Coefficients

COEFFICIENTS, a mathematical concept. In algebra, a coefficient is understood as a multiplier expressed in digits and placed before a monomial. Thus, in the monomial 6a²b²c, the number 6 is the coefficient of this monomial. A statistical coefficient, in the usual sense of the word, is understood as a value showing the frequency of the occurrence of a given phenomenon in the environment where it took place. For example, the mortality coefficient is understood as the quotient obtained by dividing the number of deaths during a given calendar period of time (usually a year) by the average population in the environment where the cases of death were observed; for the purpose of ease of comparison, this quotient is reduced to some single base, usually to unity with several zeros (1,000, 10,000, etc., or, as it is customary to call it, pro mille, pro decimille, etc.). Thus, a statistical coefficient, like a coefficient in algebra, is essentially one of the multipliers; the other multiplier is the denominator of the fraction, and their product constitutes the absolute value of the manifestations of the trait in the studied aggregate. From the point of view of derived statistical values, statistical coefficients represent special cases of the expression of so-called relative numbers of frequency or intensity.

Depending on the content of the aggregates brought into relation with each other, statistical coefficients are divided into general and special coefficients. In the case where the aggregate (environment) taken in the denominator of the ratio is in close genetic connection with the aggregate indicated in the numerator (the phenomenon under study), the coefficient has a special meaning. With a different character of the connection between the correlated aggregates, the coefficient acquires a general meaning. General coefficients serve only the purposes of general orientation, in contrast to special coefficients, which provide a more accurate measure of a phenomenon and allow for more accurate conclusions to be drawn about it. An example of a general coefficient is the birth rate, and an example of a special coefficient is the fertility rate. The number of births (excluding stillbirths) in Leningrad in 1923 was 31,906 people; the average population in the same year was 1,093,000 people; hence the birth rate coefficient is equal to 31,906 / 1,093,000 = 29.2 ‰ (pro mille). Although this coefficient gives an idea of the reproductive power of the population of Leningrad, especially in comparison with the general mortality coefficient, it is still a very conditional value, because its denominator contains a significant part of the population that does not participate in the production of births shown in the numerator (the elderly, children). Furthermore, the cited birth rate coefficient is also conditional because it does not properly reflect the family and age composition of the population, which is the main factor among all others determining the degree of fertility of the population. In order to clarify the latter, it is necessary to calculate a number of special coefficients, namely: the general fertility coefficient, the marital fertility coefficient, and age-specific marital fertility coefficients. The first of these is equal to: (number of births x 1,000) / (number of women aged 15-49).

For the city of Leningrad, this is (31,906 x 1,000) / 347,002 = 89.4 ‰; the second coefficient (marital fertility) is: (number of births in wedlock x 1,000) / (number of married women aged 15-49); for the same conditions of time and space, this is (26,821 x 1,000) / 173,627 = 154.4 ‰; the third type, the age-specific fertility coefficient, is constructed as follows: (number of births to married women of a given age x 1,000) / (number of married women of the same age).

Or for the city of Leningrad for women of a given age, it is calculated accordingly. Along with the stated, most common interpretation, the term in question was and is often used in statistics for a number of other reasons. Most often, it is used to denote the measure of the relationship between various numerical characteristics of series. Thus, in this sense, one speaks of the coefficient of variation (σ/M x 100%), the coefficient of asymmetry (μ₃/σ³), the coefficient of regression (b = r σ₁/σ₂), and the coefficient of correlation (r = Σxy / Nσ_xσ_y). - P. Kuvshinnikov. (For details on coefficients of variation and correlation, see Variation Statistics, Correlation).

Coefficient of efficiency of the human body is the ratio of the amount of mechanical work performed to the energy expended. Usually, two coefficients of efficiency are distinguished. To calculate one of them, the so-called gross coefficient of efficiency, the amount of mechanical work T is divided by the total energy expenditure D, and this coefficient of efficiency is denoted by the letter η. To calculate the net coefficient of efficiency, one proceeds somewhat differently: the energy expenditure at rest for the same period of time D_s is subtracted from the total energy expenditure D, and the amount of mechanical work is divided by the difference resulting from this subtraction, or, in other words, by the so-called dynamic energy expenditure (D - D_s = D_d). Consequently, the net coefficient of efficiency can be expressed by the formula: K = T / D_d, while the gross coefficient of efficiency is calculated by the formula η = T / D.

[formula: ... or D/D]. Of course, before division, the amount of mechanical work is expressed in large calories (by dividing the number of kg/m by 425) or, conversely, energy expenditure is recalculated into kg/m (by multiplying the number of large calories by 425). The coefficient of efficiency of any machine characterizes the quality of its work. The higher the coefficient of efficiency, the more economical and better the machine works, and the less energy, relatively, is spent on performing the same mechanical work. The healthier and more trained a person is, and the more rationalized their work, the higher the economy of their work, and the greater the coefficient of efficiency. As Herbst showed, people with heart disease, emphysematous patients, and tuberculosis patients yield a lower coefficient of efficiency during muscular work than healthy people. The value of the coefficient of efficiency is of enormous importance for the physiology of labor. On one hand, it indicates the state of the worker, and on the other, it characterizes the work itself, its organization, and the degree of rationalization. Training in any work increases the coefficient of efficiency. For example, Atzler in 1924 showed that the net coefficient of efficiency changes by days as follows: 1st day... 12.9%, 2nd... 10.3%, 3rd... 12.3%, 4th... 15.0%, 6th... 16.0%, 7th... 18.1%, 8th... 19.4%, 9th... 19.4%, 10th... 19.9%, 11th... 20.4%, 12th... 20.6%, 13th... 20.6%, 14th... 21.0%. Thus, it is evident that the coefficient of efficiency grows in parallel with training, at first rapidly, then more and more slowly, until it reaches its maximum value (21%). The increase in the coefficient of efficiency is explained by the ever-improving coordination of movements as a result of training, due to which unnecessary movements of synergistically working muscle groups are eliminated. On the second day of training, a certain decrease in the coefficient of efficiency is observed, apparently caused by the transition from previous work methods to new ones and the destruction of old coordination connections (re-learning). A beginner worker always spends more energy on the same work than an experienced adult worker. For example, Amar found that when filing, a beginner spends 4.9 large calories per 1 g of filings (coefficient of efficiency = 6.72%), while an experienced one spends only 2.5 large calories (coefficient of efficiency = 9.42%). The former works unevenly, tires often, their pulse rate increases on average by 35% (instead of the usual 20%), and respiratory rate by 54% (instead of 30%). It is clear that the increased work of the heart and respiratory apparatus increases the value of D_d and consequently reduces the coefficient of efficiency. As stated above, the coefficient of efficiency characterizes not only the state of the organism and the degree of training of the worker but also provides indications of the rationality of the organization of the work itself. A person, like a machine, works most economically only within certain limits of speed and load ("optimal zones"). As an example, one can cite the data of Chauveau relating to work on an ergometric bicycle: Speed (number of revs per 1 min.) 70, 80, 90, 90, 0; Net coefficient of efficiency 25.1%, 26.7%, 28.4%, 30.6%, 25.8%. From this, it is evident that a person works with the highest coefficient of efficiency, and consequently with the greatest economy, at a speed of 90 wheel revolutions per minute. A similar relationship exists between load and the coefficient of efficiency. This circumstance is of enormous theoretical and practical interest, as it allows for the design of work in accordance with the data of the physiology of labor. "Overload," caused by work with high speed and load, leads to a whole series of pathological processes (heart disease, overexertion, etc.). As the physiology of labor develops, the standardization of labor will be increasingly based on physiological data. The coefficient of efficiency changes during the work itself, usually reaching a maximum in the 2nd hour of work and decreasing under the influence of fatigue towards the end of the work. The effect of exercise (at the beginning of work) and fatigue (at its end) is particularly clearly reflected in the coefficient of efficiency. The value of the coefficient of efficiency for various jobs is different. On average, it can be considered that the net coefficient of efficiency is equal to 20% and only in the case of habitual work and the training of the performers can it reach 33% (walking). The gross coefficient of efficiency is much lower, and its value fluctuates between 7% and 10%. In industrial work, the coefficient of efficiency, according to some researchers, is equal to 4-6%. These figures are much lower than the coefficient of efficiency of various machines (steam engines - 12-20%, internal combustion engines - about 35%, electric motors - 90-98%), which, in connection with the low power of a person (1/10 of a horsepower) and the high cost of the "fuel" consumed by them (proteins, fats, carbohydrates), has led and is leading to the replacement of the human engine with a machine engine. K. Kekcheev. Absorption coefficient in agglutination is an indicator of the ability of a given agglutinogen to bind the agglutinin of a given serum and depends on the avidity (see) of the reacting bodies. The indicator of the absorption coefficient is the ratio of the amount of absorbed agglutinin to that which is present in the given serum (to the titer of the latter). The methodology for determining the absorption coefficient was proposed by Eisenberg and Volk, Barykin and Friese.