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<br />.
<br />
<br />FREQUENCY CURVES
<br />
<br />5
<br />
<br />bility. The last two columns of the following
<br />table are computed.
<br />
<br /> Recurrence
<br />X. P(X<X.) P(X>X.) intervaJof
<br /> eJ:ceedence
<br />,.-2.00'''' 60 0.02 0.98 1.02
<br />~-1..5...= 70 .07 .83 1.08
<br />,.-1.0...= 80 .10 .S< l.Hl
<br />p.- .8el'_ 84- .21 .79 1.27
<br />,,- .6.,.= 90 .31 .69 1.'-'
<br />,..- .2.,."" 96 .42 .68 1.72
<br />11 _100 .60 .60 2.00
<br />"'+ .2...=104 .68 .42 2.83
<br />.1&+ .5cr=110 .69 .31 3.22
<br />...+ .8.,._116 .79 .'1 4.76
<br />,.+1.0v=12O .S< .16 6.25
<br />,.+1.5cr...180 .83 .07 14.4
<br />,.+2.0.,..,140 .OS .02 5<>.0
<br />
<br />If the sample is drawn from a time series of
<br />annual values, the computed recurrence inter-
<br />val is in years. The same results could have
<br />been obtained from a plot on normal probability
<br />paper. The mean is plotted at 0.5 probability,
<br />the standard devit'ion is plotted plus and minus
<br />from the mean at probabilities of 0.16 and 0.84,
<br />respectively, a straight line is drawn through
<br />the plotted points, and probabilities at selected
<br />levels are read from the line.
<br />
<br />.
<br />
<br />Three-parameter distributions
<br />
<br />Compute mean, X, standard deviation, S,
<br />and skew coefficient, C" by the following
<br />equations:
<br />
<br />X=2:,X/N
<br />S,_2:,X'-(2:,X)'/!!
<br />N-1
<br />
<br />G _ N'2:,X"-3NL,XL,X"+2(2:,X)3
<br />,- N(N-1)(N-2)S3 '
<br />
<br />where X is the magnitude of an event, and
<br />N is the number of events in the sample. These
<br />sample parameters are treated as though they
<br />were the population parameters in fitting a
<br />distribution. These parameters could be sub-
<br />stitu ted in the formula for the distribution to
<br />be used, but the distribution cannot be inte-
<br />grated directly. Therefore, the relation between
<br />magnitude and probability of exceedence (or
<br />nonexceedence) is commonly determined from
<br />a table of frequency factors for the chosen
<br />. distribution and from the general formula
<br />
<br />X=X+KS,
<br />
<br />where X is the variable, X is the mean of the
<br />sample, S is the standard deviation of the
<br />sample, and K is the frequency factor. For
<br />example, in the table under the section on
<br />"Normal distribution," the coefficients of cr in
<br />the first column are frequency factors, K, for
<br />the normal distribution.
<br />Frequency factors for the lognormal distri-
<br />bution are given by Chow (1964, p. 8--26).
<br />Recurrence interval is the reciprocal of the
<br />probability given in the table. A table by Hazen
<br />(1930) has been widely used, but it was devel-
<br />oped empirically and contains some values
<br />which differ from the theoretical ones. Chow's
<br />table shows a definite theoretical relation be-
<br />tween the coefficient of variation, C" defined as
<br />
<br />G,=SjX
<br />
<br />and the coefficient of skew, C,. Values of C,
<br />and C. computed from a sample will rarely be
<br />related according to theory because the coeffi-
<br />cient of skew computed from a few items is
<br />notably unreliable. If C, and C, define a much
<br />different relation than prescribed by theory,
<br />the lognormal distribu tion may provide poor
<br />fit to the data. Matalas and Benson (personal
<br />commun., 1964) show the standard error of the
<br />skew coefficient for N from 4 to 100.
<br />Frequency factors for the Pearson Type III
<br />distribution, adapted from a table by Beard
<br />(1962), are given in table 1. Chow (1964) shows
<br />
<br />Table 1.-Frequency factors for Pearson Type III distributions
<br />lFrOlIl Beard, 19621
<br />
<br />C'
<br />
<br />Reeummce interval (years)
<br />
<br />100 20
<br />
<br />10 3.33
<br />
<br />2
<br />
<br />1.43 1.11 1.<Hi
<br />
<br />1.0 3.03 1.87 1.il< 0.83 -0.16 -0.61 -1.12 -un
<br />.S 2.90 1.83 1.34 .42 -.13 -.60 -1.16 -1,38
<br />.6 ,77 1.79 1.33 .'-' -." -.69 -1.19 -1.46
<br />.. ... 1.7. 1.32 ... -." -.67 -1.22 -1.61
<br />.2 2." 1.69 1.30 .51 -.03 -." -1.25 -1.68
<br />0.0 .33 1." 1.2S ." 0.00 -,62 -1.28 -1.84
<br />-.2 2.18 I.5S 1." ." .03 -.51 -1.30 -1.69
<br />-.' 2.03 l.lU 1.22 .57 .06 -." -1.32 -1.7.
<br />-.' 1.88 1.'-' 1.19 .69 .09 -.'-' -1.83 -1.7g
<br />-.S 1.74 1.83 1.16 .60 .13 -.<2 -1.34 -1.83
<br />-1.0 1.69 1.31 1.12 .61 .1' -.83 -1.14 -1,87
<br />
<br />the plotted relation of K to recurrence interval,
<br />T, for the Pearson Type III distribution for 5
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