CSSS/SOC/STAT 221 Summer 2020 The gray histogram presents the sample distribution of all 440 valid responses. The solid black, red, green, and blue lines present smoothed representations of each quarter’s sample distribution (they are similar to the hollow histograms discussed in chapter 2 of your textbook). As you can see, the location, scale, and shape of all four quarter-specific sample distributions are very similar to one another. Their medians vary between 10 and 12.5; their first quartiles vary between -5 and 0; their third quartiles vary between 20 and 24; and their interquartile ranges vary between 20 and 29. All four samples are also clearly left-skewed. When combined into a single sample, the quartiles {Q0, Q1, Q2, Q3, Q4 } for all 440 valid cases are {-60, -1, 12, 22, 60} degrees F, implying an interquartile range of 23 degrees. The dashed line in the above graph represents a new parametric probability distribution model for continuous numerical variables called the Gumbel distribution, which has been fitted to the combined student data. The Gumbel model belongs to a set of models called “extreme value distributions,” which are widely used to approximate the probability distributions underlying “lowest value from median” or “highest value from median” variables, for example scores for a particular sporting event observed over multiple years; daily website user traffic; market anomalies; etc. These distributions tend to be left-skewed for “lowest value per observational unit” data or right-skewed for “highest value per observational unit.” When applied to left-skewed data such as the temperature variable described above, the Gumbel model has two parameters, the location parameterα (the Greek letter alpha) and the scale parameter
