A dependence between random variables not necessarily expressed by a rigorous functional relationship. Unlike functional dependence, a correlation is, as a rule, considered when one of the random variables depends not only on the other (given) one, but also on several random factors. The dependence between two random events is manifested in the fact that the conditional probability of one of them, given the occurrence of the other, differs from the unconditional probability. Similarly, the influence of one random variable on another is characterized by the conditional distributions of one of them, given fixed values of the other. Let and be random variables with given joint distribution, let and be the expectations of and , let and be the variances of and , and let be the correlation coefficient of and . Assume that for every possible value the conditional mathematical expectation of is defined; then the function is known as the regression of given , and its graph is the regression curve of given . The dependence of on is manifested in the variation of the mean values of as varies, although for each fixed value , remains a random variable with a well-defined spread. In order to determine to what degree of accuracy the regression reproduces the variation of as varies, one uses the conditional variance of for a given or its mean value (a measure of the spread of about the regression curve):
If and are independent, then all conditional mathematical expectations of are independent of and coincide with the unconditional expectations: ; and then also . When is a function of in the strict sense of the word, then for each the variable takes only one definite value and . Similarly one defines (the regression of given ). A natural index of the concentration of the distribution near the regression curve is the correlation ratio
if, moreover, , then is related to through an exact linear dependence; but if , there is no functional dependence between and . There is an exact functional dependence of on , other than a linear one, if and only if . With rare exceptions, the practical use of the correlation coefficient as a measure of the lack of dependence is justifiable only when the joint distribution of and is normal (or close to normal), since in that case implies that and are independent. Use of as a measure of dependence for arbitrary random variables and frequently leads to erroneous conclusions,
since may vanish even when a functional dependence exists. If the joint distribution of and is normal, then both regression curves are straight lines and uniquely determines the concentration of the distribution near the regression curves: When the regression curves merge into one, corresponding to linear dependence between and ; when one has independence.
When studying the interdependence of several random variables with a given joint distribution, one uses multiple and partial correlation ratios and coefficients. The latter are evaluated using the ordinary correlation coefficients between and , the totality of which form the correlation matrix. A measure of the linear relationship between and the totality of the other variables is provided by the multiple-correlation coefficient. If the mutual relationship of and is assumed to be determined by the influence of the other variables , then the partial correlation coefficient of and with respect to is an index of the linear relationship between and relative to .
Mathematical statisticians have developed methods for estimating coefficients that characterize the correlation between random variables or tests; there are also methods to test hypotheses concerning their values, using their sampling analogues. These methods are collectively known as correlation analysis. Correlation analysis of statistical data consists of the following basic practical steps: 1) the construction of a scatter plot and the compilation of a correlation table; 2) the computation of sampling correlation ratios or correlation coefficients; 3) testing statistical hypothesis concerning the significance of the dependence. Further investigation may consist in establishing the concrete form of the dependence between the variables (see Regression ).
Among the aids to analysis of two-dimensional sample data are the scatter plot and the correlation table. The scatter plot is obtained by plotting the sample points on the coordinate plane. Examination of the configuration formed by the points of the scatter plot yields a preliminary idea of the type of dependence between the random variables (e.g. whether one of the variables increases or decreases on the average as the other increases). Prior to numerical processing, the results are usually grouped and presented in the form of a correlation table. In each entry of this table one writes the number of pairs with components in the appropriate grouping intervals. Assuming that the grouping intervals (in each of the variables) are equal in length, one takes the centres (or ) of the intervals and the numbers as the basis for calculation.
For more accurate information about the nature and strength of the relationship than that provided by the scatter plot, one turns to the correlation coefficient and correlation ratio. The sample correlation coefficient is defined by the formula