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in statistics, a number characterizing the generalized typical value of a group of qualitatively homogeneous values that differ quantitatively. K. Marx wrote: “In every industry, each individual laborer, be he Peter or Paul, differs from the average laborer. These individual differences, or ’errors’ as they are called in mathematics, compensate one another, and vanish, whenever a certain minimum number of workmen are employed together” (K. Marx and F. Engels, Soch., 2nd ed., vol. 23, p. 334). Average values play an important role in economics. For example, the determination of cost requires consideration of average labor; the average organic composition of capital is of great importance in the analysis of the rate of profit; and determinations of depreciation are based on the average period of service of the equipment in question.
Several types of average values exist. When the individual values do not exhibit a high degree of variation, the type of average value used is not of great importance. In the case of a high degree of variation, however, the choice of the type of average value is governed by the nature of the entity. For example, when average productivity of labor is computed, there must be taken into account both its direct proportionality to the amount produced and its inverse proportionality to the work time expended in production. The arithmetic mean, or arithmetic average, is therefore used in calculating the average daily output of the workers, and the harmonic mean, or harmonic average, is used in determining the average time spent by the workers per unit output. Computations of the average annual rate of growth of, for example, output or population are based on the following principle: the ratio of the level finally reached to the initial level in a given series is equal to the product of quantities of the form 1 + ti, where ti, is the rate of growth for a particular (ith) year. The geometric mean is therefore determined from these quantities, and 1 is subtracted from it to obtain the average rate.
Average values must be distinguished from indiscriminate averages, which cannot be legitimately used to characterize a set of heterogeneous units. This distinction was first made by V. I. Lenin in The Development of Capitalism in Russia (1896–99). He argued against constructions based on a nonscientific use of averages and showed that the heterogeneous mass of peasant households could not be characterized by a single average, since such an average would be an indiscriminate average instead of a value genuinely characteristic for all households.
Closely associated with average values is the law of large numbers (seeLARGE NUMBERS, LAW OF). When a random element is present in the individual values, the greater the number of individual values embraced by the average value, the greater the extent to which the random element is canceled out in the average value.