Averaging should be carried out for each sampling time. Again, only in the most carefully
designed and executed studies is such information available. More typically, an analyst may have
three to seven lateral measurements along a transect to compare with fixed monitor information.
In this case, the analysis can be performed, but the analyst must be aware that those limited data
lessen the weight that may be given to any conclusions.
Statistical Comparison
At this point, the verification data set should contain n pairs of data (Xm,j , Xs,j), each
containing a monitor observation Xmj and an average stream value Xsj for time j, where m and s
indicate that the observation came from the monitor or stream, respectively. Next, one tests the
relationship between the two locations using a paired t-test (following Hines and Montgomery
1980). This test assumes that the samples each come from a normally distributed, independent
distribution. However, moderate departures from normality should not adversely affect the
analysis (Pollard 1977). The difference between each pair of observations, Dj = Xmj Xsj , should
come from a normally distributed independent distribution.
To verify that the data come from a normally distributed population, either of two methods
can be used. The easiest method is to plot the data on normal probability paper or use a
statistics software package to generate a normal probability graph. A second method is to use a
quantitative test such as the Kolmogorov-Smirnov test or Lillefore's test. Further details of these
tests can be found in Hines and Montgomery (1980) and Pollard (1977). Within this technical
note, normal plots are used; these were generated using SPSS (SPSS, Inc., Chicago, IL), a
statistical analysis software package.
Once it has been determined that the data come from a normal or nearly normal distribution,
one can begin the comparison by stating the hypotheses. The null hypothesis is that the mean of
the differences between pairs, D , is zero. This implies that monitor value agrees with stream
values and is representative. The alternative hypothesis is that the mean of the differences is not
zero; that is, the monitor values do not agree with stream values and are not representative. This
is stated as follows:
H0 : D = 0
(2)
H1: D ≠ 0
(3)
These hypotheses are tested with the following statistic:
D
t0 =
,
SD
n
(4)
where
n
∑D
j
j =1
D=
,
(5)
n
4
Water Quality Technical Note AM-03 (January 1998)
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