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13 FREQUENCY CURVES . notably streamflow, are continuous-but re- ported as discrete-and the theoretical proba- bility of occurrence of any particular value in a continuous distribution is zero. Therefore it seems desirable to use only "of exceeding." The interpretations of frequency curves given above will not answer questions such as the probability of an event of 1O-year recurrence interval being exceeded in a 10-year period. Intuitively, one might expect that probability to be 0.5, but it is not. The correct probability can be computed as follows. Since the proba- bility of exceeding the 1O-year event in 1 year is 0.1, the probability of not exceeding it in 1 year is 0.9. Then, the probability of not ex- ceeding it in 10 years is, by the multiplicative law of probabilities (0.9)10=0.35, and the probability of one or more events ex- ceeding the 10-year event in 10 years is 0.65. A more complete interpretation of a fre- quency curve is given by Riggs (1961) who also proposed in that reference, that the fre- quency curve be used as a basis for a family of curves giving the probability of events ex- ceeding certain magnitudes in given periods of years (design periods). Figure 8 shows a flood- frequency curve and the design-probability curves computed from it. It should be noted that the above discussion does not apply to frequency curves based on the Beard (1943) method of computing plotting positions. For such curves the n-year event has a probability of 0.5 of not being exceeded in n years. See Langbein (1960) for a further dis- cussion of Beard's plotting position. Although frequency curves are used as though they Were accurate representations of the popu- lation distribution, we know that they may not be. Benson (1960) sampled from a known dis- tribution and showed a wide range in shape and position of frequency curves defined by different samples of the same size. Another way of assessing the reliability of a frequency curve is by computing the confidence limits. Chow (1964, p. 8-31) describes a method. These computations indicate that the frequency curve is most reliable in the vicinity of the mean. In comparing frequency curves for two dif- ferent streams, the analyst should keep in mind ( . . 30 ~ ::~ ~ 15 = ~1O ~ 5 u '" B ~ o ~ o z ;)i ~ o ~ ~ '" "' o = ~ 30 ~ a 1.1 1.31.52 3 5710 20 30 50 100 RECURRENCE INTERVAl, IN YEARS Pis the probability that the curve value I wilt be e~ceeded the stated number of times in the corresponding design period {p(NOnel=O.so P(lormore)=O.50 P(2ormore)=0.09 to 0.13 ~ 25 ~ ~ ~ ~ 20 z ~ P (None) =0.75 P (1 or more) =0.25 \ P(2 or more) =0.02 to 0.03 =-rp(None)~O.25 l.P(lorrnore)=0.75 15 10 5 7 10 20 304050 100 DESIGN PERIOD, IN YEARS Fi9ure 8.-Design-probability curves (lower graph) and the frequency curve on which they ore based (upper graph). that the curves may differ because of chance or because of the effects of different basin char- acteristics, or both. The population distribution of a flow characteristic of one stream may be considerably different from the population dis- tribution of that characteristic for another stream. Riggs (1965) discusses some basin char- acteristics which influence the shape of the frequency curve of annual minimum flows. Special Cumulative Frequency Curves Frequency curves are often useful for defiu- ing the distribution of events even though the events are not entirely independent of each other (that is, they are serially correlated), in which case the probability interpretation must be somewhat modified and cannot be precisely stated. An example of a frequency curve based on serially correlated data is that of the annual minimum flows of South Fork Obion River,