Workshop Statistics: Discovery with Data, Second Edition

Topic 4: Measures of Center

Activity 4-1: Supreme Court Service

(a)

(b) Answers will vary from student to student, e.g., middle value is 11.
(c) 13.44 years
(d) more than mean: 3; less than mean: 6
(e) 11 years
(f) more than median: 4; less than median: 4
(g) fifth
(h) n = 5: 3; n = 7: 4; n = 9: 5; n = 11: 6; n = 13: 7
(i) if n is odd, then location of median at the (n + 1)/2 observation.
(j) There is no one definite middle point in an even number, such as 8 justices. If ordered, the middle will fall between the fourth and fifth justice.
(k) 10 years
(l) The mean and median of the tenures of the current justices do not accurately estimate the mean and median for all previous justices. They are underestimates. This makes sense because justices have their positions until they die or resign. Since the current justices have obviously not died at the point of this data collection, they may still serve for a few more years.

Activity 4-2: Properties of Averages

(a) Answers will vary from student to student.
(b)

	small cars	family cars	large cars
mean	32.520	28.964	27.167
median	33	29	27

	Senators' service	% urban residents	textbook prices
mean	11.09	68.2	73.27
median	10	68	79.50

(i) The mean is close to the median with symmetric distributions. The mean is greater than the median with skewed right distributions. The mean is less than the median with skewed left distributions. The mean follows the tail.
(j) The mean gender does not make sense because one is either male or female, we can't calculate an "average" value. The median party of the Senators does not make sense because they are either Republicans or Democrats, there is no "middle party." The mode gender and the mode party make sense because they would reveal the most frequent representation.

Activity 4-3: Rowers' Weights (cont.)

(a), (d), (f), (g)

	whole team	without coxswain	also without lightweights	with max at 324	with max at 2224
mean	191.85	194.68	206.32	211.58	312
median	202.50	205.00	207.00	207.00	207

(b) No, this information can only give us a partial understanding because we do not know how the team's weights are distributed about the mean and median (e.g. shape and variability).
(c) Answers will vary from student to student.
(d) Answers regarding prediction will vary from student to student.
(e) Answers will vary from student to student.
(f) Mean was affected more than median. Answers regarding prediction will vary from student to student.
(h) All of the rowers weigh less than the mean, except the rower mistakenly entered as 2224. The mean value is extreme enough to call attention to the typographical error, but the median is not.
(i) The median is resistant because it is the middle point in an ordered set. If you were to increase the largest point, the median would still be the middle point as it stands. The mean is not resistant because it is more dependent on the values of the other points. If you were to increase the largest point, it would increase the arithmetic average, which is equal to the mean.
(j) The mean will not change any more than the single changed value has (but if we increase it forever, the mean can also increase forever).

Activity 4-4: Readability of Cancer Pamphlets

(a) Those patients with reading levels below 3 or above 12 are not recorded exactly, so the mean cannot be calculated.
(b) 9
(c) 9
(d) These medians are equal.
(e) The medians do not indicate that the pamphlets are well matched to the patients' reading levels. Seventeen out of the 63 patients cannot read even the simplest pamphlet. Also, six pamphlets are written above level 12. Only 17 out of the 63 patients can read above level 12, and it has not been determined how far above level 12 they are. So, it is possible that all 17 read at level 13, in which case 4 of the pamphlets would be too difficult for even the best readers.
(f) .2698