This post is a supplement to the discussion of extreme value statistics at the end of Section 5.3 of the book. You can find an online extreme value distribution calculator, provided by South Dakota State University, at
This calculator fits a Gumbel distribution (a form of generalized extreme value distribution) to a data set. It uses the language of river floods because that is what the authors are interested in, but the underlying mathematics applies to many different situations.
To use the calculator one provides a data series consisting of extreme values. For instance, one might provide the data series
12.1; 11; 2; 1.8; 16.4; 6.7; 8; 3; 4; 9;
which represents the biggest value of some variable (the depth of the deepest flood, the windspeed of the strongest storm, or whatever) in each of 10 successive years. The output of the calculator is a table giving a probability distribution. It has five columns: the key ones are labeled T (return period), P (probability) and Q (“flood discharge” for this calculator, but it refers to whatever variable we are modeling). Here is part of the output for the data series above:
|Return period T, year
||Probability P, percent
This tells us (based on the data provided) that, for instance, the value \(Q=28\) will be exceeded only once in a hundred years; the value \(Q=25\) will be exceeded only once in fifty years; and so on.
This is a sample case study
Here is a link to the Washington Post article The world is about to install 700 million air conditioners. Here’s what that means for the climate, which is referenced in one of the Case Study exercises.
The article begins,
As summer temperatures finally settle in, many in the United States take it for granted that they can dial down the thermostat: Americans use 5 percent of all of their electricity cooling homes and buildings. In many other countries, however — including countries in much hotter climates — air conditioning is still a relative rarity. But as these countries boom in wealth and population, and extend electricity to more people even as the climate warms, the projections are clear: They are going to install mind-boggling amounts of air conditioning, not just for comfort but as a health necessity.
The first week of class is critical for a course like this. Many students may be skeptical about ever enjoying a mathematics course. For some their attitude toward mathematics may be deeply negative. Some may not be willing to apply themselves beyond the level needed to get a passing grade; some may be apprehensive about the course content or the writing requirement.
For the best outcome, our expectations and attitudes toward the course need to be reset during the first week. By creating a classroom community,
where learning is shared and students are accountable to one another, we believe that the instructor can help craft a course which is more enjoyable for everyone. This is the work of the first week. Continue reading “Classroom Community”
The Mathematics for Sustainability textbook supports a course that has been taught at Penn State for several years. A very significant component of the course as taught has been the requirement that students develop extended pieces of writing which use the mathematical skills developed in the course in support of advocacy or analysis of specific sustainability-related proposals. See the introduction to the Case Studies section (7.1) for more detail about this. Continue reading “Introduction to Writing Assignments”