N
σ =
N
Formulas Relating a Value, the Mean, and the Standard Deviation:
• value = mean + (#ofSTDEVs)(standard deviation), where #ofSTDEVs = the number of standard devi-
ations
• x = x+ (#ofSTDEVs)(s)
• x = µ + (#ofSTDEVs)( σ)
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CHAPTER 2. DESCRIPTIVE STATISTICS
2.11 Practice 1: Center of the Data14
2.11.1 Student Learning Outcomes
• The student will calculate and interpret the center, spread, and location of the data.
• The student will construct and interpret histograms an box plots.
2.11.2 Given
Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one
week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve
generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.
2.11.3 Complete the Table
Data Value (# cars)
Frequency
Relative Frequency
Cumulative Relative Frequency
Table 2.11
2.11.4 Discussion Questions
Exercise 2.11.1
(Solution on p. 116.)
What does the frequency column sum to? Why?
Exercise 2.11.2
(Solution on p. 116.)
What does the relative frequency column sum to? Why?
Exercise 2.11.3
What is the difference between relative frequency and frequency for each data value?
Exercise 2.11.4
What is the difference between cumulative relative frequency and relative frequency for each data
value?
2.11.5 Enter the Data
Enter your data into your calculator or computer.
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2.11.6 Construct a Histogram
Determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram
below. Label the horizontal and vertical axes with words. Include numerical scaling.
2.11.7 Data Statistics
Calculate the following values:
Exercise 2.11.5
(Solution on p. 116.)
Sample mean = x =
Exercise 2.11.6
(Solution on p. 116.)
Sample standard deviation = sx =
Exercise 2.11.7
(Solution on p. 116.)
Sample size = n =
2.11.8 Calculations
Use the table in section 2.11.3 to calculate the following values:
Exercise 2.11.8
(Solution on p. 116.)
Median =
Exercise 2.11.9
(Solution on p. 116.)
Mode =
Exercise 2.11.10
(Solution on p. 116.)
First quartile =
Exercise 2.11.11
(Solution on p. 117.)
Second quartile = median = 50th percentile =
Exercise 2.11.12
(Solution on p. 117.)
Third quartile =
Exercise 2.11.13
(Solution on p. 117.)
Interquartile range (IQR) = _____ - _____ = _____
Exercise 2.11.14
(Solution on p. 117.)
10th percentile =
Exercise 2.11.15
(Solution on p. 117.)
70th percentile =
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CHAPTER 2. DESCRIPTIVE STATISTICS
Exercise 2.11.16
(Solution on p. 117.)
Find the value that is 3 standard deviations:
a. Above the mean
b. Below the mean
2.11.9 Box Plot
Construct a box plot below. Use a ruler to measure and scale accurately.
2.11.10 Interpretation
Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or
concentrated in some areas, but not in others? How can you tell?
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2.12 Practice 2: Spread of the Data15
2.12.1 Student Learning Outcomes
• The student will calculate measures of the center of the data.
• The student will calculate the spread of the data.
2.12.2 Given
The population parameters below describe the full-time equivalent number of students (FTES) each year
at Lake Tahoe Community College from 1976-77 through 2004-2005. (Source: Graphically Speaking by Bill
King, LTCC Institutional Research, December 2005 ).
Use these values to answer the following questions:
• µ = 1000 FTES
• Median = 1014 FTES
• σ = 474 FTES
• First quartile = 528.5 FTES
• Third quartile = 1447.5 FTES
• n = 29 years
2.12.3 Calculate the Values
Exercise 2.12.1
(Solution on p. 117.)
A sample of 11 years is taken. About how many are expected to have a FTES of 1014 or above?
Explain how you determined your answer.
Exercise 2.12.2
(Solution on p. 117.)
75% of all years have a FTES:
a. At or below:
b. At or above:
Exercise 2.12.3
(Solution on p. 117.)
The population standard deviation =
Exercise 2.12.4
(Solution on p. 117.)
What percent of the FTES were from 528.5 to 1447.5? How do you know?
Exercise 2.12.5
(Solution on p. 117.)
What is the IQR? What does the IQR represent?
Exercise 2.12.6
(Solution on p. 117.)
How many standard deviations away from the mean is the median?
Additional Information: The population FTES for 2005-2006 through 2010-2011 was given in an updated
report. (Source: http://www.ltcc.edu/data/ResourcePDF/LTCC_FactBook_2010-11.pdf). The data are re-
ported here.
Year
2005-06
2006-07
2007-08
2008-09
2009-10
2010-11
Total FTES
1585
1690
1735
1935
2021
1890
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CHAPTER 2. DESCRIPTIVE STATISTICS
Table 2.12
Exercise 2.12.7
(Solution on p. 117.)
Calculate the mean, median, standard deviation, first quartile, the third quartile and the IQR.
Round to one decimal place.
Exercise 2.12.8
Construct a boxplot for the FTES for 2005-2006 through 2010-2011 and a boxplot for the FTES for
1976-1977 through 2004-2005.
Exercise 2.12.9
(Solution on p. 117.)
Compare the IQR for the FTES for 1976-77 through 2004-2005 with the IQR for the FTES for 2005-
2006 through 2010-2011. Why do you suppose the IQRs are so different?
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2.13 Homework16
Exercise 2.13.1
(Solution on p. 117.)
Twenty-five randomly selected students were asked the number of movies they watched the pre-
vious week. The results are as follows:
# of movies
Frequency
Relative Frequency
Cumulative Relative Frequency
0
5
1
9
2
6
3
4
4
1
Table 2.13
a. Find the sample mean x
b. Find the sample standard deviation, s
c. Construct a histogram of the data.
d. Complete the columns of the chart.
e. Find the first quartile.
f. Find the median.
g. Find the third quartile.
h. Construct a box plot of the data.
i. What percent of the students saw fewer than three movies?
j. Find the 40th percentile.
k. Find the 90th percentile.
l. Construct a line graph of the data.
m. Construct a stem plot of the data.
Exercise 2.13.2
The median age for U.S. blacks currently is 30.9 years;
for U.S. whites it is 42.3
years.
((Source: http://www.usatoday.com/news/nation/story/2012-05-17/minority-births-
census/55029100/1) )
a. Based upon this information, give two reasons why the black median age could be lower than
the white median age.
b. Does the lower median age for blacks necessarily mean that blacks die younger than whites?
Why or why not?
c. How might it be possible for blacks and whites to die at approximately the same age, but for
the median age for whites to be higher?
Exercise 2.13.3
(Solution on p. 118.)
Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X
= the number of pairs of sneakers owned. The results are as follows:
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CHAPTER 2. DESCRIPTIVE STATISTICS
X
Frequency
Relative Frequency
Cumulative Relative Frequency
1
2
2
5
3
8
4
12
5
12
7
1
Table 2.14
a. Find the sample mean x
b. Find the sample standard deviation, s
c. Construct a histogram of the data.
d. Complete the columns of the chart.
e. Find the first quartile.
f. Find the median.
g. Find the third quartile.
h. Construct a box plot of the data.
i. What percent of the students owned at least five pairs?
j. Find the 40th percentile.
k. Find the 90th percentile.
l. Construct a line graph of the data
m. Construct a stem plot of the data
Exercise 2.13.4
600 adult Americans were asked by telephone poll, What do you think constitutes a middle-class
income? The results are below. Also, include left endpoint, but not the right endpoint. (Source:
Time magazine; survey by Yankelovich Partners, Inc.)
NOTE: "Not sure" answers were omitted from the results.
Salary ($)
Relative Frequency
< 20,000
0.02
20,000 - 25,000
0.09
25,000 - 30,000
0.19
30,000 - 40,000
0.26
40,000 - 50,000
0.18
50,000 - 75,000
0.17
75,000 - 99,999
0.02
100,000+
0.01
Table 2.15
a. What percent of the survey answered "not sure" ?
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b. What percent think that middle-class is from $25,000 - $50,000 ?
c. Construct a histogram of the data
a. Should all bars have the same width, based on the data? Why or why not?
b. How should the <20,000 and the 100,000+ intervals be handled? Why?
d. Find the 40th and 80th percentiles
e. Construct a bar graph of the data
Exercise 2.13.5
(Solution on p. 118.)
Following are the published weights (in pounds) of all of the team members of the San Francisco
49ers from a previous year (Source: San Jose Mercury News)
177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241;
223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230;
250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265
a. Organize the data from smallest to largest value.
b. Find the median.
c. Find the first quartile.
d. Find the third quartile.
e. Construct a box plot of the data.
f. The middle 50% of the weights are from _______ to _______.
g. If our population were all professional football players, would the above data be a sample of
weights or the population of weights? Why?
h. If our population were the San Francisco 49ers, would the above data be a sample of weights
or the population of weights? Why?
i. Assume the population was the San Francisco 49ers. Find:
i. the population mean, µ.
ii. the population standard deviation, σ.
iii. the weight that is 2 standard deviations below the mean.
iv. When Steve Young, quarterback, played football, he weighed 205 pounds. How many
standard deviations above or below the mean was he?
j. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard
deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team,
who was lighter, Smith or Young? How did you determine your answer?
Exercise 2.13.6
An elementary school class ran 1 mile with a mean of 11 minutes and a standard deviation of 3
minutes. Rachel, a student in the class, ran 1 mile in 8 minutes. A junior high school class ran 1
mile with a mean of 9 minutes and a standard deviation of 2 minutes. Kenji, a student in the class,
ran 1 mile in 8.5 minutes. A high school class ran 1 mile with a mean of 7 minutes and a standard
deviation of 4 minutes. Nedda, a student in the class, ran 1 mile in 8 minutes.
a. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he?
b. Who is the fastest runner with respect to his or her class? Explain why.
Exercise 2.13.7
In a survey of 20 year olds in China, Germany and America, people were asked the number of
foreign countries they had visited in their lifetime. The following box plots display the results.
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CHAPTER 2. DESCRIPTIVE STATISTICS
a. In complete sentences, describe what the shape of each box plot implies about the distribution
of the data collected.
b. Explain how it is possible that more Americans than Germans surveyed have been to over eight
foreign countries.
c. Compare the three box plots. What do they imply about the foreign travel of twenty year old
residents of the three countries when compared to each other?
Exercise 2.13.8
One hundred teachers attended a seminar on mathematical problem solving. The attitudes of
a representative sample of 12 of the teachers were measured before and after the seminar. A
positive number for change in attitude indicates that a teacher’s attitude toward math became
more positive. The twelve change scores are as follows:
3; 8; -1; 2; 0; 5; -3; 1; -1; 6; 5; -2
a. What is the mean change score?
b. What is the standard deviation for this population?
c. What is the median change score?
d. Find the change score that is 2.2 standard deviations below the mean.
Exercise 2.13.9
(Solution on p. 118.)
Three students were applying to the same graduate school. They came from schools with different
grading systems. Which student had the best G.P.A. when compared to his school? Explain how
you determined your answer.
Student
G.P.A.
School Ave. G.P.A.
School Standard Deviation
Thuy
2.7
3.2
0.8
Vichet
87
75
20
Kamala
8.6
8
0.4
Table 2.16
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Exercise 2.13.10
Given the following box plot:
a. Which quarter has the smallest spread of data? What is that spread?
b. Which quarter has the largest spread of data? What is that spread?
c. Find the Inter Quartile Range (IQR).
d. Are there more data in the interval 5 - 10 or in the interval 10 - 13? How do you know this?
e. Which interval has the fewest data in it? How do you know this?
I. 0-2
II. 2-4
III. 10-12
IV. 12-13
Exercise 2.13.11
Given the following box plot:
a. Think of an example (in words) where the data might fit into the above box plot. In 2-5 sen-
tences, write down the example.
b. What does it mean to have the first and second quartiles so close together, while the second to
fourth quartiles are far apart?
Exercise 2.13.12
Santa Clara County, CA, has approximately 27,873 Japanese-Americans. Their ages are as follows.
(Source: West magazine)
Age Group
Percent of Community
0-17
18.9
18-24
8.0
25-34
22.8
35-44
15.0
45-54
13.1
55-64
11.9
65+
10.3
Table 2.17
a. Construct a histogram of the Japanese-American community in Santa Clara County, CA. The
bars will not be the same width for this example. Why not?
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CHAPTER 2. DESCRIPTIVE STATISTICS
b. What percent of the community is under age 35?
c. Which box plot most resembles the information above?
Exercise 2.13.13
Suppose that three book publishers were interested in the number of fiction paperbacks adult
consumers purchase per month. Each publisher conducted a survey. In the survey, each asked
adult consumers the number of fiction paperbacks they had purchased the previous month. The
results are below.
Publisher A
# of books
Freq.
Rel. Freq.
0
10
1
12
2
16
3
12
4
8
5
6
6
2
8
2
Table 2.18
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Publisher B
# of books
Freq.
Rel. Freq.
0
18
1
24
2
24
3
22
4
15
5
10
7
5
9
1
Table 2.19
Publisher C
# of books
Freq.
Rel. Freq.
0-1
20
2-3
35
4-5
12
6-7
2
8-9
1
Table 2.20
a. Find the relative frequencies for each survey. Write them in the charts.
b. Using either a graphing calculator, computer, or by hand, use the frequency column to construct
a histogram for each publisher’s survey. For Publishers A and B, make bar widths of 1. For
Publisher C, make bar widths of 2.
c. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
d. Would you have expected the