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<br />. <br /> <br />FREQUENCY CURVES <br /> <br />7 <br /> <br />are shown to indicate the fit of the computed <br />line. <br /> <br />Type III extreme-value distribution <br />Examples of fitting this type of distribution <br />to low-flow data are given by Gumbel (19Mb). <br /> <br />Graphical Fitting <br /> <br />Graphical fitting requires no assumption as <br />to the type or characteristics of the distribu- <br />tion. In the graphical method, each individual <br />in the sample is assigned a probability or re- <br />currence interval. Then magnitudes of the in- <br />dividuals are plotted against probabilities or <br />recurrence intervals, and a line is drawn to <br />properly interpret the points. <br />Assignment of probabilities is by means of <br />a plotting-position formula. Various formulas <br />may be used, each based on a different assump- <br />tion as to the characteristics of the sample. <br />Langbein (1960) relates the better-known plot- <br />ting-position formulas to their underlying as- <br />sumptions. Benson (1962b) compares the ro- <br /> <br />. ~ 1400 <br /> 0 <br /> " <br /> ~ <br /> "' <br /> 0: <br /> ~ 1200 <br /> t;; <br /> w <br /> ~ <br /> ~ 1000 <br /> :;) <br /> " <br /> ~ <br /> 0 <br /> "' soo <br /> 0 <br /> z <br /> <( <br /> "' <br /> :;) <br /> 0 <br /> I 600 <br /> I- <br /> '!O <br /> w <br /> " <br /> 0: 400 <br /> <( <br /> I <br />l " <br />"' .. <br />Ci <br />" <br /> 8 200 <br /> ~ <br /> ~ <br /> ~ <br /> <( <br /> :;) <br /> z 0 <br /> z 1.01 1.2 <br /> <( <br /> <br /> <br /> <br />sults of using various plotting positions on the <br />economics of engineering planning. The Geo- <br />logical Survey uses the formula <br /> <br />T=1/p=(n+1)/m, <br /> <br />where T is recurrence interval in years, p is <br />probability of an exceedence in anyone year, <br />n is the number of items in the sample, and <br />m is the order number of the individual in the <br />sample array (Dalrymple, 1960). Upper case <br />symbols, P, N, and M are often used alterna- <br />tively. The sample data may be arrayed- <br />arranged in order of magnitude-beginning <br />with the largest as No.1, or with the smallest <br />as No.1, according to whether the frequency <br />curve is to describe the probability of exceed- <br />ence or of nonexceedence. A distribution curve <br />can be cumulated from either end, and in the <br />graphical method this effect is accomplished <br />by selecting the direction in which the data <br />are arrayed. <br />The next step is plotting magnitude against <br />recurrence interval (or probability) on a graph. <br />If arithmetic coordinates are used, an S-curve <br /> <br />. <br /> <br /> <br />2 <br /> <br />5 10 <br />RECURRENCE INTERVAL, IN YEARS <br /> <br />20 <br /> <br />50 <br /> <br />100 <br /> <br />. <br /> <br />Figure: S.--Gumbel frequency curve of annual floods on Columbia River near The Oolles, Oreg., showing agreement with the <br />plotted points. <br />