Workshop Statistics: Discovery with Data and Fathom

Topic 4: Measures of Center

Activity 4-1: Supreme Court Service

(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: third;  n = 7: fourth;  n = 9: fifth;  n = 11: sixth;  n = 13: seventh
(i) if n is odd, then the location of median is 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

(c)

	small cars	family cars	large cars
mean	32.52	28.96	27.17
median	33	29	27

(d) - (h) Answers will vary from student to student.
(i)

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

(j) 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.
(k) 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	203.70	208.70	304
median	202.50	205.00	206.00	206.00	206

(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) A change in one person's weight will always affect the mean, but a change in one person's weight will only effect the median a little, if at all  As long as the person's weight began above the median and stayed above the median (or began below the median and stayed below) the change in that one person's weight will have no effect on the median.
(i)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.
(j) 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.
(k) 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