Statistics for Environmental Engineers

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Making the reference distribution is conceptually and mathematically simple. No particular knowledge of statistics is needed, and the only mathematics used are counting and simple arithmetic. Despite this simplicity, the concept is statistically elegant, and valid judgments about statistical significance can be made.

Constructing an External Reference Distribution


The first 130 observations in Figure 6.1 show the natural background pH in a stream. Table 6.1 lists the data. Suppose that a new effluent has been discharged to the stream and someone suggests it is depressing the stream pH. A survey to check this has provided ten additional consecutive measurements: 6.66, 6.63, 6.82, 6.84, 6.70, 6.74, 6.76, 6.81, 6.77, and 6.67. Their average is 6.74. We wish to judge whether this group of observations differs from past observations. These ten values are plotted as open circles on the right-hand side of Figure 6.1. They do not appear to be unusual, but a careful comparison should be made with the historical data.


The obvious comparison is the 6.74 average of the ten new values with the 6.80 average of the previous 130 pH values. One reason not to do this is that the standard procedure for comparing two averages, the t-test, is based on the data being independent of each other in time. Data that are a time series, like these pH data, usually are not independent. Adjacent values are related to each other. The data are serially correlated (autocorrelated) and the t-test is not valid unless something is done to account for this correlation. To avoid making any assumption about the structure of the data, the average of 6.74 should be compared with a reference distribution for averages of sets of ten consecutive observations.

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