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<br />n <br /> <br />Sx = { ~ <br /> <br />\' <br />L (Xi <br />i=l <br /> <br />} 1/2 <br />-2 <br />- x) <br /> <br />tv <br />~ Often, e~uation(2)is modified by replacing the divisor n by n-l <br />-;1 to make the sample standard deviation squared on unbiased esti- <br />o mate for! the population variance. <br /> <br />For the ~urposes of the generation performed here, means and standard <br />deviations are estimated for each month individually. The mean, <br />repre8entin~ the central tendency, and the standard deviation, measuring <br />the degr~e of dispersion, are two well-known properties frequently <br />used to characterize a data set. <br /> <br />Population monthly mean values are estimated for months 1 through 12 <br />using the expression: <br /> <br />1 <br />mj -L <br /> <br />L <br />'\ <br />L.- Xi,j <br />i-I <br /> <br />(j - 1,12) <br /> <br />In the above equation, L is the number of years, j is the month, and <br />Xi,j is the value observed in month j of year i. <br /> <br />Similarly, population monthly standard deviations, Sj' may be esti- <br />mated from: <br /> <br />s,. { 1 <br />J - <br />L <br /> <br />L <br />'\ <br />L (xi,j <br />i-l <br /> <br />1/2 <br />mj)2 } <br /> <br />(j - 1,12) <br /> <br />This series of 12 values is also tested to determine the existence of <br />a cyclic'pattern, <br /> <br />Yi,j - <br /> <br />Xi,j - mj <br />Sj <br /> <br />When the,monthly means and standard deviations have been examined, <br />the results may be useful in reducing the original set of data to a <br />second-order stationary series provided no trends, jumps, or other <br />abnormalities are present in the data. To this end, the adjustment <br /> <br />is made. This yields a residual set of data (Yi ) which has been <br />standardized, having an average of 0.0 and a staAdard deviation of <br />1.0 for each month, <br /> <br />7 <br /> <br /> <br />(2 ) <br /> <br />(3) <br /> <br />(4) <br /> <br />(5) <br />