Mean is the average observation.The arithmetic mean or average is the sum of numbers included in a given sample divided by the number of observations. If n is the number of sample observations, X1 is the first observation, X2 is the second observation, X3 is the third observation, and so on, then the sample mean is calculated as
Alternatively, the arithmetic mean or average is sometimes expressed as n
where the greek letter sigma, , is referred to as the mathematical summation sign. signals to sum over the sample observations from i = 1, the first sample observation, to i = n, the last sample observation.
To illustrate, consider the net profit, profit margin, and sales revenue data contained in Table for a hypothetical sample of small regional markets for a leading provider of telecommunications services. Profit margin, defined as net profit divided by sales revenue, is the rate of profitability expressed as a percentage of sales. Although the data are hypothetical, they are representative of actual figures. Both net profit and profit margin, expressed in percentage terms, are common measures of firm performance. Sales revenue is a commonly used measure of firm size. Each row of information shows relevant data for each market in 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.
Annual Net Profit, Profit Margin, and Sales Revenue in 25 Regional Tele communications Services Markets
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.
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