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be calculated 
for quantitative data only, and which can be calculated for both quan-
titative and qualitative data? Illustrate with examples.
3.5 Which of the five measures of center (the mean, the median, the 
trimmed mean, the weighted mean, and the mode) can assume more 
than one value for a data set? Give an example of a data set for which 
this summary measure assumes more than one value.
3.6 Is it possible for a (quantitative) data set to have no mean, no 
median, or no mode? Give an example of a data set for which this 
summary measure does not exist.
3.7 Explain the relationships among the mean, median, and mode for 
symmetric and skewed histograms. Illustrate these relationships with 
graphs.
3.8 Prices of cars have a distribution that is skewed to the right with 
outliers in the right tail. Which of the measures of center is the best to 
summarize this data set? Explain.
3.9 The following data set belongs to a population:
5 −7 2 0 −9 16 10 7
Calculate the mean, median, and mode.
APPLICATIONS
3.10 The following data give the 2014 profits (in millions of 
dollars) of the top 10 companies listed in the 2014 Fortune 500 
(source: www.fortune.com).
Company
2014 Profits 
(millions of dollars)
Wal-Mart Stores 16,022
Exxon Mobil 32,580
Chevron 21,423
Berkshire Hathaway 19,476
Apple 37,037
Phillips 66 3726
General Motors 5346
Ford Motor 7155
General Electric 13,057
Valero Energy 2720
Find the mean and median for these data. Do these data have a mode? 
Explain.
3.2 Measures of Dispersion for Ungrouped Data 89
139 151 138 153 134 136 141 126 109 144
111 150 107 132 144 116 159 121 127 113
a. Calculate the mean, median, and mode for these data.
b. Calculate the 10% trimmed mean for these data.
3.19 In a survey of 640 parents of young children, 360 said that 
they will not want their children to play football because it is a very 
dangerous sport, 210 said that they will let their children play 
football, and 70 had no opinion. Considering the opinions of these 
parents, what is the mode?
3.20 A statistics professor based her final grades on quizzes, in-class 
group work, homework, a midterm exam, and a final exam. However, 
not all of the assignments contributed equally to the final grade. John 
received the scores (out of 100 for each assignment) listed in the table 
below. The instructor weighted each item as shown in the table.
Assignment
John’s Score 
(Total Points)
Percentage of Final Grade 
Assigned by the Instructor
Quizzes 75 30
In-class group work 52 5
Homework 85 10
Midterm exam 74 15
Final exam 81 40
Calculate John’s final grade score (out of 100) in this course. 
(Hint: Note that it is often easier to convert the weights to decimal 
form for weighting, e.g., 30% = .30; in this way, the sum of weights 
will equal 1, making the formula easier to apply.)
3.21 A clothing store bought 8000 shirts last year from five different 
companies. The following table lists the number of shirts bought from 
each company and the price paid for each shirt.
Company
Number of 
Shirts Bought
Price per Shirt 
(dollars)
Best Shirts 1200 30
Top Wear 1900 45
Smith Shirts 1400 40
New Design 2200 35
Radical Wear 1300 50
Calculate the weighted mean that represents the average price paid for 
these 8000 shirts.
3.22 One property of the mean is that if we know the means and 
sample sizes of two (or more) data sets, we can calculate the com-
bined mean of both (or all) data sets. The combined mean for two 
data sets is calculated by using the formula
Combined mean = x =
n1x1 + n2x2
n1 + n2
where n1 and n2 are the sample sizes of the two data sets and x1 and x2 
are the means of the two data sets, respectively. Suppose a sample of 
10 statistics books gave a mean price of $140 and a sample of 8 math-
ematics books gave a mean price of $160. Find the combined mean. 
(Hint: For this example: n1 = 10, n2 = 8, x1 = $140, x2 = $160.)
3.23 For any data, the sum of all values is equal to the product of the 
sample size and mean; that is, ∑x = nx. Suppose the average amount 
of money spent on shopping by 10 persons during a given week is 
$105.50. Find the total amount of money spent on shopping by these 
10 persons.
3.24 The mean 2015 income for five families was $99,520. What 
was the total 2015 income of these five families?
3.25 The mean age of six persons is 46 years. The ages of five of 
these six persons are 57, 39, 44, 51, and 37 years, respectively. Find 
the age of the sixth person.
3.26 Seven airline passengers in economy class on the same flight 
paid an average of $361 per ticket. Because the tickets were purchased 
at different times and from different sources, the prices varied. The 
first five passengers paid $420, $210, $333, $695, and $485. The sixth 
and seventh tickets were purchased by a couple who paid identical 
fares. What price did each of them pay?
3.27 When studying phenomena such as inflation or population 
changes that involve periodic increases or decreases, the geometric 
mean is used to find the average change over the entire period under 
study. To calculate the geometric mean of a sequence of n values x1, 
x2,..., xn, we multiply them together and then find the nth root of this 
product. Thus
Geometric mean =
n
√x1 · x2 · x3 · . . . · xn
Suppose that the inflation rates for the last five years are 4%, 3%, 
5%, 6%, and 8%, respectively. Thus at the end of the first year, the 
price index will be 1.04 times the price index at the beginning of 
the year, and so on. Find the mean rate of inflation over the 5-year 
period by finding the geometric mean of the data set 1.04, 1.03, 
1.05, 1.06, and 1.08. (Hint: Here, n = 5, x1 = 1.04, x2 = 1.03, and so 
on. Use the x1/n key on your calculator to find the fifth root. Note that 
the mean inflation rate will be obtained by subtracting 1 from the 
geometric mean.)
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The measures of center, such as the mean, median, and mode, do not reveal the whole picture of 
the distribution of a data set. Two data sets with the same mean may have completely different 
spreads. The variation among the values of observations for one data set may be much larger or 
smaller than for the other data set. (Note that the words dispersion, spread, and variation have 
similar meanings.) Consider the following two data sets on the ages (in years) of all workers at 
each of two small companies.
Company 1: 47 38 35 40 36 45 39
Company 2: 70 33 18 52 27
3.2 Measures of Dispersion for Ungrouped Data
90 Chapter 3 Numerical Descriptive Measures
 The mean age of workers in both these companies is the same, 40 years. If we do not know the 
ages of individual workers at these two companies and are told only that the mean age of the work-
ers at both companies is the same, we may deduce that the workers at these two companies have a 
similar age distribution. As we can observe, however, the variation in the workers’ ages for each of 
these two companies is very different. As illustrated in the diagram, the ages of the workers at the 
second company have a much larger variation than the ages of the workers at the first company.
Company 1 36 39
 35 38 40 45 47
Company 2
 18 27 33 52 70
 Thus, a summary measure such as the mean, median, or mode by itself is usually not a suf-
ficient measure to reveal the shape of the distribution of a data set. We also need a measure that 
can provide some information about the variation among data values. The measures that help us 
learn about the spread of a data set are called the measures of dispersion. The measures of center 
and dispersion taken together give a better picture of a data set than the measures of center alone. 
This section discusses four measures of dispersion: range, variance, standard deviation, and coef-
ficient