Extreme value problems

In some chemical engineering situations and in many design problems for environmental engineers, extremes have overriding importance. For example, an engineer designing an impoundment to serve as a reservoir for drinking water wants it to be large enough to hold enough water to last through a period of drought. The normal rainfall has some importance but the periods of low rainfall are the key to the design. There is no way to predict exactly what rainfall will occur in any future year, but we can make projections based on probability theory.

Floods tend to follow a Pearson3 distribution with a lower limit but no maximum. The following numbers are the largest storm (in cubic meters per second) during a given year along the Susquehanna river at Harrisburg, PA. They are in numerical order and not chronological order to help you sort them.

20956  9345  7136  6088
20022  8892  6995  6060
16284  8722  6995  6003
13990  8722  6938  5947
12715  8439  6938  5947
12602  8269  6910  5833
12460  8212  6910  5833
11866  8212  6853  5635
11837  8127  6740  5579
11667  8014  6740  5295
11639  7986  6598  5097
11441  7872  6598  5040
10959  7703  6570  4956
10704  7618  6570  4701
10280  7618  6541  4701
10110  7589  6485  4644
10081  7533  6258  4587
9827   7363  6230  4106
9402   7249  6202  3993

A solution that we found for this example grouped the numbers with class intervals. A reasonable interval is 500 cubic meters per second as you go from one group to the next. However, there seems to be no need to group because the abscissa is cumulative frequency. There are 76 data points, so the frequency steps are 1/76 = 0.013158. A portion of the calculation is:

Datum         Cumulative Percentage
3993     1/76 = 0.013158 = 1.32 per cent
4106     2/76 = 0.026316 = 2.63
4587     3/76 = 0.039474 = 3.95