142 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



MEASURES OF PRECISION AND THEIR 

 INTERPRETATION. 



In 16 of the columns of Table 32 the values 

 of capacity before and after adjustment consti- 

 tute data suitable for the computation of 

 probable errors. As the number of residuals 

 is in each case small, the particukr values of 

 probable errors (recorded at the bottom of the 

 table) are not themselves of high precision; but 

 the averages have greater claim to attention. 

 It should be added that the residuals were 

 treated as of equal weight, despite the fact 

 that in the adjustment which determined them 

 the relative weights of the observational data 

 were recognized. 



The probable errors of the upper line, marked 

 " C," pertain to the adjusted values of capacity 

 taken from Table 12, which are here treated as 

 observations. The probable errors of the lower 

 line, marked " C r ," pertain to the groups of 

 readjusted values of capacity. 



The average probable error (or, strictly, the 

 mean of the nine tabulated errors) of the read- 

 justed capacities is 1.3 per cent. The cor- 

 responding average for the capacities before 

 readjustment is 2.64 per cent. 



It is of interest to compare the last figure 

 with the previously computed average probable 

 error of the adjusted values of capacity in 

 Table 12, namely, 2.50 per cent. The earlier 

 estimate applies to 66 series of values, the later 

 to 36 values taken from 36 of the series. The 

 36 values may be regarded as properly repre- 

 sentative of the series from which they come. 

 Of the unrepresented series, a portion escaped 

 because the values they furnished to Table 32 

 fell in groups of less than four, and such a 

 group did not afford suitable data for computa- 

 tion of probable error. The others were omitted 

 because the groups they constituted involved 

 incongruities so great that they were rejected in 

 the readjustment. The omission of incongru- 

 ous groups evidently had the effect of lowering 

 the estimate of average probable error, and to 

 give validity to the comparison due allowance 

 should be made for that effect. The discarded 

 values were accordingly treated so far as neces- 

 sary to compute their probable errors, and it was 

 found that by the inclusion of these the estimate 

 of average probable error was raised from 2.64 to 

 3.2 per cent. The revised estimate is believed 



to be properly comparable with the earlier esti- 

 mate of 2.50 per cent. It will be recalled that 

 the original observations were characterized by 

 accidental errors, ascribed chiefly to rhythm, and 

 by systematic errors, ascribed chiefly to methods 

 of observation. The nature of the first adjust- 

 ment was such that its computed probable 

 errors were little affected by the errors of the 

 second class. The adjustment was condi- 

 tioned by a formula involving the assumption 

 that capacity varies as a power of (S-a), and 

 also by various assumptions involved in the 

 arrangement of a system of values for a. What- 

 ever errors were introduced in connection with 

 these assumptions tended to increase the esti- 

 mates of probable error; but they may also be 

 supposed to have aggravated somewhat the 

 errors of the class not covered by the computa- 

 tions of probable error. 



The errors falling outside the estimates of 

 probable error were largely of such nature as to 

 affect an observational series in its entirety, 

 and it was expected that they would be re- 

 vealed in the failuie of groups of quantities 

 taken from different series to exhibit an orderly 

 sequence. Abundant evidence of their exist- 

 ence has been encountered in various discus- 

 sions, including the control by conditions of the 

 sensitiveness of capacity to slope, of the sen- 

 sitiveness of capacity to form ratio, of the 

 value of the optimum form ratio, and of the 

 value of the constant ; but the present dis- 

 cussion is the only one affording an estimate of 

 their magnitude. 



Assuming that the estimate of 2.5 per cent 

 represents the influence of a restricted class of 

 errors, and that the estimate of 3.2 per cent 

 represents the joint influence of that class and a 

 second class, the independent influence of the 

 second class is represented by 



V3.2 2 -2.5 2 = 2.0 per cent 



The indication is that the two classes of errors 

 are of nearly equal importance. 1 



These results are qualified by the fact that 

 the estimate of 3.2 per cent includes not only 

 the errors of the adjusted values of capacity in 



1 The considerations making it probable that 2.5 per cent is an over- 

 estimate for the average error of the first class (p. 74) do not apply to the 

 estimate of 3.2 for the combined error. If the average error for the first 

 class is as low as 2.0 por cent, the computed average for the second 

 class becomes 2.5 percent. 



