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<br />" <br /> <br />fraction of time the magnitude was exceeded historically. For example, if the mean <br />daily discharge at a given location exceeds 80 cfs in 6015 of 8766 days, the relative <br />frequency is 0.68. The estimated probability of exceedance of 80 cfs is 0.68. <br /> <br />The reliability of a relative frequency model improves as the sample size <br />increases; with an infinite sample size, relative frequency equals exactly probability. <br />Unfortunately, sample sizes available for streamflow frequency analysis are small by <br />scientific standards. Thus, relative frequency generally is not a reliable estimator of <br />probability for hydrologic engineering purposes. <br /> <br />The alternative to the empirical relative frequency model is a theoretical <br />frequency model. With a theoretical model, the relationship of magnitude and <br />probability for the parent population is hypothesized. The relationship is represented <br />by a frequency distribution. A cumulative frequency distribution is an equation that <br />defines probability of exceedance as a function of specified magnitude and one or <br />more parameters. An inverse distribution defines magnitude as a function of specified <br />probability and one or more parameters. <br /> <br />2.3. DISTRIBUTION SELECTION AND PARAMETER ESTIMATION. In' <br />certain scientific applications, one distribution or another may be indicated by the <br />phenomena of interest. This is not so in hydrologic engineering applications. Instead, <br />a frequency distribution is selected because models well the data that are observed. <br />The parameters for. the model are selected to optimize the fit. A graphical or numerical <br />technique can be used to identify the appropriate distribution and to estimate the <br />parameters. <br /> <br />3. GRAPHICAL TECHNIOUES. <br /> <br />Some of the early and simplest methods of frequency analysis were <br />graphical techniques. These techniques permit inference of the parent population <br />characteristics with a plot of observed magnitude vs. estimated exceedance probability <br />of that data. If a best-fit lines drawn on the plot, the probability of exceeding various <br />magnitudes can be estimated. Also, any desired quantiles can be estimated. <br />Graphical representations also provide a useful check of the adequacy of a <br />hypothesized distribution. <br /> <br />3.1. PLOTTING-POSITION ESTIMATES OF PROBABILITY. Graphical <br />techniques rely on plotting positions to estimate exceedance probability of observed <br />events. The median plotting position estimates the exceedance probability as: <br /> <br />7-92 <br />