Example 6 critical thinking and measures of center

 Example 6: Critical Thinking and Measures of Center
See each of the following illustrating situations in which the mean and median are
not meaningful statistics.
a. Zip codes of the Gateway Arch in St. Louis, White House, Air Force Division
of the Pentagon, Empire State Building and Statue of Liberty: 63102, 20500,

20330, 10118, 10004. (The zip codes don’t measure or count anything. The
numbers are just labels for geographic locations.)
b. Ranks of selected national universities of Harvard, Yale, Duke, Dartmouth,
and Brown (from U.S. News & World Report): 2, 3, 7, 10, 14. (The ranks
reflect an ordering, but they don’t measure or count anything.)
c. Numbers on the jerseys of the starting defense for the Seattle Seahawks when
they won Super Bowl XLVIII: 31, 28, 41, 56, 25, 54, 69, 50, 91, 72, 29. (The
numbers on the football jerseys don’t measure or count anything; they are just
substitutes for names.)
d. Top 5 incomes of chief executive officers (in millions of dollars): 131.2, 66.7,
64.4, 53.3, 51.5. (Such “top 5” or “top 10” lists include data that are not at all
representative of the larger population.)
e. The 50 mean ages computed from the means in each of the 50 states. (If you
calculate the mean of those 50 values, the result is not the mean age of people
in the entire United States. The population sizes of the 50 different states must
be taken into account, as described in the weighted mean.
-In the spirit of describing, exploring, and comparing data, we provide the table below, which
summarizes the different measures of center for the smartphone data speeds referenced in the
beginning of the chapter.
Verizon Sprin
t
AT&T T-Mobile
Mean 17.60 3.71 10.70 10.99
Median 13.90 1.60 8.65 9.70
Mode 4.5, 11.1 0.3 2.7 3.2, 4.4, 5.1, 13.3, 15.0, 16.7,
27.3
Midrang
e
39.30 15.30 19.80 14.00
3.1B – Beyond the Basics of Measures of Center
-Calculating the Mean from a Frequency Distribution
 The formula below consists of the same calculation for the mean that was presented by
the formula
Mean =
∑
x
n
, but it incorporates this approach: When working with data
summarized in a frequency distribution, we make calculations possible by pretending that
all sample values in each class are equal to the class midpoint
´
x =
∑
( f ∙ x )
∑
f

 Example 7: Computing the Mean from a Frequency Distribution
The first two columns of the table below shown here are the same as the
frequency distribution of McDonald’s Lunch SERVICE Times.
Time (seconds) Frequency
f
Class Midpoint
x
f · x
75-124 11 99.5 1094.5
125-174 24 149.5 3588.0
175-224 10 199.5 1995.0
225-274 3 249.5 748.5
275-324 2 299.5 599.0
Totals:
∑
f = 50
∑
(
f ∙ x
)
= 8025.0
Remember, when working with data summarized in a frequency
distribution, we make calculations possible by pretending that all sample
values in each class are equal to the class midpoint. For example, consider
the first class interval of 75-124 with a frequency of 11. We pretend that
each of the 11 service times is 99.5 sec (the class midpoint). With the
service time of 99.5 repeated 11 times, we have a total of 99.5 · 11 =
1094.5, as shown in the last column of the table given above. We can then
add those results to find the sum of all sample values. The bottom row of
the table given above shows the two components we need for the
calculation of the mean:
∑
f = 50
and
∑
( f ∙ x )= 8025.0
. We