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6 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . o x Figure 7.-Hypothetical sampling distribution of means. latter condition the probability is small of obt~ing an X of such size from a population havmg a mean of zero. Therefore, we would reject the hypothesis and would state that the result was ~ignificant at a certain probability level, meanmg that the results obtained differ significantly from the hypothesis. A common problem is the test of significance ?f a r~gression coefficient. The null hypothesis IS aga~n t~at the true value of the regression coeffiCient IS zero, and the test may be made in the same way as before. However, the pro- cedure commonly used is somewhat different. The confidence limits about the theoretical value are computed. If the regression coefficient is b, its standard error S" and its population value 13, then the limits are found to be b-tS,<13<b+tS,. where t is the appropriate value for the chosen significance level and sample size. If the limits !nclude zero, the hypothesis is accepted, that IS, the regression coefficient is not significantly different from zero. If the limies are both on one side of zero, the hypothesis is rejected and t~e regressi?n coefficient is considered sig- nific~ntly different from zero, that is, it is considered meaningful. Many other tests of significance are available but all parametric tests are based on the theor; of sampling and follow the general procedure described above. A less powerful group of non- pa;~met?c .test~ may be used when the prob- ability distributiOn of the statistic is not known. (See Siegel, 1956.) Correlation and Regression The distinctions between correlation and regression must be recognized in order to apply and interpret either of the methods. These distinctions are very marked although they may seem of little'importance because of the simi- larity of the computation proqedures. Dixon and Massey (1957, p. 189) made the following distinction between the two: "A regression problem considers the fre- quency distribution of one variable when another is held fixed at each of several levels. A correlation problem considers the joint varia- tion of two measurements, neither of which is restricted by the experiment." Correlation is a process by which the degree of association between samples of two variables is defined. The correlation coefficient is a mathematical definition of that association. It is, of course, possible to compute a correlation coefficient from any two sets of data. The mathematical definition of association implies no cause-and-effect relation nor even that the relation between the two variables results from a common cause. Correlation theory requires that the data be drawn randomly from a bivariate normal dis- tribution. However, McDonald (1957) reported that experimental sampling studies show the nonnormality effects, usually regarded as dis- turbing by statisticians, to be of incousequential magnitude geophysically. A further require- ment of correlation is that both variables X and Y be without error due to measurement. Nothing can be measured without error, so the above requirement is one of degree. The question of the error allowable is subject to arbitrary decisions, particularly since the true error of the data is never known. The end product of the process of correlation is the correlation coefficient; it is not an equa- tion. The equations which describe Y as a function of X, and X as a function of Y, are regression equations, not correlation equations. Another way of stating the distinction between correlation and regression is that correlation measures the degree of association between two variables, whereas regression prmrides equations for estimating individual values of one variable from given values of the other. Reliability of correlation results depends on the number of items used to compute the correlation coefficient and the mllgnitude of the computed correlation coefficient. Confidence . .