# Statistics for Environmental Engineers

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Imagine ways in which the errors of the nitrate measurements might be nonrandom. Suppose, for example, that the measurement process drifted such that early measurements tended to be high and later measurements low. A plot of the errors against time of analysis would show a trend (positive errors followed by negative

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FIGURE 2.7 Plot of nitrate measurement errors indicates randomness.

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FIGURE 2.8 Plot of nitrate residuals in order of sample number (not order of observation) and differentiated by chemist.

errors), indicating that an element of nonrandomness had entered the measurement process. Or, suppose that two different chemists had worked on the specimens and that one analyst always measured values that tended too high, and the other always too low. A plot like Figure 2.8 reveals this kind of error, which might be disguised if there is no differentiation by chemist. It is a good idea to check randomness with respect to each identifiable factor (day of the week, chemist, instrument, time of sample collection, etc.) that could influence the measurement process.

Independence means that the simple multiplicative laws of probability work (that is, the probability of the joint occurrence of two events is given by the product of the probabilities of the individual occurrence). In the context of a series of observations, suppose that unknown causes produced experimental errors that tended to persist over time so that whenever the first observation y1 was high, the second observation y2 was also high. In such a case, y1 and y2 are not statistically independent. They are dependent in time, or serially correlated. The same effect can result from cyclic patterns or slow drift in a system. Lack of independence can seriously distort the variance estimate and thereby make probability statements based on the normal or t distributions very much in error.

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