Statistics for Environmental Engineers

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Random Variable

The term random variable is widely used in statistics but, interestingly, many statistics books do not give a formal definition for it. A practical definition by Watts (1991) is “the value of the next observation in an experiment.” He also said, in a plea for terminology that is more descriptive and evocative, that “A random variable is the soul of an observation” and the converse, “An observation is the birth of a random variable.”

Experimental Errors

A guiding principle of statistics is that any quantitative result should be reported with an accompanying estimate of its error. Replicated observations of some physical, chemical, or biological characteristic that has the true value n will not be identical although the analyst has tried to make the experimental conditions as identical as possible. This relation between the value n and the observed (measured) value yt is yt = n + e, where ei is an error or disturbance.

Error, experimental error, and noise refer to the fluctuation or discrepancy in replicate observations from one experiment to another. In the statistical context, error does not imply fault, mistake, or blunder. It refers to variation that is often unavoidable resulting from such factors as measurement fluctuations due to instrument condition, sampling imperfections, variations in ambient conditions, skill of personnel, and many other factors. Such variation always exists and, although in certain cases it may have been minimized, it should not be ignored entirely.

Example 2.1

A laboratory’s measurement process was assessed by randomly inserting 27 specimens having a known concentration of ц = 8.0 mg/L into the normal flow of work over a period of 2 weeks. A large number of measurements were being done routinely and any of several chemists might be assigned any sample specimen. The chemists were ‘blind’ to the fact that performance was being assessed. The ‘blind specimens’ were outwardly identical to all other specimens passing through the laboratory. This arrangement means that observed values are random and independent. The results in order of observation were 6.9, 7.8, 8.9, 5.2, 7.7, 9.6, 8.7, 6.7, 4.8, 8.0, 10.1, 8.5,

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