# Statistics for Environmental Engineers

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The elegantly simple dot diagram and the time series plot have beautifully described the data. No numerical summary could transmit the same information as efficiently and clearly. Assuming a “normallike” distribution and reporting the average and standard deviation would be very misleading.

### Probability Plots

A probability plot is not needed to interpret the data in Table 5.1 because the time series plot and dot diagrams expose the important characteristics of the data. It is instructive, nevertheless, to use these data to illustrate how a probability plot is constructed, how its shape is related to the shape of the frequency distribution, and how it could be misused to estimate population characteristics.

The probability plot, or cumulative frequency distribution, shown in Figure 5.4 was constructed by ranking the observed values from small to large, assigning each value a rank, which will be denoted by i, and calculating the plotting position of the probability scale as p = i/ (n + 1), where n is the total number of observations. A portion of the ranked data and their calculated plotting positions are shown in Table 5.2. The relation p = i/(n + 1) has traditionally been used by engineers. Statisticians seem to prefer p = (i — 0.5)/n, especially when n is small.
2 The major differences in plotting position values computed from these formulas occur in the tails of the distribution (high and low ranks). These differences diminish in importance as the sample size increases.

Figure 5.4(top) is a normal probability plot of the data, so named because the probability scale (the ordinate) is arranged in a special way to give a straight line plot when the data are normally distributed. Any frequency distribution that is not normal will plot as a curve on the normal probability scale used in Figure 5.4(top). The abcissa is an arithmetic scale showing the BOD concentration. The ordinate is a cumulative probability scale on which the calculated p values are plotted to show the probability that the BOD is less than the value shown on the abcissa.

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