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FREQUENCY CURVES 11 . A simpler definition of 0, is the ratio of the third moment about the mean to the 3/2 power of the second moment about the mean; that is L:,(Q-Q)'/N 08- S3 ' f which, for this example, gives 0,=0.94 if S is not adjusted by the factor N/N -1. The differ- ence between results of the two formulas for 0, will increase as N decreases. Plotting positions are computed by Q=Q+KS, where Q and S are as computed above, and K is obtained from table 1 for various recurrence intervals at a 0, of 1.0. For example, for the 20-year recurrence interval, . Q=347+ 1.87(77.8) =492. It can be seen from table 1 that the frequency factor is little affected by changes of a tenth or so in the coefficient of skew; therefore, a simple method of computing 0, should be adequate. The theoretical Pearson Type III curve defined by the data is plotted on figure 6. Computation of the coefficient of skew re- quires cubing the individual values, which is time consuming on a desk computer. Theoret- ical fitting, therefore, is most easily done on a digital computer. Use of historical data " The graphical method of defining a frequency curve is readily adaptable to inclusion of cer- tain historical data. Essentially, an estimate of the recurrence interval of each historical event is made on the basis of available information, and the event plotted at this recurrence inter- val. Occasionally, the historical information may indicate the need to modify the recurrence interval of a flood within the period of record. Dalrymple (1960, p. 16-18) describes the procedure. Comparison of Mathematical and Graphical Fitting Mathematical fitting has the following ad- vantages: . 1. For the same theoretical distribution, every analyst using a given set of data would get the same answer. 2. Fitting can be done by electronic computer. 3. The result can be completely described by a few parameters. Mathematical fitting has the following dis- advantages: 1. Selection of the theoretical distribution is arbitrary. 2. No one theoretical distribution will fit all data of one type such as flood peaks. 3. Characteristics of a set of data tend to be obscured if they are not plotted; however, plotting can be done separately. 4. No objective method is available for incor- porating historical information ill the computation. Graphical fitting provides the following ad- vantages : 1. The procedure is simple and can be done quickly. 2. No assumption as to the particular form of the distribution need be made. 3. Relation of the curve to the points is readily seen. 4. Historical data may be included. Graphical fitting has the following dis- advantages : 1. Even though the same plotting position formula is used, different analysts will draw somewhat different frequency curves. 2. The result cannot be described by two or three parameters. The above comparisons of mathematically and graphically fitted frequency curves are based on statistical considerations. If there were one underlying distribution for a parti- cular streamflow characteristic, such as annual flood peaks, then fitting that distribution mathematically to all sets of annual f1ood- peak data would be desirable. Under that con- dition it is assumed that the type of population distribution is known, and the sample is used to estimate the parameters. Among several basins of equal size the differences in computed parameters would be due to sam pIing errors only. But basins are not only not the same size, they differ also in their flood-generating character- istics and, consequently, would have different population frequency distributions. Thus, the