This is arithmetic average of the x values and is usually referred to simply as the mean.

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1 prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout summarizes the formulas for computig the statistics (sample estimators) commoly used i describig of raifall or streamflow populatios. These textbook formulas are useful for had calculatios, but you should ote that roudoff problems (particularly the iability to retai a large eough umber of sigificat digits) may lead to sigificat errors i the results. Properly desiged computer algorithms avoid these problems. If possible, I recommed the use of a major statistics package (StatView, DataDesk, JMP, Systat or MINITAB o the Macitosh; MINITAB, SPSS, SAS, or Systat o other platforms) to carry out these calculatios. Usig spreadsheets for statistics is ot geerally advisable because of roudoff errors ad use of iadequate algorithms. KEY TO SYMBOLS USED: x P k x i x i total umber of observatios sample mea of x sample stadard deviatio of x k'th percetile (value of x for which k % of the data are smaller) sum of values of x from x 1 to x sum of squared x values from x 1 to x sum of cubed x values from x 1 to x product of x values from x 1 to x I. MEASURES OF CENTRAL TENDENCY These statistics are measures of where the data distributio is cetered, i.e., they attempt i some way to describe quatitatively where the "middle" of the data lies. If the uderlyig data distributio is symmetrical, all these measures will give essetially the same result. Where the uderlyig distributio is asymmetric (skewed), each of these measures will yield a differet value. a. sample arithmetic mea x = This is arithmetic average of the x values ad is usually referred to simply as the mea. b. sample geometric mea x g = 1 This is the th root of the product of terms. Note that the logarithm of the geometric mea is equal to the arithmetic mea of the logarithms of the idividual x values.

2 c. sample harmoic mea x h = 1 xi This is the reciprocal of the mea value of the reciprocals of idividual values. d. sample media x md = 50th percetile of data This is the middle value of a data series: half of the values are larger tha this umber, ad half are smaller. If the data series cosists of a eve umber of values, the media is the average of the two middle values. e. sample mode x mo = most frequet value This is the value which occurs most frequetly i a data series. If there is o most frequet value, the data series is modeless. If there are two or more most frequet values, the distributio is bimodal or multimodal. The arithmetic mea is the most commoly used measure of cetral tedecy o accout of its computatioal simplicity ad geeral samplig stability. The US Weather Bureau uses the mea as as the precipitatio ormal. However, i sigificatly skewed distributios the mea may be misleadig ad the media is a better idicator of the ceter of the distributio. II. MEASURES OF VARIABILITY OR DISPERSION a. sample stadard deviatio = - x - 1 = - The sample stadard deviatio is the square root of the mea squared differece betwee each observatio ad the sample mea; it is defied by the left-most formula above. This formula is awkward for had computatios, but miimizes roudoff errors. The equatio is usually reorgaized ito the form o the right for had calculatios. - 1 b. sample variace = square of the sample stadard deviatio

3 c. graphical stadard deviatio S gr = P 84 - P 16 The graphical stadard deviatio is a quick approximatio of the sample stadard deviatio based o the cumulative curve. P 84 ad P 16 are the 84th ad 16th percetiles of the data as read from the cumulative frequecy curve plotted o probability paper P P % idicated value d. sample rage sample rage = x max -x mi The sample rage is the differece betwee the maximum ad miimum values i a data set. e. sample coefficiet of variatio C v = x The sample coefficiet of variatio is the ratio of the sample stadard deviatio to the sample mea. It is useful i comparig data sets where the variability of the data icreases markedly with a icrease i the mea.

4 III. MEASURES OF ASYMMETRY OR SKEWNESS a. sample skewess a = x = x + x The sample skewess measures the degree to which a distributio is asymmetric. Symmetric distributios have zero skewess. Distributios with a tail to the right yield positive (+) values of skew, while those with a left tail yield egative ( ) values. Values of raw skewess are ofte very large ad hard to iterpret; the sample coefficiet of skewess, below, is more commoly used. b. sample coefficiet of skewess C s = a The sample coefficiet of skewess is the ratio of the sample skewess to the cube of the sample stadard deviatio. For symmetrical distributios C s = 0. If C s > 0, the distributio is asymmetric with a tail extedig to the right (right skew); if C s < 0, the distributio is asymmetric with a tail extedig to the left (left skew). The larger the absolute value of C s, the more asymmetric the distributio. c. Pearso's sample skewess S k = x - x mo This is aother commoly used measure of skewess. It is much easier to calculate by had tha the ordiary sample skewess. mea media mode symmetrical distributio: skew = 0

5 mode media mea asymmetric distributio: positive (right) skew IV. CONFIDENCE LIMITS a. (1- ) x 100% cofidece limits o the mea CI = x ± t 1,α / t 1,α / is the value of the t-statistic from tables, where -1 is the degrees of freedom ad (α/) is typically 0.05 (i.e, α = 0.05). If is "large eough" (typically 5-0) we ca substitute z α/ i place of the t - value. Cofidece limits o the mea allow us to assess the degree of ucertaity likely i our usig the sample mea x to estimate the populatio mea µ. For example, if we costruct a (1-α) = 95% cofidece iterval aroud the sample mea, we ca have 95% cofidece that the true populatio mea lies i that iterval. That is, 95 times out of 100 a iterval costructed from our sample i this fashio will cotai the true populatio mea.