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2 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . frequencies indicates that the binomial distri- bution is applicable to this problem. Table 1.-Re:sults of dice-tossing [AI.... Kendall ('''')J experiment No. of Fro- Relative sixes quency frequency Theoretical relative frequency Oun..n.... 447 0, 109 0.112 L'.Uh..n. 1,145 .280 .269 2u.__.n.... 1, 181 ,288 .296 3U.h.n.n. 796 ,194 .197 4un..n.n. 380 ,093 .089 5...n..u.n 115 ,028 .029 6..n...un. 24 .006 ,007 7 and over____ 8 .002 .001 TotaLn..n 4,096 1.00 1.00 The binomial distribution is a discrete dis- tribution, that is, it can take values only at specific points along a scale. In the dice-tossing experiment, it is possible to obtain an integer number of sixes only; there is no such thing as 5.5 or 3.2 sixes. More commonly, a variable may take any value along a scale. Such a variable and its distribution are known as continuous. A variable may be classified as continuous if it can take any value along a scale even though the limitations of measurement restrict the observations to discrete values. This condition exists with most natural phenomena. To aid in understanding a distribution, con- sider 1,000 tree-ring indices ranging in size from 2 to 240. If these are grouped by six-unit in- crements of size, a histogram, or frequency distribution, is obtained (fig. 1). The irregu- larity of the profile of this distribution is due to the small (in a statistical sense) number of indices used in its preparation. The greater the number used, the smoother would be the profile of the frequency distribution. If the number of observations approaches infinity and the size increment approaches zero, the enveloping line of the frequency distribution will approach a smooth curve. Then if the ordinat6 values are divided by a number such that the area under the curve becomes one, the resulting curve is a probability density curve, or probability dis- tribution, such as figure 2. The process just described requires the additional assumption that the variable can take any value within the range, that the variable is continuous, not discrete. 80 '" '" 260 D ~ ~40 '" '" '" " ~o => z o o 30 60 90 120 150 180 210 240 WIDTH INDEX Figure 1.-Histogram, or frequency distribution, of 1,000 tree-ring indices. ~l I- ::; a; "" '" o '" Q. o 30 60 90 120 150 180 210 240 WIDTH INDEX .1 Figure: 2.~robability density curve of 1,000 tree-ring indices. A theoretical probability distribution de- scribes the relation between size (or some other other characteristic) and probability. For this relation to be valid, the individuals must occur randomly or be drawn randomly. The size of any individual drawn should not depend on the size of anyone previously drawn, Probabil. ity, in the concept of frequency distributions, is defined as relative frequency. The distribu. tion of the number of sixes obtained from ree peated tosses of 12 dice may be illustrated by plotting the theoretical relative frequencies of table 1. The relative frequencies of each of the 6.unit increments in figure 1 could likewise be com'puted. In the first of these examples, a probability is associated with each possible outcome. In the second, a probability is associ. ated with each increment of size; here the probability is of obtaining not a specific indi- vidual but any individual within the increment of size. This interpretation is required for con- tinuous distribution's because there is an infinite number of possible values and, thus, no proba- bility of occurrence of a particular individual. .