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<br />f <br /> <br />. <br /> <br />FREQUENCY CUllVES <br /> <br />3 <br /> <br /> <br /> <br />o <br />~. 0 <br />-g] :l <br />"" " <br /> <br /> <br /> <br />o <br />.!S! IV <br />" " <br />~ 0 <br />" " <br /> <br />. <br /> <br />Figure 3.-Relative positions of the: mean, median, and <br />mode for right-skewed (vpper) and 112Ft-skewed <br />(lower) distributions. <br /> <br />ces, or nonexceedenc8s, of a particular magni- <br />tude. It is also defined as the reciprocal of the <br />probability of exceedence (Gumbel, 1954a; <br />Langbein, 1960, p. 48). Recurrence intervals <br />of hydrologic events are usually stated in years <br />and thus are reciprocals of probabilities of ex- <br />ceedence in one year. Further discussion of the <br />meaning of recurrence interval is given in the <br />section on "Interpretation of frequency curves." <br /> <br />. <br /> <br />Distributions Used in Hydrology <br /> <br />Hydrologists have long sought one theoretical <br />distribution that would describe flood events. <br />If there were such a universal distribution, the <br />observed distributions of flood events at various <br />sites would differ only in the parameters in that <br />universal distribution, and in sampling error. <br />Basin characteristics influence the distribution <br />of floods, so that it seems unlikely that anyone <br />theoretical distribution would be generally ap- <br />plicable. It is well established that the distri- <br />butions of annual minimum flows are highly <br />dependent on basin characteristics (Riggs 1965). <br />A sample of only 20 or 30 items may define <br /> <br />~ <br />o <br />=> <br />>- <br />z <br />'-' <br /><{ <br />" <br /> <br /> <br /> <br />(Right skew) <br /> <br />B <br /> <br />'( 0.5 <br />PROBABILITY OF EXCEEDANCE (A) <br />OR NONEXCEEDANCE (B) <br /> <br />~ <br />o <br />=> <br />>- <br />z <br />'-' <br /><{ <br />" <br /> <br /> <br />A <br /> <br />Mean <br /> <br />B <br /> <br />I( 0.5 <br />PROBABILITY OF EXCEEDANCE (A) <br />OR NONEXCEEOANCE (~ <br /> <br />Figure: 4.-Frequency curves showing effect of direction <br />of skew and direction of cumulation On position of <br />the mean with respect to the median. <br /> <br />a frequency curve which differs greatly from <br />the population frequency curve. Thus, fre- <br />quency curves based on small samples may <br />appear very dissimilar, and yet the correspond- <br />ing population frequency curves may be <br />similar. <br />As a consequence of the variability of char- <br />acteristics from basin to basin and of the <br />sampling variability in time, several theoreti- <br />cal distributions are used in hydrology. Char- <br />acteristics of the more common ones are <br />described below. In addition, graphically de- <br />fined distributions (those having no known <br />underlying formula) are widely used. <br /> <br />Normal Distribution <br /> <br />The equation of the normal probability <br />density curve is <br /> <br />f(X)=(I/...../'h) e-(X-",'/2,,', <br />