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SOME STATISTICAL TOOLS IN HYDROLOGY 33 . chlLIlce if there were no real difference between periods. The double asterisk on the mean square for between periods (in the analysis-of- varilLIlce table) denotes statistical significance above the 0.01 level. Now consider a similar problem, to deter- mine whether mean annual precipitation at three stations is different. The data are given in the following table. Precipitation, in inches, at- Ye." Site 1 Site 2 Site 3 I' 1945_____________ 1946_____________ 1947_____________ 1948_____________ 1949_____________ 1950_____________ 1951_____________ 1952_____________ 47.5 348 42.2 59.9 51. 7 42.5 40.5 47.8 366.9 45.9 40.6 36. 1 37.5 52. 3 42. 2 40.6 38. 3 45. 8 333.4 41. 7 48.2 40.2 37.8 58.2 43.3 41. 4 42.3 48.2 359.6 45. 0 Sums__n_n ynnn_n_______ . From the data in the table we ClLIl make the following calculations: T=1,059.9 T'fN=46,807.8 2:Y~=47,784.0 2:T,'fn=46,885.4 The lLIlalysis-of-varilLIlce table is Source Sum of squares Mean square Degrees of freedom Among sites_______ Within sitee_______ TotaL___n 77.6 898. 6 976. 2 2 38. 8 21 42. 9 23 nn___.. . F2.21=38,8f42.9<1; therefore there is no dif- ference statistically among the three means. A perusal of hydrologic literature will turn up very few applications of analysis of variance. Most analyses of varilLIlce are based on data from a designed experiment, lLIld it is this ap- lication for which the best results are obtained. Hydrologic data are usually parts of a time series which may not be stationary. Thus the individual values may not be entirely inde- pendent as required for a valid lLIlalysis of variance. In the example comparing mean run- offs for two periods of record, it was concluded that a real difference existed between periods. But there is no physical reason to expect a change in this basin. The earlier period was one of high precipitation; the later period in- cluded the drought of the thirties. It is also possible that some of the annual rnnoffs were serially correlated. Thus the characteristics of the data tend to discredit the results of this particular application of the analysis of variance. In the last example, the precipitation at site 2 is greater than that at site 1 for every year .shown, yet the lLIlalysis of variance shows no difference in melLIlS. (An analysis of variance between site 1 and site 2, only, shows a dif- ference at a probability level of about 0.25.) The annual precipitations at a site may be independent, but the precipitations at the sev- eral sites for the same year are not. Therefore the requirements of the method are not met, and the results must be accepted with reservation. The two examples given utilize a very simple statistical model. For more complicated prob- lems, several models may be considered. Selec- tion of the appropriate one is difficult for the "part-time" statisticilLIl. Many statistics texts treat lLIlalysis of variance in detail. See Bennett lLIld Franklin (1954), Brownlee (1960), and Dixon and Massey (1957). In general, an lLIlalysis of varilLIlce made by someone not thoroughly familiar with the process should be reviewed by a statisticilLIl for suitability of the model and correctness of the interpretation. Analysis of covariance The lLIlalysis of variance of the runoff data for two periods (see p. 32) indicated that the population means were probably different, yet other information, particularly precipitation records, leads to the opposite conclusion. The precipitation data can be incorporated in the lLIlalysis by using lLIl analysis of covariance. This method includes concepts from analysis of variance and regression and is applicable where a variable represents a measurement for each individual as opposed to a variable which can only be separated into a few categories.