Large Numbers, Law of

a general principle by virtue of which the collective effect of a large number of random factors leads, under certain very general conditions, to a result that is almost independent of chance. The precise formulation and conditions of applicability of the law of large numbers are given in the theory of probability. The law of large numbers is one of the expressions of the dialectical connection between chance and necessity. The first precisely proved theorem was given by Jakob Bernoulli (published after his death in 1713). Bernoulli’s theorem was generalized by S. Poisson, who first used the term “law of large numbers” in his paper Investigation of the Probability of Judgment (1837). A considerably more general conception of this term is based on P. L. Chebyshev’s work On Average Values (1867). In this modern conception, the law of large numbers asserts that under certain conditions liable to a precise designation, the arithmetic average

x̅ = (x₁ + x₂ + ... + x_n)/n

of a sufficiently large number n of random values X_k with a probability as close as desired to unity differs as little as desired from its mathematical expectation a = E (x̅). Chebyshev’s proposed method of proving the law of large numbers using the so-called Chebyshev inequality proved to be new and very fruitful.

For independent random variables having identical probability distributions and finite mathematical expectation a, the law of large numbers asserts that for any ∊ >0, the probability of the inequality ǀx - aǀ < ∊ converges to unity as n → ∞. The order of deviation of x from a is indicated by the limit theorems of the theory of probability. In typical cases the deviations are on the order of 1/√n. Correspondingly, the random deviations of the sumx = x₁ + x₂ + ... + x_n = nx̅ from its mathematical expectation na grows as √n. This fact (called in popular terminology “the law of the square root of n”) gives a certain, although coarse, idea of the character of the operation of the law of large numbers.

A descriptive explanation of the meaning and value of the law of large numbers is given by the following example. We have N molecules of a gas in a closed container. In accordance with the kinetic theory, each molecule moves randomly within the container, undergoing numerous collisions with other molecules and the walls of the container. By striking some area σ of the wall during a selected time interval of t seconds, an individual molecule imparts the momentum f_k, to this area. Momentum f_k is a typical random variable because the state of the gas under consideration determines only the mathematical expectation a - E (f_k)) of this momentum; the real value of the momentum of a given molecule in a given time interval can be quite varied (beginning with zero—in the case when, for a given time interval, the given molecule does not strike within the area σ). The sum F = ∑^N_K = ₁f_k of the momentums of all the molecules that collide with the area σ in the given time interval is also a random variable with mathematical expectation equal to A = Na. However, by virtue of the law of large numbers, which appears here with unusual accuracy because the number N is very large, F in reality turns out to be almost independent of the random circumstances of the motion of individual molecules, namely, almost exactly equal to its mathematical expectation A. This, from the viewpoint of the kinetic theory, explains the fact that the pressure of the gas on the area σ is practically constant and does not fluctuate randomly.

One often has to apply the law of large numbers in situations where the number of random addends is not as great as in the example with the gas molecules; then the deviations of the sum of random values from its mathematical expectation can be considerable. In this case, it is extremely important to be able to estimate the size of these deviations. Let us, for example, randomly select from 1,000 batches of a certain item, 100 pieces in each, ten pieces at a time from each batch for testing; from the tested 10,000 pieces, we discover 125 defective items. If one denotes by n_K the number of defective items in the _k th batch, then the total number of defective items is equal to n = the mathematical expectation of the number of defective items among those tens that are taken for testing from the_k th batch is equal to S_K = (10/100)n_K, and the mathematical expectation of the total number of defective items in 1,000 samples of ten pieces each is equal to S = . By virtue of the law of large numbers, it is natural to assume that n/10 ~ 125, that is, that among 100,000 items in all the batches, there are approximately 1,250 defective ones. A more accurate study using the theory of probability leads to the following result: if the selection of items from each batch was actually random, then it would be safe to assert with sufficient confidence that in fact 1,000 < n< 1,500, but even the estimate 1,100 1,400 would not be sufficiently reliable, and the estimate 1,200 < n < 1,300 has no serious basis. One can obtain a more accurate estimate for n only by testing a larger number of items.

If the condition of independence of terms in most applications of the law of large numbers is fulfilled, it is only with one or another approximation. Thus, even in the first example, the motions of individual molecules of gas, strictly speaking, cannot be considered independent. Therefore, it is important to investigate the conditions of applicability of the law of large numbers to the case of dependent terms. The basic mathematical work in this area has been done by A. A. Markov, S. N. Bernshtein, and A. Ia. Khinchin. Qualitatively, their studies have shown that the law of large numbers is applicable if the dependence between addends with numbers far removed is sufficiently weak. Such, for example, is the situation in a series of meteorological observations of the temperature or pressure of air. In applying the law of large numbers, it is necessary to thoroughly verify the conformity of the conditions of its applicability to real circumstances.

in economic science and in socioeconomic statistics, the manifestation of one of the most important objective laws accompanying the formation of the patterns related to mass socioeconomic processes.

The patterns of homogeneous aggregates consisting of random solitary phenomena are manifested (and consequently can be studied) only in a sufficiently large number of units (cases). These patterns can be expressed quantitatively only in the form of average numbers (for example, average levels, average proportions of features or groups in an aggregate, various coefficients, and other generalizing characteristics). The larger the number of units of the phenomenon encompassed, the more accurately the patterns are expressed by average numbers. The deviations caused by random factors of these individual units to one side or another of the characteristics of the overall pattern of the entire phenomenon will, with a sufficiently large number of units, almost cancel each other out. In any mass phenomenon, along with factors which are common to the entire mass of units, random factors are also at work—that is, factors that can be different in individual instances; the action of these factors can be directed to various sides, since there is a certain degree of mutual independence between these units. As a result of the canceling out of the action of random factors, the action of factors common to the phenomenon is manifested—that is, the necessity for the regularity of the entire mass phenomenon is evinced.

The law of large numbers is not related to the second group of factors (causes), or consequently, to the essence of the mass phenomenon. It does not create either the patterns themselves that are manifested in the average or their overall average measure for the mass of units of the phenomenon (for example, the level of value or of the productivity of labor, the average profit norm, the probability of illness, and so forth). Consequently, the law of large numbers is not capable of changing the average level of a phenomenon, causing the stability of the dynamic series of levels, predicting the scale of the deviations from the average level, or particularly, serving to explain the real causes for the occurrence of the level itself or the deviations from it. Hence, the complete invalidity of the antiscientific attempts by certain bourgeois scientists to ascribe to the law of large numbers a miraculous and almost mystical capability of creating regularity from the chaos of every random factor can clearly be seen; this capability supposedly exists even if there is no internal necessity or internal regularity in the factors, as if the “large number” of the units purportedly in and of itself would lead to the occurrence of a regularity in the “large number,” regardless of the essence of the mass phenomenon. The law of large numbers does not form a regularity but merely directs its manifestation.

J. Graunt (1662), W. Petty, E. Halley (1693), J. Sussmilch (1741), and A. Quételet in their demographic and statistical research worked from an intuitive recognition of the law of large numbers. In the 19th century the interpretation of economic phenomena as mass phenomena, with the accompanying action of the law of large numbers, took on wider and wider recognition. In the works of K. Marx, and particularly in Das Kapital, all categories of economic reality and economic science appear as average values (the average socially necessary working time, simple average labor, the average level of the ability and intensity of labor in a given society, the average rate of monetary circulation, the average profit rate, and so forth).

In Marx’s conception, any economic laws and regularities (which under capitalism operate “blindly” or spontaneously) likewise can be manifested only as average levels or only as an average. At the same time, Marx and Engels repeatedly wrote about the specific form for the manifestation of economic laws and regularities: “The aggregate movement of this disorder is its order” (K. Marx, in K. Marx and F. Engels, Soch., 2nd ed., vol. 6, p. 438; the issue was the movement of prices). Or “The general laws are implemented, . . . only as a predominant tendency, as some never firmly established average of constant fluctuations” (K. Marx, ibid., vol. 25, part 1, p. 176). Or “The internal law which makes its way through these random factors and which regulates them becomes visible only when they are encompassed in large masses, and ... it therefore remains invisible and incomprehensible for the most isolated agents of production” (ibid., vol. 25, part 2, p. 396). Engels wrote “about economic laws generally” that “all of them do not have any reality except as an approximation, as a tendency, as an average, but not in an immediate reality” (ibid., vol. 39, p. 355). Marx interpreted deviations in a multiplicity of prices from value as a form of the manifestation of the law of value: “The possibility of a deviation of a price from the amount of value is contained in the form itself of a price. And this is not a shortcoming of this form; on the contrary, it is precisely this distinguishing feature that makes it an adequate form for that method of production whereby the rule can make its way through disorderly chaos only as a blindly operating law of average numbers” (ibid., vol. 23, p. 112).

Later, V. I. Lenin wrote about the same concept in somewhat different terms: “Quite naturally, in a society of separated commodity producers linked only by the market, a regularity cannot be manifested except as an average, a social or mass regularity with the canceling out of individual deviations to one side or another” (Poln. sobr. soch., 5th ed., vol. 26, p. 68). There can be no doubt that both Marx and Lenin here were speaking about the law of large numbers; however, Marx used another term for it, namely, Durchschnittsgesetz (the “law of averaging,” the “averaging law,” or the “law of average numbers”). The reason for this difference must be seen in that Marx considered the fact of the manifestation of any law in the form of an average value to be more essential than the fact of its manifestation merely in a large number of cases. Thus originates the identification which has become established in Soviet statistical science of the concepts and terms of “the law of large numbers” and “the law of average numbers,” and often “the law of large (average) numbers.”

The canceling out of random deviations of individual units from the average level of the entire mass phenomenon through the action of the law of large numbers must be strictly distinguished from the purely algebraic balancing of the total positive and total negative deviations in calculating any arithmetical average. Total positive and total negative deviations are equalized because of the very law of calculating an average with completeness, both in the case of the typical average for a homogeneous aggregate (when the individual deviations are truly random), and in the case of a purely fictitious “indiscriminate” average for an obviously heterogeneous aggregate (when substantive and random elements are intertwined in the individual deviations), as well as with any number of individual values which are united by an arithmetical average. But the action of the law of large numbers consists in the canceling out of random deviations from the level corresponding to the regularity of the mass phenomenon and only approximately reflected by the average value. Hence such a canceling out cannot be complete, and it depends upon the number of solitary phenomena constituting the mass.

The significance of the fact of the action of the law of large numbers is great for any modern science, specifically and especially for the scientific elaboration of the theory of statistics and the methods of statistical cognition. The action of the law of large numbers has universal significance for the objects of statistical study themselves—that is, for the statistical aggregates with their summary features and mass regularities. An important statistical method of sampling is based upon the planned use of the action of the law of large numbers with the random choice of the units from the mass aggregate that will form the sampling.