# Statistics for Environmental Engineers

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1.    Generate four random, independent, normally distributed numbers having n = 2 and a = 1.

2.    Transform the normal variates into lognormal variates x = exp( у).

3.    Average the four values to estimate the 4-day average ( x4).

4.    Repeat steps 1 and 2 one thousand times, or until the distribution of x4 is sufficiently clear.

5.    Plot a histogram of the average values.

Figure 50.1(a) shows the frequency distribution of the 4000 observations actually drawn in order to compute the 1000 simulated 4-day averages represented by the frequency distribution of Figure 50.1(b). Although 1000 observations sounds likes a large number, the frequency distributions are still not smooth, but the essential information has emerged from the simulation. The distribution of 4-day averages is skewed, although not as strongly as the parent lognormal distribution. The median, average, and standard deviation of the 4000 lognormal values are 7.5, 12.3, and 16.1. The average of the 1000 4-day averages is 12.3; the standard deviation of the 4-day averages is 11.0; 90% of the 4-day averages are in the range of 5.0 to 26.5; and 50% are in the range of 7.2 to 15.4.

### Case Study: Percentile Estimation

A state regulation requires the 99th percentile of measurements on a particular chemical to be less than 18 Rg/L. Suppose that the true underlying distribution of the chemical concentration is lognormal as shown in the top panel of Figure 50.2. The true 99th percentile is 13.2 Rg/L, which is well below the standard value of 18.0. If we make 100 random observations of the concentration, how often will the 99th percentile

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