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12 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS variability among a group of annual flood-peak frequency curves is due to sampling error and differences in basin characteristics; any advan- tage of mathematical fitting is reduced because physical characteristics may produce a fre- quency curve that is unlike any theoretical one. The large influence of basin characteristics on the shape of certain low-flow frequency curves is described by Riggs (1965). Likewise, a flood-frequency curve for a stream draining a basin composed of a humid mountainous part and a semiarid lowland part would be a composite having an irregular shape not closely represented by any theoretical frequency distribution. Describing Frequency Curves It is sometimes desired to characterize fre- quency curves by means of numerical indexes for comparing several curves or for use in hy- drologic analyses. The mean or median is a commonly used index of central tendency. Variability is usually described by the coeffi- cient of variation, C,=S/X, which is the standard deviation divided by the mean, and thus is dimensionless. The mathematical fitting process provides values of the mean and standard deviation. If a three-parameter distribution is fitted, the coefficient of skew is also computed. These three parameters completely describe the distribution. The mean and standard deviation of graphi- cally fitted frequency curves are readily ob- tained from the graph if the frequency curve is a straight line on normal or lognormal probability paper. The usual graphically fitted frequency curve is not a straight line on the plotting paper used; consequently, the mean is not at a known probability or recurrence interval, the standard deviation cannot be accurately determined, and the curvature indicates the existence of skew- ness. For such curves it is customary to use the median flow as the characteristic of central tendency. The Lane-Lei (Lane and Lei, 1950) variability index may be used to descrihe the variability. The Lane-Lei index is an approxi- mation of the standard deviation and is com- puted as follows from a plot on lognormal probability paper: 1. Values of discharge at 10 percent intervals from 5 to 95 percent are read from the curve (a probability scale rather than a recurrence interval scale is used). 2. The logarithms of these values are found. 3. The standard deviation of the logarithms VE, (log X - (log X)'/N -1 is the varia- bilit y index. A coefficient of variation can also be defined as the variability index divided by the median. It is not customary to estimate the skewness of a graphical frequency curve because skew- ness is of little use in characterizing such a curve. More specific comparisons of frequency curves can be made by considering magni- tudes at particular probability levels or re- currence intervals. The mean is such a one, of course. Others commonly used are the annual minimum flow at 2-year recurrence interval and flood peaks at many recurrence intervals (Benson, 1962a). . Interpretation of Frequency Curves . A frequency curve based on random homo- geneous data is an estimate of the cumulative probability distribution of the population from which the sample was drawn. The following interpretations of the frequency curve require the assumption that the curve is a good repre- sentation of the population distribution. Referring to the graphical curve of figure 6, the recurrence interval of 500 cfs is 16 years. This means that the annual maximum will ex- ceed 500 cfs at intervals averaging 16 years in length, or that the probability of the annual maximum exceeding 500 cfs in anyone year is 1/1~. From figure '1 the recurrence interval of 250 cfs is 13 years. Thus, the annual minimum dis- charge will be less than 250 cfs at intervals aveniging 13 years in length, and the proba- bility that the minimum discharge in anyone year will be less than 250 cfs is 1/13. Many interpretations of frequency curves in hydrology have heen stated in terms of the probahility "of equaling or exceeding" a se- lected value. Most variables in hydrology, .