Statistics for Environmental Engineers

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How Much Testing and Sampling are Needed?


The variance of an average of n independent observations is obtained by dividing the variance of the data by n. Also, the variances of summed independent variances are additive. If we take ns specimens from a batch and do nt tests on each specimen, the total number of measurements used to calculate the batch average is nsnt and the variance of the batch average is:


_2 2 at + a t


ns    nsn


The reason at is divided by nn and not by nt is that the total of nsnt measurements would be averaged.


Some Possible Sampling and Testing Schemes with Their Variance and Cost


Scheme


Number of Specimens


(ns)


Number of Tests per Specimens («,)


Variance


Total Cost ($)


1


1


1


17.3


21


2


5


1


4.5


25


3


8


1


3.3


28


4


16


1


2.3


36


5


32


2


1.1


72

Computed using of = 1.3, a2, = 16.0, Cs = $1, and Ct = $20.


Let us experiment with this formula using a] = 0.5 and a] = 5.0. The variance of the average of results coming from nt tests on each of ns specimens is a2 = П0 + П.5-. If we take one specimen and test it once, then ns = 1 and nt = 1 and a2 = 50 + 05 = 5.5 and a = 2.34. If instead we test this one specimen ten times, then ns = 1 and nt = 10, giving a2 = — + и = 5.05 and a = 2.24. The nine extra tests reduced the standard deviation by only 4%. This is hardly a worthwhile investment.

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