm size differences. As might be expected, the range in market size as measured by sales revenue is substantial. The \$49.7 million in sales revenue earned in the largest market is roughly 150 percent greater than the \$20.3 million size of the smallest market in the sample.

Range has intuitive appeal as a measure of dispersion because it identifies the distance between the largest and smallest sample observations. Range can be used to identify likely values that might be associated with "best case" and "worst case" scenarios. Although range is a popular measure of variability that is easy to compute, it has the unfortunate characteristic of ignoring all but the two most extreme observations. As such, the range measure of dispersion can be unduly influenced by highly unusual outlying observations. The effects of outlyers are sometimes minimized by relying on interquartile or percentile range measures. For example, the interquartile range identifies the spread that bounds the middle 50th percent of sample observations by measuring the distance between the first and third quartiles. Similarly, by measuring the distance between the 90th and 10th percentile of sample observations, the bounds on the middle 80 percent of sample observations can be determined. Both interquartile and percentile range measures are attractive because they retain the ease of calculation and intuitive appeal of the range measure of dispersion. However, like any range measure, they do not provide detailed information on the degree of variation among all sample observations. For this reason, range measures are often considered in conjunction with measures of dispersion that reflect the average deviation among all sample observations.

Variance and Standard Deviation population variance

### Average squared deviation from the overall

Despite their ease of calculation and intuitive interpretation, the usefulness of range measures of dispersion is limited by the fact that only two data points, the high and low observations, are reflected. For this reason, range measures of dispersion are often supplemented by measures that reflect dispersion through the sample or entire population. A measure of dispersion throughout the population is given by the population variance, or the arithmetic mean of the squared deviation of each observation from the overall mean. The squared deviation of each observation from the overall mean is considered in order to give equal weight to upside as well as downside variation within the population. Without this squaring process, positive and negative deviations would tend to cancel and result in an understatement of the degree of overall variability. Population variance is calculated using the following expression:

mean

2 = (X! - ^)2 + (X2 - ^)2 + ■ ■ ■ + (XN - U)2