Statistics for Environmental Engineers

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7.6    32    5.0    4.2    14    18    2.3 52    10    3.3    38    3.4    4.3    0.05 0.05 0.10

0.10    0.05    0.05    0.05    0.0    0.05    1.2    0.10    0.10    0.10    0.10    0.10    0.23    4.4    0.42 0.10

16.    2.0    2.0    1.0    3.2    0.43    1.4    0.10    5.9    0.10    0.10    0.23    0.29    5.3    5.7    0.10

7.4    Are Transformations Necessary? Which of the following are correct reasons for transforming data? (a) Facilitate interpretation in a natural way. (b) Promote symmetry in a data sample.

(c) Promote constant variance in several sets of data. (d) Promote a straight-line relationship between two variables. (e) Simplify the structure so that a simple additive model can help us understand the data.

7.5    Power Transformations. Which of the following statements about power transformations are correct? (a) The order of the data in the sample is preserved. (b) Medians are transformed to medians, and quartiles are transformed to quartiles. (c) They are continuous functions. (d) Points very close together in the raw data will be close together in the transformed data, at least relative to the scale being used. (e) They are smooth functions. (f) They are elementary functions so the calculations of re-expression are quick and easy.


Estimating Percentiles

KEY WORDS confidence intervals, distribution free estimation, geometric mean, lognormal distribution, normal distribution, nonparametric estimation, parametric estimation, percentile, quantile, rank order statistics.

The use of percentiles in environmental standards and regulations has grown during the past few years. England has water quality consent limits that are based on the 90th and 95th percentiles of monitoring data not exceeding specified levels. The U.S. EPA has specifications for air quality monitoring that are, in effect, percentile limitations. These may, for example, specify that the ambient concentration of a compound cannot be exceeded more often than once a year (the 364/365th percentile). The U.S. EPA has provided guidance for setting aquatic standards on toxic chemicals that require estimating 99th percentiles and using this statistic to make important decisions about monitoring and compliance. They have also used the 99th percentile to establish maximum daily limits for industrial effluents (e.g., pulp and paper). Specifying a 99th percentile in a decision-making rule gives an impression of great conservatism, or of having great confidence in making the “safe” and therefore correct environmental decision. Unfortunately, the 99th percentile is a statistic that cannot be estimated precisely.

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