Statistics for Environmental Engineers

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a X(y,-n)2 a = N


The standard deviation of the population is a measure of spread that has the same units as the original measurements and as the mean. The standard deviation is the square root of the variance:


a


X(yirj)2 N


The true values of the population parameters a and a2 are often unknown to the experimenter. They can be estimated by the sample variance:


„2 = X(y,-y)2 s = n-1


where n is the size of the sample and y is the sample average. The sample standard deviation is the square root of the sample variance:


s


X(y,-y)2


n-1


Here the denominator is n — 1 rather than n. The n — 1 represents the degrees of freedom of the sample. One degree of freedom (the -1) is consumed because the average must be calculated to estimate s. The deviations of n observations from their sample average must sum exactly to zero. This implies that any n — 1 of the deviations or residuals completely determines the one remaining residual. The n residuals, and hence their sum of squares and sample variance, are said therefore to have n — 1 degrees of freedom. Degrees of freedom will be denoted by the Greek letter v. For the sample variance and sample standard deviation, v = n — 1.


Most of the time, “sample” will be dropped from sample standard deviation, sample variance, and sample average. It should be clear from the context that the calculated statistics are being discussed. The Roman letters, for example s2, s, and y , will indicate quantities that are statistics. Greek letters (a2, a, and n) indicate parameters.

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