Statistics for Environmental Engineers

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Example 8.4


A sample of n = 575 daily BOD observations is available to estimate the 99th percentile by the nonparametric method for the purpose of setting a maximum limit in a paper mill’s discharge permit. The 11 largest ranked observations are:


Rank 575    574    573    572    571    570    569    568    567    566    565


BOD    10565    10385    7820    7580    7322    7123    6627    6289    6261    6079    5977


The 99th percentile is located at observation number p(n + 1) = 0.99(575 + 1) = 570.24. Because this is not an integer, interpolate between the 570th and 571st largest observations to estimate


y 0.99 = 7171.


The disadvantage of this method is that only the few largest observed values are used to estimate the percentile. The lower values are not used, except as they contribute to ranking the large values. Discarding these lower values throws away information that could be used to get more precise parameter estimates if the shape of the population distribution could be identified and used to make a parametric estimate.


Another disadvantage is that the data set must be large enough that extrapolation is unnecessary. A 95th percentile can be estimated from 20 observations, but a 99th percentile cannot be estimated with less than 100 observations. The data set should be much larger than the minimum if the estimates are to be much good. The advisability of this is obvious from a probability plot, which clearly shows that greatest uncertainty is in the location of the extreme quantiles (the tails of the distribution). This uncertainty can be expressed as confidence limits.


The confidence limits for quantiles that have been estimated using the nonparametric method can be determined with the following formulas if n > 20 observations (Gilbert, 1987). Compute the rank order of two-sided confidence limits (LCL and UCL):

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