Homework #1 — Solutions
(1) (a) At the right is a histogram produced in JMP’s Graph Builder (modified from the default to have bin widths of 2).
We see a strong skew with a long right tail and no left tail. There is a strong peak in the lowest bin (0 to 2 metric tons per person) and a smaller peak from 8 to 10, though that second peak may not be very evident if fewer, wider bins are used.
The bimodal (2‐peaked) shape makes description of the “center” of the distribution somewhat questionable, but I would say it’s around 4.
The three highest countries (the United States, Canada, and Australia) appear to be outliers; apart from those countries, the distribution is spread from 0 to 11 metric tons per person.
(b) The five‐number summary (in units of metric tons per person) is (from JMP; quartiles from other software may differ slightly):
Min = 0, Q1 = 0.725, M = 3.2, Q3 = 7.9, Max = 19.9. The evidence for the skew is in the larger differences between the higher numbers; that is, the differences Q3M and MaxQ3 are large compared to Q1Min and MQ1.
(c) x = 4.595833, s = 4.8221545. The mean is nearly half‐again as large as the median, reflecting the skew: the long right tail pulls the mean above the median. (d) Because of the skew the 5‐number summary better describes these data.
(e) Using JMP’s quartiles, the IQR is Q3  Q1 = 7.175, so outliers would be less than 10.0375 or greater than 18.6625. According to this rule, only the United States quali‐
fies as an outlier, but Canada and Australia seem high enough to also include them. (Because there are several ways to compute the quartiles, results may vary somewhat among different software packages, but the conclusion should not be affected.)
(f) The features most apparent in the NQQ plot are the strong skew, shown by the bend in the pattern at around 1 metric ton per person, and the three high outliers, which are clearly sepa‐
rated from the other points. The slight S bend between about 5 and 7 units reflects the bimo‐
dality (the second peak), but this is pretty subtle.
(2)
(3)–(5)