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2 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS 8 8 14 20 MAGNITUOE 26 32 32 . 26 w Cl ::> ~ 20 '-' " '" 14 8 0.9 0.8 0.7 0.5 0.3 0.2 0.1 PROBABILITY THAT A RANDOMLY DRAWN INDIVIDUAL WILL EXCEED THE INDICATED MAGNITUDE A 8 c Figure 1.-Two normal distributions end their cumulative forms. 10 15 20 MAGNITUDE A. Right skewed B. Normal C. Left skewed 30 30 w Cl ::> ~ 15 '-' " '" . o 0.9 0.8 0.7 0.5 0.3 0.2 0.1 PROBABILITY THAT A RANDOMLY DRAWN INDIVIDUAL WilL EXCEED THE INDICATED MAGNITUDE Figure 2.-Normal and skewed distributions and their cumulative forms on a normal-probability plot. distributions. The relative positions of the mean, median, and mode for skewed distri- butions are shown in figure 3. Only the median value can be determined from the cumulative plot. The position of the mean with respect to the median on the cumulative plot depends on the degree of skewness, the direction of skew- ness, and the direction in which the frequency distribution is cumulated. For example, the mean of a particular right (positive)-skewed distribu tion will be exceeded 43 percent of the time; but 57 percent of the time it will not be exceeded. Thus, if the distribution is cumulated from the high end, the mean is to the right of the median; if cumulated from the low end, the mean is to the left of the median. These relations are reversed for a left-skewed distri- bution. Figure 4 illustrates the relations. The probability scales of the two plots of figure 4 are different. Each is designed for the particular distribution plotted. Frequency curves of a time series commonly relate magnitude to recurrence interval or re- turn period instead of to probability of exceed- ence or nonexceedence. Recurrence interval is the average length of time between exceeden- .