Mean,Median,ModeFor Exercise1, Questions 8 & 10 do histogram and Polygon instead of stem leaf display. All the pictures have the questions and the tables in order to solve them.Homework must be done in PDF or Word.doc636 III CHAPTER 12 Statistics
3. Construct two histograms, one from the expected
frequency distribution and one from your observed
For Further Thought (cont.)
In an actual experiment, we could obtain observed
frequencies (or empirical frequencies), which
would most likely differ somewhat from the expected
frequencies. But 64 repetitions of the experiment
should be enough to provide fair consistency between
expected and observed values.
frequency distribution.
4. Compare the two histograms.
Observed
Number of Expected
Heads Frequency
Frequency
O
e
х
2
0
10
1
O Nm
20
For Group or Individual Investigation
Toss five coins a total of 64 times, keeping a record of
the results.
1. Enter your experimental results in the third column
of the table at the right, producing an observed
frequency distribution.
2. Compare the second and third column entries.
20
10
4
2
12.1 EXERCISES
In Exercises 3-6, use the given data to do the following:
(a) Construct grouped frequency and relative frequency dis-
tributions, in a table similar to Table 3. (Follow the sug-
gested guidelines for class limits and class width.)
(b) Construct a histogram.
(c) Construct a frequency polygon.
3. Exam Scores The scores of the 54 members of a sociol-
ogy lecture class on a 70-point exam were as follows.
In Exercises 1 and 2, use the given data to do the following:
(a) Construct frequency and relative frequency distribu-
tions, in a table similar to Table 2.
(b) Construct a histogram.
(c) Construct a frequency polygon.
1. Preparation for Summer According to Newsmax (May,
2010, page 76), the following are five popular “mainte-
nance” activities performed as summer approaches.
1. Prep the car for road trips.
2. Clean up the house or apartment.
3. Groom the garden.
4. Exercise the body.
5. Organize the wardrobe.
The following data are the responses of 30 people who
were asked, on June 1st, how many of the five they had
accomplished.
60 63 64 52 60 58 63 53 56
64 48 54 64 57 51 67 60 49
59 54 49 52 53 60 58 60 64
52 56 56 58 66 59 62
66 59 62 50 58
53 51 65 62 61 55 59 52 62
58 61 65 56 55 50 61 55 54
Use five classes with a uniform class width of 5 points,
and use a lower limit of 45 points for the first class.
1 1 3 1 0 3 0 0 2 1
2 2 0 0 5 3 4 0 1 0
4 2 0 2 0 1 0 1 2 3
2. Responses to “Pick a Number” The following data are
the responses of 28 people asked to “pick a number
from 1 to 10.”
4.) Charge Card Account Balances The following raw data
represent the monthly account balances (to the nearest
dollar) for a sample of 50 brand-new charge card users.
78 175 46 138 79 118 90 163 88 107
126 154 85 60 42 54 62 128 114 73
67 119 116 145 129 130 81 105 96 71
100 145 117 60 125 130 94 88 136 112
85 165 118 84 74 62 81 110 108 71
Use seven classes with a uniform width of 20 dollars,
where the lower limit of the first class is 40 dollars.
4 7 2 7 6 3 1
7
4 9 8 5 6 10
4 10 8 9 5 4
5
9 2 6 6 6 8 7
12.1
Visual Displays of Data III 637
the daily high temperatures (in degrees
Fahrenheit) for
5. Daily High Temperatures The following data represent
the month of June in a southwestern U.S. city.
9. Distances to School The following data are the daily
round-trip distances to school (in miles) for 30 ran-
domly chosen students attending a community college
in California.
90 82
16 30 10 11 18 26 34 18 8 12
21 14 5 22 4 25 9 10 6 21
12 18 9 16 44 23 4 13 36 8
Ohe
79 84 88 96 102 104
99 97 92 94
85 92 100 99 101 104 110 108 106 106
74 72 83 107 111 102 97 94
Use nine classes with a uniform width of 5 degrees,
where the lower limit of the first class is 70 degrees.
6. iQ Scores of College Freshmen The following data repre-
sent IQ scores of a group of 50 college freshmen.
113 109 118 92 130 112 114 117 122 115
127 107 108 113 124 112 111 106 116 118
121 107 118 118 110 124 115 103 100 114
104 124 116 123 104 135 121 126 116 111
96 134 98 129 102 103 107 113 117 112
10. Yards Gained in the National Football League The follow-
ing data represent net yards gained per game by
National Football League running backs who played
during a given week of the season.
25 19 36 73 37 88 67 33 54 123 79
19 39 45 22 58 7 73 30 43 24 36
65 43 33 55 40 29 112 60 94 86 62
52 29 18 25 41 3 49 102 16 32 46
Use nine classes with a uniform width of 5, where the
lower limit of the first class is 91.
Federal Government Receipts The graph shows U.S. govern-
ment receipts and outlays (both on-budget and off-budget) for
2001–2011. Refer to the graph for Exercises 11-15.
In each of Exercises 7-10, construct a stem-and-leaf display
for the given data. In each case, treat the ones digits as the
leaves. For any single-digit data, use a stem of 0.
FISCALLY STRAINED
UNCLE SAM
7. Games Won in the National Basketball Association Approaching
midseason, the teams in the National Basketball Asso-
ciation had won the following numbers of games.
4,000
27 20 29 11 26 11 12 7 26 18
22 19 14 13 22 9 25 11 10 15
38 10 22 23 31 8 24 15 24 15
3,600
3,200
Billions of Dollars
Receipts
2,800
Outlays
2,400
2,000
1,600
2001 ’02’03’04 ’05 ’06 407 408 ’09 ’10* ’11*
Fiscal Years
20
Source: Department of the Treasury, Office of
Management and Budget.
* Data are estimates.
laking
20.
11. Of the period 2001-2011, list all years when receipts
exceeded outlays.
。
GURERS
12. Identify each of the following amounts and when it
occurred.
(a) the greatest one-year drop in receipts
(b) the greatest one-year rise in outlays
13. In what years did receipts appear to climb faster than
outlays?
0
14 About what was the greatest federal deficit, and in what
year did it occur?
Accumulated College Units
The students in a biology
class were asked how many college units they had accu-
mulated to date. Their responses are shown below.
12 4 13 12 21 22 15 17 33 24
32 42 26 11 53 62 42 25 13 8
54 18 21 14 19 17 38 17 20 10
15. Plot a point for each year and draw a line graph show-
ing the federal surplus (+) or deficit (-) over the years
2001-2011.
Measures of Central Tendency Ill 649
12.2
Some helpful points of comparison follow.
1. For distributions of numeric data, the mean and median will always exist, while
tive” measure.
the mode may not exist. On the other hand, for
nonnumeric data, it may be that
none of the three measures exists, or that only the mode exists.
2. Because even a single change in the data may cause the mean to change, while
the median and mode may not be affected at all, the mean is the most “sensi-
3. In a symmetric distribution, the mean, median, and mode (if a single mode
exists) will all be equal. In a nonsymmetric distribution, the mean is often
unduly affected by relatively few extreme values and, therefore, may not be a
good representative measure of central tendency. For example, distributions of
salaries, family incomes, or home prices often include a few values that are
much higher than the bulk of the items. In such cases, the median is a more use-
ful measure.
4. The mode is the only measure covered here that must always be equal to one of
the data items of the distribution. In fact, more of the data items are equal to
the mode than to any other number. A fashion shop planning to stock only
one hat size for next season would want to know the mode (the most com-
mon) of all hat sizes among their potential customers. Likewise, a designer of
family automobiles would be interested in the most common family size. In
examples like these, designing for the mean or the median might not be right
for anyone.
For Further Thought
For Group or Individual Investigation
Simpson’s Paradox
In baseball statistics, a player’s “batting average” gives the
average number of hits per time at bat. For example, a
player who has gotten 84 hits in 250 times at bat has a
batting average of 250 = .336. This “average” can be
interpreted as the empirical probability of that player’s
getting a hit the next time at bat.
The following are actual comparisons of hits and
at-bats for two major league players in the 1995, 1996,
and 1997 seasons. The numbers illustrate a puzzling
statistical occurrence known as Simpson’s paradox.
The example below, involving Dave Justice and Derek
Jeter, was referred to in the “Conspiracy Theory”
episode of the television series NUMB3RS. (Source:
www.wikipedia.org)
1. Fill in the twelve blanks in the table, giving batting
averages to three decimal places.
2. Which player had a better average in 1995?
3. Which player had a better average in 1996?
4. Which player had a better average in 1997?
5. Which player had a better average in 1995, 1996,
and 1997 combined?
6. Did the results above surprise you? How can it be
that one player’s batting average leads another’s for
each of three years, and yet trails the other’s for the
combined years?
Dave Justice
Derek Jeter
Batting
Average
Batting
Average
Hits
At-bats
At-bats
Hits
12
48
104
411
1995
183
582
45
140
1996
190
654
163
495
1997
Combined
(1995-1997)
650 III CHAPTER 12 Statistics
12.2 EXERCISES
year are reduced by 4 fatal accidents and 265 fatalities, which
The year 2001 was clearly an anomaly. If the data for that
of the three measures change and what are their new values
For each list of data, calculate (a) the mean, (b) the median,
and (c) the mode or modes (if any). Round mean values to
the nearest tenth.
for each of the following?
1. 7, 9, 12, 14, 34
14. Exercise 12
20, 27, 42, 45, 53, 62, 62, 64
15. Exercise 13
16. Following 2001, in what year did airline departures start
3. 218, 230, 196, 224, 196, 233
to increase again?
@ 26, 31, 46, 31, 26, 29, 31
Spending by U.S. Travelers The table shows the top five U.S.
states for domestic traveler spending in 2007.
5. 3.1, 4.5, 6.2, 7.1, 4.5, 3.8, 6.2, 6.3
6. 14,320, 16,950, 17,330, 15,470
Spending
(billions of dollars)
7. 0.78, 0.93, 0.66, 0.94, 0.87, 0.62, 0.74, 0.81
State
8.) 0.53, 0.03, 0.28, 0.18, 0.39, 0.28, 0.14, 0.22, 0.04
$96.2
California
68.9
9. 128, 131, 136, 125, 132, 128, 125, 127
Florida
51.3
New York
10.8.97, 5.64, 2.31, 1.02, 4.35, 7.68
47.4
Texas
34.5
Nevada
Source: The World Almanac and Book of Facts 2010.
Airline Fatalities in the United States The table pertains to
scheduled commercial carriers. Fatalities data include those
on the ground except for the September 11, 2001, terrorist
attacks. Use this information for Exercises 11-16.
Find each of the following quantities for these five states.
U.S. Airline Safety, 1999–2008
17. the mean spending
Departures
(millions)
Fatal
Accidents
Year
Fatalities
18. the median spending
1999
10.9
2
12
2000
11.1
2
89
Measuring Elapsed Times While doing an experiment, a
physics student recorded the following sequence of elapsed
times (in seconds) in a lab notebook.
2001
10.6
6
531
2002
10.3
0
0
2003
2.16, 22.2, 2.96, 2.20, 2.73, 2.28, 2.39
10.2
2
22
2004
10.8
1
13
19. Find the mean.
2005
10.9
3
22
20. Find the median.
2006
10.6
2
50
2007
10.7
0
0
2008
10.6
0
0
Source: The World Almanac and Book of Facts 2010.
The student from Exercises 19 and 20, when reviewing the
calculations later, decided that the entry 22.2 should have
been recorded as 2.22, and made that change in the listing.
21. Find the mean for the new list.
For each category in Exercises 11-16, find (a) the mean, (b) the
median, and (c) the mode (if any).
22) Find the median for the new list.
11. departures
12 fatal accidents
23. Which measure, the mean or the median, was affected
more by correcting the error?
13. fatalities
In general, which measure, mean or median, is affected
less by the presence of an extreme value in the data?
Measures of Central Tendency III 651
12.2
following
Scores on Management Examinations Rob Bates earned the
32. Units
Grade
scores on his six management exams last semester.
79, 81, 44, 89, 79, 90
2
A
6
B
scores.
25. Find the mean, the median, and the mode for Rob’s
5
С
26. Which of the three averages probably is the best indica-
tor of Rob’s ability?
Most Populous Countries The table gives population (2009)
and land area for the world’s five most populous countries.
27. If Rob’s instructor gives him a chance to replace his
score of 44 by taking a “make-up” exam, what must he
score on the make-up to get an overall average (mean)
Area
(Thousands of
square miles)
Population
(millions)
of 85?
Country
China
Exercises 28 and 29 give frequency distributions for sets of
data values. For each set find the (a) mean (to the nearest
tenth), (b) median, and (c) mode or modes (if any).
Frequency
India
United States
1339
1157
307
3601
1148
3537
741
3265
28. Value
Indonesia
240
Brazil
199
12
3
Source: World Almanac and Book of Facts 2010.
14
1
8
16
18
4
Use this information for Exercises 33–36.
33. Find the mean population (to the nearest million) for
these 5 countries.
29. Value
Frequency
17
34. Find the mean area (to the nearest thousand square
miles) for these 5 countries.
7
9
615
590
605
579
586
600
35. For each country, find the population density (to the
nearest whole number of persons per square mile).
14
6
36. For the 5 countries combined, find the mean population
density.
Personal Computer Use Just six countries account for over
half of all personal computers in use worldwide. The table
shows figures for 2008. Use this information for Exercises 37
and 38.
30. Average Employee Salaries A company has
5 employees with a salary of $19,500,
11 employees with a salary of $23,000,
7 employees with a salary of $28,300,
2 employees with a salary of $31,500,
4 employees with a salary of $38,900,
1 employee with a salary of $147,500.
Find the mean salary for the employees (to the nearest
hundred dollars).
PCs in use
(millions)
Population
(millions)
Country
U.S.
264.10
China
303.8
1330.0
127.3
82.4
60.9
Japan
Germany
UK
Grade-point Averages Find the grade-point average for each
of the following students. Assume A = 4, B = 3, C = 2,
D = 1, and F = 0. Round to the nearest hundredth.
98.67
86.22
61.96
47.04
43.11
France
64.7
Source: The World Almanac and Book of Facts 2010.
31. Units
Grade
37. Estimate the mean number of PCs in use in 2008 for
these six countries.
4.
С
7
B
3
A
38. U.S. use was 22.19% of the worldwide total. How many
PCs were in use in the world in 2008?
3
F
650 III CHAPTER 12 Statistics
12.2 EXERCISES
year are reduced by 4 fatal accidents and 265 fatalities, which
The year 2001 was clearly an anomaly. If the data for that
of the three measures change and what are their new values
For each list of data, calculate (a) the mean, (b) the median,
and (c) the mode or modes (if any). Round mean values to
the nearest tenth.
for each of the following?
1. 7, 9, 12, 14, 34
14. Exercise 12
20, 27, 42, 45, 53, 62, 62, 64
15. Exercise 13
16. Following 2001, in what year did airline departures start
3. 218, 230, 196, 224, 196, 233
to increase again?
@ 26, 31, 46, 31, 26, 29, 31
Spending by U.S. Travelers The table shows the top five U.S.
states for domestic traveler spending in 2007.
5. 3.1, 4.5, 6.2, 7.1, 4.5, 3.8, 6.2, 6.3
6. 14,320, 16,950, 17,330, 15,470
Spending
(billions of dollars)
7. 0.78, 0.93, 0.66, 0.94, 0.87, 0.62, 0.74, 0.81
State
8.) 0.53, 0.03, 0.28, 0.18, 0.39, 0.28, 0.14, 0.22, 0.04
$96.2
California
68.9
9. 128, 131, 136, 125, 132, 128, 125, 127
Florida
51.3
New York
10.8.97, 5.64, 2.31, 1.02, 4.35, 7.68
47.4
Texas
34.5
Nevada
Source: The World Almanac and Book of Facts 2010.
Airline Fatalities in the United States The table pertains to
scheduled commercial carriers. Fatalities data include those
on the ground except for the September 11, 2001, terrorist
attacks. Use this information for Exercises 11-16.
Find each of the following quantities for these five states.
U.S. Airline Safety, 1999–2008
17. the mean spending
Departures
(millions)
Fatal
Accidents
Year
Fatalities
18. the median spending
1999
10.9
2
12
2000
11.1
2
89
Measuring Elapsed Times While doing an experiment, a
physics student recorded the following sequence of elapsed
times (in seconds) in a lab notebook.
2001
10.6
6
531
2002
10.3
0
0
2003
2.16, 22.2, 2.96, 2.20, 2.73, 2.28, 2.39
10.2
2
22
2004
10.8
1
13
19. Find the mean.
2005
10.9
3
22
20. Find the median.
2006
10.6
2
50
2007
10.7
0
0
2008
10.6
0
0
Source: The World Almanac and Book of Facts 2010.
The student from Exercises 19 and 20, when reviewing the
calculations later, decided that the entry 22.2 should have
been recorded as 2.22, and made that change in the listing.
21. Find the mean for the new list.
For each category in Exercises 11-16, find (a) the mean, (b) the
median, and (c) the mode (if any).
22) Find the median for the new list.
11. departures
12 fatal accidents
23. Which measure, the mean or the median, was affected
more by correcting the error?
13. fatalities
In general, which measure, mean or median, is affected
less by the presence of an extreme value in the data?
Measures of Central Tendency III 651
12.2
following
Scores on Management Examinations Rob Bates earned the
32. Units
Grade
scores on his six management exams last semester.
79, 81, 44, 89, 79, 90
2
A
6
B
scores.
25. Find the mean, the median, and the mode for Rob’s
5
С
26. Which of the three averages probably is the best indica-
tor of Rob’s ability?
Most Populous Countries The table gives population (2009)
and land area for the world’s five most populous countries.
27. If Rob’s instructor gives him a chance to replace his
score of 44 by taking a “make-up” exam, what must he
score on the make-up to get an overall average (mean)
Area
(Thousands of
square miles)
Population
(millions)
of 85?
Country
China
Exercises 28 and 29 give frequency distributions for sets of
data values. For each set find the (a) mean (to the nearest
tenth), (b) median, and (c) mode or modes (if any).
Frequency
India
United States
1339
1157
307
3601
1148
3537
741
3265
28. Value
Indonesia
240
Brazil
199
12
3
Source: World Almanac and Book of Facts 2010.
14
1
8
16
18
4
Use this information for Exercises 33–36.
33. Find the mean population (to the nearest million) for
these 5 countries.
29. Value
Frequency
17
34. Find the mean area (to the nearest thousand square
miles) for these 5 countries.
7
9
615
590
605
579
586
600
35. For each country, find the population density (to the
nearest whole number of persons per square mile).
14
6
36. For the 5 countries combined, find the mean population
density.
Personal Computer Use Just six countries account for over
half of all personal computers in use worldwide. The table
shows figures for 2008. Use this information for Exercises 37
and 38.
30. Average Employee Salaries A company has
5 employees with a salary of $19,500,
11 employees with a salary of $23,000,
7 employees with a salary of $28,300,
2 employees with a salary of $31,500,
4 employees with a salary of $38,900,
1 employee with a salary of $147,500.
Find the mean salary for the employees (to the nearest
hundred dollars).
PCs in use
(millions)
Population
(millions)
Country
U.S.
264.10
China
303.8
1330.0
127.3
82.4
60.9
Japan
Germany
UK
Grade-point Averages Find the grade-point average for each
of the following students. Assume A = 4, B = 3, C = 2,
D = 1, and F = 0. Round to the nearest hundredth.
98.67
86.22
61.96
47.04
43.11
France
64.7
Source: The World Almanac and Book of Facts 2010.
31. Units
Grade
37. Estimate the mean number of PCs in use in 2008 for
these six countries.
4.
С
7
B
3
A
38. U.S. use was 22.19% of the worldwide total. How many
PCs were in use in the world in 2008?
3
F
652 III CHAPTER 12
Statistics
In Exercises 46 and 47, use the given stem-and-leaf display to
(a) the mean, (b) the median, and (c) the mode (if any)
identify
Crew, Passengers, and Hijackers on 9/11 Airliners The table
shows, for each hijacked flight on September 11, 2001, the
numbers of crew members, passengers
, and hijackers (not
included as passengers). For each quantity in Exercises 39-41,
find
(a) the mean, and (b) the median.
for the data represented.
repair shops for a new alternator (installed). Give
(to the nearest dollar) charged by 23 different auto
Flight
Crew
Passengers
Hijackers
answers to the nearest cent.
7
American #11
11
76
9
5
2
4
4
United #175
10
9
51
5
5
7
9
American #77
6
53
5
10
4 4
3
1
United #93
7
33
4
11
11
8 8 9
5
5
Source: www.911research.wtc7.net
12
0
4
4
7 7 9
12
5
39. number of crew members per plane
13
8
40 number of passengers per plane
41. total number of persons per plane
47. Scores on a Biology Exam The display here represents
scores achieved on a 100-point biology exam by the 34
members of the class.
4 1 7
Olympic Medal Standings The top ten medal-winning nations
in the 2010 Winter Olympics at Vancouver, Canada, are
shown in the table. Use the given information for Exercises
42-45.
5
1
3 6
6
2
5
5 6
7
8
8
7.
4
Medal Standings for the 2010 Winter Olympics
0
5
7
6
8 8 8 8 9
7
8
0
1
1
1
3
4.
5 5
Nation
Gold
Silver
Bronze
Total
9
0
0 0 1 6
United States
9
15
13
37
10
Germany
13
7.
30
Canada
14
7.
5
26
48. Calculating a Missing Test Score Katie Campbell’s Busi-
ness professor lost his grade book, which contained
Katie’s five test scores for the course. A summary of the
scores (each of which was an integer from 0 to 100)
indicates the following:
The mean was 88.
9
8
Norway
6
23
Austria
4.
6
16
6
Russia
3
5
15
7
South Korea
6
6
2
14
The median was 87.
Sweden
5
2
4
11
China
5
2
4
11
The mode was 92.
(The data set was not bimodal.) What is the least possible
number among the missing scores?
2
3
6
France
11
Source: www.nbcolympics.com
49. Explain what an “outlier” is and how it affects measures
of central tendency.
Calculate the following for all nations shown.
42, the mean number of gold medals
50. Consumer Preferences in Food Packaging A food processing
company that packages individual cups of instant soup
wishes to find out the best number of cups to include in a
package. In a survey of 22 consumers, they found that five
prefer a package of 1, five prefer a package of 2, three pre
fer a package of 3, six prefer a package of 4, and three pre-
fer a package of 6.
43. the median number of silver medals
44.) the mode, or modes, for the number of bronze medals
(a) Calculate the mean, median, and mode values for
preferred package size.
45. each of the following for the total number of medals
(a) mean
(b) median
(c) mode or modes
(b) Which measure in part (a) should the food process-
ing company use?
(c) Explain your answer to part (b).
a given item, x, occurred f times, then the relative frequency of
x is . Example 1 illustrates these ideas.
EXAMPLE 1 Constructing Frequency and Relative Frequency Distributions
The 25 members of a psychology class were polled as to the number of siblings in
their individual families. Construct a frequency distribution and a relative frequency
distribution for their responses, which are shown here.
2, 3, 1, 3, 3, 5, 2, 3, 3, 1, 1, 4, 2, 4, 2, 5, 4, 3, 6, 5, 1, 6, 2, 2, 2
SOLUTION
The data range from a low of 1 to a high of 6. The frequencies (obtained by inspec-
tion) and relative frequencies are shown in Table 2.
Table 2 Frequency and Relative Frequency
Distributions for Numbers of Siblings
Number x Frequency f Relative Frequency i
1
4
25 · 16%
2
7
7
25
23
= 28%
3
6
5 =
24%
25
4
3
33 = 12%
5
3
3
25
= 12%
6
2
23 = 8%
The numerical data of Table 2 can more easily be interpreted with the aid of a
histogram. A series of rectangles, whose lengths represent the frequencies, are
placed next to one another as shown in Figure 1. On each axis, horizontal and ver-
tical, a label and the numerical scale should be shown.
The information shown in the histogram in Figure 1 can also be conveyed by a
frequency polygon, as in Figure 2. Simply plot a single point at the appropriate height
or each frequency, connect the points with a series of connected line segments, and
mplete the polygon with segments that trail down to the axis beyond 1 and 6.
no
12.1
Visual Displays of Data III 633
Table 3 Grouped Frequency and Relative Frequency
Distributions for Weekly Study Times
f
Class Limits Tally Frequency f Relative Frequency
10-19
N|
6
6
40
15.0%
20-29
8 89 |
|
11
40
27.5%
40
30-39
TH III
9
9
= 22.5%
40
关 ||
7.
40
40-49
7
17.5%
4
10.0%
50-59
40
2
2
40
5.0%
11
60-69
1
40 = 2.5%
70-79
Total: n = 40
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