N

sample variance

Average squared deviation from the sample mean

sample standard deviation

Square root of the population variance

Like the population variance, the population standard deviation reflects both upside and downside variation throughout the entire population. Because the population standard deviation is expressed in the same units as individual observations, it is also a measure of dispersion that has a very simple and intuitive interpretation. For both reasons, it is possibly the most commonly employed measure of dispersion that managers rely on.

Of course, it is often too expensive and impractical to measure the variance or standard deviation of the entire population. When a subset or sample of the overall population is analyzed, a slightly different formula must be employed to properly calculate variance and standard deviation. The sample variance is given by the expression s2 = (X! - X)2 + (X2 - X)2 + ■ ■ ■ + (Xn - X)2 n - 1

where X denotes mean for a sample of n observations. The sample standard deviation is given by the expression

Three differences between these formulas and those for the population variance and standard deviation are obvious: The sample mean X is substituted for the population mean squared deviations are measured over the sample observations rather than over the entire population, and the denominator is n-1 rather than n. The answer as to why n-1 is used rather than n is quite complex, but reflects the fact that dispersion in the overall population would be underestimated if n were used in the denominator of the sample variance and standard deviation calculations. It is therefore necessary to rely on the population variance and standard deviation formulas when calculating measures of dispersion for an entire population. If the list of markets in the telecommunications services example comprises a complete list of the markets served by a given firm, then it would be appropriate to calculate the dispersion in net profits, profit margins, and sales revenue using formulas for the population variance and standard deviation. If this list comprised only a sample or subset of all markets served by the firm, then it would be appropriate to calculate the dispersion in net profits, profit margins, and sales revenue using formulas for the sample variance and standard deviation.

From a practical standpoint, when a relatively large number of sample observations is involved, only a modest difference results from using n-1 versus n in the calculation of variance and standard deviation. Table 3.1 shows variance and standard deviation calculations based on the assumptions that the list of telecommunications services markets comprises only a subset or sample of relevant markets versus the overall population. When as few as 25 observations are considered, only modest differences would be noted between the population parameter calculations for variance and standard deviation and the relevant sample statistics.

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