Correlation

CORRELATION (from French correlation-relationship) in statistics is understood as the relationship between the statistical quantities, series, and groups being studied; to determine the presence or absence of C., statistics uses a special method. The method of C. is applied to determine the parallelism-direct or inverse-in the changes of numbers in the compared series. By finding the correlation coefficient, statistics approaches the measurement of the very measure or degree of parallelism between phenomena. But internal causal connections between different factors are not found in this way, since the question of causality is not resolved by mere calculations and the techniques of the abstract statistical method. The task of statistics is not so much to discover causal dependencies as to help other sciences in discovering these dependencies. "The statistician must be content with revealing numerical relationships-and only, leaving the explanation of these differences and relationships to physiology, meteorology, and other special branches of knowledge" (Westergaard). But if statistics itself is unable to establish causality of connections, then statistical groupings and constructions significantly facilitate the possibility of establishing causal dependencies: thanks to the numerical expression of phenomena and facts, representatives of special knowledge find a more easy possibility of all kinds of comparisons and comparisons (S. A. Novoselsky). The coefficient of C. was introduced by Galton and Pearson (Galton, Pearson) into the theory of variations and inheritance when accounting for connections of anthropological traits in individual individuals and relatives (eye and hair color, fertility of mothers and daughters, etc.). Further, the method of C. finds application in the study of phenomena of human physical development (growth size, weight, chest circumference, etc.). Finally, it extends to the areas of study of socio-economic phenomena (social position and mortality, fertility, nuptiality, tuberculosis-ages and occupations, etc.). The school of Pearson and his followers carries out a wide development of the theory of correlation and its mathematical justifications. It must be noted, however, that among some prominent statistician specialists there exists a certain skepticism regarding the prospects of the method of C. in the area of complex phenomena of social order. "The great labor which is connected with the calculations of C. for numerous series is not justified by the results, since the same is achieved by simple comparisons of series with the correct distribution of material" (Westergaard).- In any case, it is recognized by all that the establishment and measurement of the parallelism of the studied phenomena, given by the methods of C., represents in itself a great scientific value. It contributes to special research in the area of studying the laws of causality and the connection of phenomena in the various areas of scientific research. True, in the discovery of parallelism in series of numbers, a preliminary analysis is still necessary to clarify whether this parallelism is a random result of the coexistence of events in the absence of a real connection between them. Logic and acquaintance with the studied phenomena make it possible to avoid erroneous conclusions in this direction. Logical analysis comes to the decisive place in these cases; the statistical method helps in the evaluation of facts of connection (Novoselsky). The method of C. The basic material for determining the coefficient of C. is given in the form of statistical series expressing the indications of the compared factors over the course of time, place, etc. The task is to trace the variability of these series and their mutual connection in this variability: to what extent the increase of numbers of one series is accompanied by an analogous increase of another series or, conversely, is accompanied by a decrease of another series, or generally no connection of series is observed, or it is observed to a weak degree. It is clear that the basis of such determinations must be laid as an assessment of deviations from some norm of the members of the compared numerical series. Deviations from the arithmetic mean value and square deviations must be used in necessary combinations. This method can be checked on a simple example.- A task is given: in the city of Vienna there is a significant spread of overcrowded dwellings, and at the same time tuberculosis is widespread among the population. Is there a correlational connection between these two factors (components)? If it exists, then to what extent? The city has 19 districts; in each of them the percentage of overcrowded dwellings (premises with 4 or more people living per room) and the indicators of mortality from tuberculosis (per thousand inhabitants) are known. The arrangement of the basic material and the procedure for determining C. for resolving the task are presented in the table (st. 783-784) (F. Prinzing-"Methods of sanitary statistics"). All city districts are arranged on the table in descending order by the height of the indicator of overcrowding; correspondingly, the numbers of tuberculosis mortality are arranged for the districts. Even a simple comparison of these series (2 and 3) reveals an obvious parallelism between them (especially clearly, if these data are plotted on a graph). For both series, arithmetic mean values are sought (the sum of all numbers of the series is divided by the number of members); these values (4.3% for the first and 3.7°/₀₀ for the second series) are laid as the basis for determining deviations for the members of each series. These deviations (with a sign + in cases of exceeding the mean and with a sign - in cases of not reaching the mean level) are depicted in the two following vertical rows of the table (4 and 5). Next, for each member of the series, these two values are multiplied: in the products both factors enter, so to speak, into mutual connection (row 6); the sum of all products is determined (51.06). In the last rows, deviations are raised to the 2nd degree (D|, _D$) in order to obtain square deviations (rows 7 and 8) (see Variational statistics). All these manipulations have as their goal to put the indications of both components-overcrowding and tuberculosis mortality-in a numerical connection for all districts of the city. The very determination of the coefficient of C. is produced by the formula proposed for this purpose by the mathematician Bravais (Bravais, 1846): 2(Da;.D«) f =-----------------L- . N.o_x.o_y The numerator of the fraction corresponds to the product of the deviations of both series taken in their sum; the denominator is the product of the square deviations of both series multiplied by the number of members of the series. Substituting the numerical values of our example table, we obtain: Numerator: V(D_x.D_y)= +51.06 Denominator: JV= 19 > (tab. row 6) _ЛГ гву _/"2з74Т . _{л ЛЛГ}г ⁰^У ~ir=y -i9-^{= + 1}'^110a + 2.7139 (row 7) (row 8). Hence the sought coefficient of C. is equal to + 51.06________ „ oq^ o 19x2.7139x1.1105^-U'^0yiD- Such is the simple path for determining the coefficient of C., which in this case equals a fraction close to one. The value of this coefficient generally fluctuates between 0 and ±1. In cases when it equals 0, there is no C. between the compared series, when it equals +1 or -1, there is a complete positive or negative C. between the series. For intermediate values, the following conditional scale, proposed by Chaddock (R. E. Chaddock), can be adopted: Value of g"

According to this scheme, in the example of the relationships between housing overcrowding and tuberculosis mortality in Vienna, statistics establish a high degree of correlation. From a socio-hygienic point of view, this position must undoubtedly be interpreted in the sense that the complex complex of severe living and domestic conditions, usually associated with overcrowded and unhealthy housing, insufficient nutrition, excessive labor, unemployment, etc., is a factor favoring high tuberculosis mortality in the exploited classes of the population. This is the method of solving the problem of determining correlation under the simplest conditions (2 components); the mechanism becomes more complicated with a larger number of comparable series. It is necessary to make one more note: the determination of the correlation coefficient requires the calculation of its mean error (verification); the formula for the mean error (s) with a correlation coefficient r and the number of series members N is as follows: s = + -- ~ V N ' in our case we will have: 1-0,89162 i/i9 = ±0,047. Adding three times the mean error to the calculated correlation coefficient and subtracting the same value from it determine the limits of its possible values; this position requires that the correlation coefficient exceed its corresponding mean error by at least 3 times, otherwise it loses its significance; in our example this excess reaches almost 20 times.

P. Kurkin. Correlation in psychology is applied in many cases. 1. For determining the connection between various psychological functions. For example, if psychological research proves a correlation between observance and attention, then the presence of observance gives grounds to expect the existence of attention in a given person. 2. For determining the connection between psychological functions and somatic, constitutional, and physiological signs: the study of somatic constitutions by Kretschmer proved a correlation between them and known psychological traits, and therefore one can judge the character of a given person from the somatic status. For example, a pyknic constitution allows one to assume a certain character in a given subject. The same applies to the motor constitution: a known type of movement can characterize to a certain degree the character of a given person. 3. For determining the connection between the results of test tests and practical achievements, for characterizing individual tests in terms of their reliability: if a significant correlation is obtained between them during the study of known psychological qualities by means of tests, then this test can be considered reliable for revealing this quality in others, which we see in professional selection.