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4 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . where j(X) is the probability-density function, X is the event described, and the parameters of the distribution are the mean, 1', and the variance, u2 (commonly reported as u, the standard deviation). The density curve is bell shaped and symmetrical; therefore the mean and median are the same. Values of X corre- sponding to various cumulative probabilities are tabulated in most statistics texts. Normal probability plotting paper, available commer- cially, is designed so that any cumulative normal distribution plots as a straight line on it. The range of the normal distribution is from minus to plus infinity. Lognormal distribution Lognormal distribution is a normal distribu- tion of In X (Naperian logarithm of X). In terms of X, it is a three-parameter distribution having a range from zero to plus infinity. The statistical parameters of X are given by Chow (1964). In practice, X is plotted on common- logarithmic probability paper, and the parame- ters given by Chow using N apierian logarithms do not apply. The lognormal distribution can be treated simply ss a normal distribution of logarithms, or in a complex manner as a skewed distribution of the untransformed data. Type I extreme-value distribution (Gumbel) Type I extreme-value distribution, the first asymptotic distribution (Gumbel, 1958), has two parameters, but it has a fixed skew of 1.139 and therefore is not symmetrical about the mean. Use of this distribution for annual floods wss proposed by Gumbel (1941). Powell (1943) prepared the plotting paper based on this distribution; the Geological Survey Form 9-179a is a slight modification of Powell's plot. The mean of the distribution occurs at the 2.33-year recurrence interval when the distribu- tion is cumulated from the upper end. Chow (1954) shows by his table 3 that for practical purposes the extreme-value law is but a special esse of the log-probability law. Type III extreme-value distribution Type III extreme-value distribution is also called the Weibull distribution after the man who first used it in analysis of strength of materials. Gumbel (1954b) hss applied it to drought-frequency analysis. The distribution has three parameters, a lower limit (which may be zero or a finite value greater than zero), a characteristic value which hss a recurrence in- terval of 1.58 years, and a parameter which defines skewness. Pearson Type 111 distribution The Pearson Type III distribution is a flex- ible distribution in three parameters with a limited range in the left direction and unlim- ited range to the right. Plotting paper is not available for this distribution because skewness varies. This distribution is commonly fitted to the logarithms of flood magnitudes rather than to the magnitudes themselves because this re- sults in a smaller skew. The Pearson Type III distribution having zero skew is identical to the normal distribution. The gamma distributions, sometimes used in hydrology, are in effect the Type III curves of Pearson (Mood, 1950, p. 118). . Graphically defined distributions Distributions defined by graphical means may conform to some theoretical distribution but ordinarily do not. Mathematical Curve, Fitting Normal distribution Compute mean and standard deviation of sample. Using these, the detailed characteris- tics of the distribution can be extracted from a table of the cumulative normal distribution. For example, suppose the mean and standard deviation are computed ss 100 and 20, respec- tively. We sssume that these sample parame- ters, X and S, are equal to their respective population parameters, I' and q. Then the mag- nitude of the variable at selected levels, call it X., can be computed from I' and q; and the probabilities that a random value of X will be less than these values of X. are taken from table A-4, page 382, Dixon and Massey (1957). In this table, "area" is equivalent to probs- .