## Info

the sample, when sample markets are numbered in sequential order. Average net profit per market is \$5 million, the average profit margin is 14.8 percent, and average sales revenue is \$33.7 million. In each instance, the sample average reflects a simple sum of each respective value over the entire sample of n = 25 markets, all divided by 25, the total number of sample observations. In this particular sample, no individual observation has exactly the sample average level of net profit or sales revenue. With a net profit of \$4.9 million, regional market C comes closest to the sample average net profit. With \$32.8 million in sales, regional market C is also closest to the sample average revenue. Regional market P has exactly the sample average net profit margin of 14.8 percent.

Any individual observations may coincide with averages for the overall sample, but this is mere happenstance. When profit, profit margin, and sales revenue data are measured in very small increments, it is quite rare to find individual observations that exactly match sample averages. Based on the sample mean criterion, each sample observation that is near sample averages can be described as typical of sample values. It is important to note, however, that there is substantial variation around these sample averages, and the chance of atypical sample values is correspondingly high.

The mean represents an attractive measure of central tendency when upward and downward divergences from the mean are fairly balanced. If the number of sample observations above the sample mean is roughly the same as the number of observations below the sample mean, then the mean provides a useful indicator of a typical observation. However, when the number of sample observations above or below the mean is unusually large, as sometimes occurs when there is a significant divergence between extremely large or extremely small observations, the sample mean has the potential to provide a biased view of typical sample values.

Median median The sample median, or "middle" observation, sometimes has the potential to provide a

"Middie" °bservati°n measure of central tendency that is more useful than the sample mean. When the number of sample observations either above or below the mean is unusually large, then the sample mean can be far different from the value for a typical observation. Such divergences exist whenever a sample includes values that are either very small or very large in relation to the typical observation. For example, annual sales revenue can range from a few million dollars per year for small- to medium-size regional competitors into the tens of billions of dollars per year for large multinational corporations such as ExxonMobil, GE, or IBM. Despite the fact that the overwhelming majority of firms in most industries are relatively small, the average level of sales per firm can be relatively high—given the influence of revenues generated by industrial giants. Not only sales revenue but also profit numbers, wealth, and many other types of important economic data tend to be skewed. It is typical to find most observations at relatively modest levels of revenue, profit, or wealth; a small and declining number can be found along a diminishing "tail" that reaches upward to the end of the sample distribution. In such instances, the sample median can provide a very useful indicator of central tendency.

To illustrate, Table 3.2 presents the net profit, profit margin, and sales revenue data contained in Table 3.1 in a new rank order from largest to smallest values. Sample observations are now simply numbered from 1 to 25, because the values in any given row no longer refer to any single market. The sample average (and standard deviation discussed later) is not affected by this new sample ordering. In Table 3.2, sample medians for net profit, profit margin, and sales revenue can be determined by simply counting from the largest to the smallest values to find the middle observation. With an overall sample size of n = 25, the middle observation occurs at the 13th sample observation, given exactly 12 larger and 12 smaller observations. For this sample of regional telecommunications services markets, median net profit is \$4.7 million, median profit margin is 14.9 percent, and median sales revenue is \$32.8 million. Based on the sample median criterion, each of these observations is typical of sample values. 