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. FREQUENCY CURVES By H. C. Riggs Abstract This manual describes graphical and mathematical procedures for preparing frequency curves from salU- pIes of hydrologic data.. It also discusses the theory of frequency curves, compares advantages of graphical and mathematical fitting, suggests methods of describ- ing graphically defined frequency curves analytically, and emphasizes the correct interpretations of a fre- quency curve. Introduction . A frequency curve relates magnitude of a variable 1,0 frequency of occurrence. The curve is an estimate of the cumulative distribution of the population of that variable and is pre- pared from a sample of data. Frequency curves have many uses in hydrol- ogy. Flood-frequency curves are widely used in the design of bridge openings, channel capaci- tIes, and ro~dbed elevations; for flood-plain zonmg; and ill studies of economics of flood- protection works. Frequency curves of annual low flows are used in design of industrial and domestic water-supply systems, classification of streams as to their potential for waste dilution definition of the probable amount of wate; available for supplemental irrigation, and main- tenance of certain channel discharges as re- quired by agreement or by law. Frequency curves of annual mean flows are sometimes used in studies of the carryover of annual storage (Beard, 1964). Frequency curves also provide a means of classifying data for use in subsequent analyses. For example, Benson (1962a) used intensity of ra.mfall for a given frequency in his regional flood-frequency analysis for New England and Riggs (1953) used a frequency curve of r~noff in excess of assured flow in a forecasting prob- lem. Many other applications have been and can be made. . Cumulative Distributions Book 4, chapter Al of the series of Techniques of W~ter-Resources Investigations (Riggs, 1967) descnbes the relation of a frequency distribu- tlOn or probability density curve to its cumu- lative form. A more detailed examination of this relation helps in understanding the cumu- lative distribution, or frequency curve. We begin with the two normal distributions shown in figure 1. Their cumulative forms can be ex- pressed as straight lines by use of the special abscissa scale which is deri ved from the charac- teristics of the normal distribution. Both dis- tributions have the same median value, 20, and these medians plot at 0.5 probability on the cumulative graph. The variability of a distri- bution is indicated by the slope of the cumu- lative distribution; that is, the greater the variability, the greater the slope. The standard deviation is half the difference between magni- tudes at probabilities of 0.16 and 0.84 (Dixon and Massey, 1957, table A -4). Many frequency distributions are nonsym- metrical. For such distributions, the mean, median, and mode have different values which consequently correspond to different probabili- ties on the cumulative graph. A nonsymmetrical distribution is classified as skewed. Skewness may be shown graphically as right or left; it, may be described mathematically by a number, either positive or negative. Two skewed disl,ri- butions and a symmetrical distribution are shown in figure 2, which also shows the corre- sponding cumulative distributions (frequency curves) . For a normal, or any symmetrical, distribu- tion the mean and median are the same value. Thus, the value corresponding to the proba- bility of 0.5 on the cumulative frequency curve is the mean as well as the median for such 1