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740
CHAPTER 15 y Descriptive Statistics
15.3
Measures of Dispersion
Objectives
1. Compute the range of a data set.
2. Understand how the standard deviation measures the spread of a distribution.
3. Use the coefficient of variation to compare the standard deviations of different
distributions.
Let’s imagine that you are a cardiac surgeon faced with deciding which of two heart pacemaker batteries you should choose. Because you must operate to replace a failing battery,
it is important that you choose the one that will last the longest. Assume that battery A lasts
for a mean time of 45,000 hours (slightly more than 5 years) and battery B has a mean time
of 46,000 hours.
On the surface, it would seem that you should choose battery B. However, suppose that we
also tell you that in testing the batteries, we found that all of A’s times were within 500 hours
of the mean, but B’s times varied widely. In fact, some of B’s times were as much as
2,500 hours below the mean. So B’s times could be as low as 46,000 - 2,500 = 43,500 hours,
whereas A’s times were never below 44,500. Based on this information, it appears that battery
A is the better choice.
The point of our story is that although the mean and median tell you something about a
distribution, they do not tell the whole story. For example, the following two distributions
both have a mean and median of 25, but Y’s values are much more spread out than X’s:
X: 24, 25, 25, 25, 25, 26
Y: 1, 2, 3, 47, 48, 49
From these examples, it is clear that we need to develop some method for measuring
the spread of a distribution.
KEY POINT
The range is a crude measure
of the spread of a data set.
The Range of a Data Set
It is clear from the discussion of pacemaker batteries that a numerical measure of the
dispersion, or spread, of a data set can provide useful information. One simple way to
describe the spread of a set of data is to subtract the smallest data value from the largest.
D E F I N I T I O N The range of a data set is the difference between the largest and
smallest data values in the set.
EXAMPLE 1
Comparing Heights
Find the range of the heights of the people listed in the accompanying table.
Person
Height
Height in Inches
Leonid Stadynk
(World’s Tallest Person)
8 feet, 6 inches
102 inches
LaDainian Tomlinson
5 feet, 10 inches
70 inches
Madge Bester
(World’s Shortest Person)
2 feet, 2 inches
26 inches
LeBron James
6 feet, 8 inches
80 inches
SOLUTION: The range of these data is
range = largest data value - smallest data value = 102 - 26 = 76 inches = 6 feet, 4 inches. ]
Copyright © 2010 Pearson Education, Inc.
15.3 y Measures of Dispersion
741
HISTORICAL HIGHLIGHT ¶ ¶ ¶
The Origins of Statistics
Ring a ring of roses,
A pocket full of posies,
Asha Asha, We all fall down.*
It may have been King Henry VII’s fear of the dreaded
Black Plague that prompted him to begin publishing weekly
Bills of Mortality in 1532. John Graunt, a merchant, noticed
patterns in these reports regarding deaths due to accident,
suicide, and disease and concluded that social phenomena
do not occur randomly. He published his observations in a
paper titled, “Natural and Political Observations . . . Made
upon the Bills of Mortality.” King Charles II, impressed by
Graunt’s work, nominated him to the Royal Society of
London even though he had few academic credentials.
Edmund Halley, the noted English astronomer, continued the work of Graunt in a paper titled “An Estimate of the
Degrees of the Mortality of Mankind, Drawn from Curious
Tables of the Births and Funerals at the City of Breslaw;
with an Attempt to Ascertain the Price of Annuities upon
Lives.” By studying the mathematics of life expectancy,
Halley helped lay the foundations for actuarial science,
which insurance companies use today to determine their
premiums.
The range is generally not useful to measure the spread of a distribution because it can
be influenced by a single outlier. For example, the range of the distribution 2, 2, 2, 2, 3, 4,
100 is 100 - 2 = 98.
Standard Deviation
KEY POINT
The standard deviation is
a reliable measure of
dispersion.
Data Value,
x
Deviation
from Mean,
x ⴚ xq
16
-1
14
-3
12
-5
21
4
22
5
Total
0
TABLE 15.11 Deviations from
the mean for data values in
distribution A.
A better way to find the spread of data is to use the standard deviation, a measure based on
calculating the distance of each data value from the mean.
D E F I N I T I O N If x is a data value in a set whose mean is xq, then x - xq is called x ’s
deviation from the mean.
To illustrate the deviation from the mean, consider distribution A: 16, 14, 12, 21, 22,
whose mean is 17. We list the deviation of each data value from the mean in Table 15.11.
It might seem reasonable to measure the spread in A by averaging the deviations from
the mean in Table 15.11. However, this will not work. As you see in Figure 15.10, for
scores above the mean that have positive deviations from the mean, there must be scores
below the mean that have negative deviations from the mean. When we add the positive
and negative deviations, they cancel each other, giving a total of zero.
To avoid this cancellation, we square each of these deviations, as shown in
Table 15.12.
(–5)
12
(–3)
14
(–1)
16
12
14
16
(+4) (+5)
21
22
18
20
22
Mean = 17
FIGURE 15.10 Values having positive and negative
deviations from the mean must balance about the mean.
*Some see this rhyme as describing the rosy rash, the pockets full of medicinal herbs, the sneezing and eventual death
associated with the Black Plague. If so, it owes its origin, like statistics, to that horrible time in European history.
Copyright © 2010 Pearson Education, Inc.
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CHAPTER 15 y Descriptive Statistics
Data Value,
x
Deviation from Mean,
x ⴚ xq
Deviation Squared,
( x ⴚ xq ) 2
16
-1
1
14
-3
9
12
-5
25
21
4
16
22
5
25
TABLE 15.12 The squares of deviations from the mean for distribution A.
If we now average* these squared deviations, we get
76
1 9 25 16 25
19.
4
51
n1
Clearly, this number is too large to represent the spread of scores from the mean. To
compensate for the fact that we squared the deviations in doing our calculations, we take
the square root of this number to get 119 L 4.36. This number is a more reasonable measurement of the way scores vary from the mean. The quantity that we just calculated is
called the standard deviation.
D E F I N I T I O N We denote the standard deviation of a sample of n data values by s,†
which is defined as follows:
s=
a (x - xq )
A n-1
2
We compute the standard deviation in four steps.
C O M P U T I N G T H E S T A N D A R D D E V I A T I O N To compute the standard
deviation for a sample consisting of n data values, do the following:
1.
Compute the mean of the data set; call it xq.
2.
Find ( x - xq )2 for each score x in the data set.
3.
Add the squares found in step 2 and divide this sum by n - 1; that is, find
2
a1x - xq2
,
n-1
which is called the variance.
4.
Compute the square root of the number found in step 3.
EXAMPLE 2
Calculating the Standard Deviation
Dunder Mifflin has hired six interns. After 4 months, their work records show the following number of work days missed for each worker:
0, 2, 1, 4, 2, 3
Find the standard deviation of this data set.
*When doing this calculation for a sample (as opposed to a population), statisticians divide by n - 1 instead of n.
There are technical reasons for doing this that are beyond the scope of this text. In doing this calculation for a
population, we would divide by n rather than n - 1.
†We represent the standard deviation of a population by s (instead of s), which we calculate using the formula
s=
A
2
a(x - m) .
n
Copyright © 2010 Pearson Education, Inc.
15.3 y Measures of Dispersion
743
0 + 2 + 1 + 4 + 2 + 3 12
=
= 2. We calculate the
6
6
squares of the deviations of the data values from the mean in the following table.
SOLUTION: The mean of this data set is
Number of
Days Missed
Deviation from
Mean
Square of Deviation
from the Mean
0
-2
4
2
0
0
1
-1
1
4
2
4
2
0
0
3
1
1
©(x - 2)2 = 10
There are six scores in the distribution, so the standard deviation is
Quiz Yourself
s=
9
Find the standard deviation of the
following sample of data values:
3, 4, 5, 6, 4, 2, 0, 8, 4
2
10
a(x - 2) =
L 1.41.
A 6-1
A5
n-1
This relatively small standard deviation shows that the data values are not spread too
far from the mean of 2.
Now try Exercises 5 to 14. ] 9
In Example 3, we calculate the standard deviation of a distribution using a frequency table.
EXAMPLE 3
Using a Frequency Table to Compute
the Standard Deviation
Anya is considering investing in WebSoft, a software company that specializes in Internet
applications. She intends to compute the standard deviation of its recent price changes to
measure how steady its stock price has been recently. The following are the closing prices
for the stock for the past 20 trading sessions:
37, 39, 39, 40, 40, 38, 38, 39, 40, 41,
41, 39, 41, 42, 42, 44, 39, 40, 40, 41
What is the standard deviation for this data set?
S O LU T I O N : In Table 15.13, we see that the sum of the 20 closing prices is 800, so the
mean is 800
20 = 40. We have added columns to the table for the deviations from the mean, the
squares of these deviations, and so on.
Closing
Price, x
Frequency,
f
Product,
x·f
Deviation,
(x ⴚ 40)
Deviation
Squared,
(x ⴚ 40)2
Product,
(x ⴚ 40)2 · f
37
1
37
-3
9
9
38
2
76
-2
4
8
39
5
195
1
5
40
5
200
-1
0
0
0
41
4
164
1
1
4
42
2
84
2
4
8
44
1
44
4
16
16
© f = 20
©(x · f ) = 800
©(x - 40)2 · f = 50
TABLE 15.13 Computations necessary to find the standard deviation of WebSoft closing prices.
Copyright © 2010 Pearson Education, Inc.
744
CHAPTER 15 y Descriptive Statistics
When calculating the standard deviation, we must multiply the squared deviation of
each price from the mean by its frequency. We list these products in the column labeled
“Product, (x - 40)2 · f,” and show the sum of these products at the bottom of the column.
We can now calculate the standard deviation. Remember that ©f is the number of data
values that we have been representing by n. The standard deviation is therefore
s=
2#
50
a(x - 40) f =
L 1.62.
A
A 19
n-1
This relatively small standard deviation indicates that the closing prices for the WebSoft stock have not been varying very much lately. If Anya is a cautious investor, she will
find the stability of WebSoft’s prices appealing.
Now try Exercises 15 to 20. ]
We summarize the method we use for finding the standard deviation in the following
formula.
F O R M U L A F O R C O M P U T I N G T H E S A M P L E S TA N DA R D D E V I AT I O N
F O R A F R E Q U E N C Y D I S T R I B U T I O N We calculate the standard deviation, s, of
a sample that is given as a frequency distribution as follows:
s=
2#
a(x - qx ) f
A n-1
where xq is the mean of the distribution, f is the frequency of data value x, and n = ©f,
the number of data values in the distribution. 10
Quiz Yourself
10
The investor in Example 3 also gathered the closing stock prices for WebRanger, another
Internet software company. We list these stock prices in Table 15.14. The mean closing price
is 42. Find the standard deviation for this distribution.
Closing Price, x
36
37
38
39
40
41
42
43
44
45
2
0
0
3
1
3
0
1
5
5
Frequency, f
TABLE 15.14 WebRanger closing stock prices for 20 days.
Comparing the standard deviation of 1.62 for WebSoft with the standard deviation of
3.01 for WebRanger in Quiz Yourself 10, we see that WebRanger’s stock is much more
volatile than WebSoft’s stock.
We have said that the standard deviation describes the spread of a distribution. In
Figure 15.11, all three distributions have a mean and median of 5; however, as the spread
of the distribution increases, so does the standard deviation.
5
4
3
2
1
0
3
4
5
6
7
s 0.82
3
4
5
6
7
s 1.29
2
3
4
5
6
s 2.38
FIGURE 15.11 As the spread of the distribution increases, so does the
standard deviation.
Copyright © 2010 Pearson Education, Inc.
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8
15.3 y Measures of Dispersion
745
HIGHLIGHT ¶ ¶ ¶
Using a Graphing Calculator to Find the Standard Deviation*
Because calculating the standard deviation can become
tedious, we often use a calculator to save work, as we show in
the accompanying TI calculator screen. We have stored the
data from Example 3 in list L1, and then used the calculator to
find the standard deviation, as we see in the screen below.
With such powerful technology available, it is very
important that you think critically to set up problems
correctly rather than just focusing on doing computations.
mean
sample standard deviation
population standard deviation
KEY POINT
We use the coefficient of
variation to compare the
standard deviations of
different data sets.
The Coefficient of Variation
In Example 3, we used the standard deviation with one set of data. If you use the standard
deviation to compare two sets of data, the data must be similar and have comparable
means. The following situation shows why.
Suppose we are comparing the body weights of two groups of people and find the standard deviation to be 3 pounds for the first group and 10 pounds for the second group. Can
we state that there is more uniformity in the first group than the second? Before answering
this question, here is more information. The first group consists of preschool children,
whereas the second is a group of National Football League linemen. Clearly, the standard
deviation of 3 is more significant, relatively speaking, for a group of preschoolers with a
mean weight of 30 pounds than the standard deviation of 10 for a group of football players
with a mean weight of 300 pounds. In order to use the standard deviation effectively to
compare different sets of data, we must first make the numbers comparable. We do this by
finding the coefficient of variation.
D E F I N I T I O N For a set of data with mean xq and standard deviation s, we define the
coefficient of variation, denoted by CV, as
CV =
s#
100%.
qx
Note that the coefficient of variation compares the standard deviation to the mean and is
expressed as a percentage. The coefficient of variation for the group of preschool children is
CV =
3 #
100% = 10%.
30
In contrast, the coefficient of variation for the group of NFL football linemen is
CV =
10 #
100% = 3.3%.
300
Because the coefficient of variation is larger for the group of preschoolers, we conclude
that there is more variation, relatively speaking, in their weights than in the weights of the
football players.
*Ask your instructor for tutorials on using technology in statistics.
Copyright © 2010 Pearson Education, Inc.
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CHAPTER 15 y Descriptive Statistics
EXAMPLE 4
Using the Coefficient of Variation
to Compare Data
Use the coefficient of variation to determine whether the women’s 100-meter race (328 feet)
or the men’s marathon (26 miles) has had more consistent times over the five Olympics
listed in Table 15.15.
Women’s 100 Meters
Men’s Marathon
2004
10.93 sec
2 h, 10 m, 55 sec (7,855 sec)
2000
10.75 sec
2 h, 10 m, 11 sec (7,811 sec)
1996
10.94 sec
2 h, 12 m, 36 sec (7,956 sec)
1992
10.82 sec
2 h, 13 m, 23 sec (8,003 sec)
1988
10.54 sec
2 h, 10 m, 32 sec (7,832 sec)
TABLE 15.15 Olympic times in women’s 100-meter race and men’s
marathon.
S O LU T I O N : Using a calculator, we found that the mean time for the 100-meter race is
10.796 and the standard deviation is approximately 0.163. For the marathon, the mean is
7,891.4 and the standard deviation is approximately 83.58.
Thus, the coefficient of variation for the women’s 100-meter race is
0.163 #
100% L 1.51%.
10.796
For the men’s marathon, the coefficient of variation is
83.5 #
100% L 1.06%.
7,891.4
Surprisingly, using the coefficient of variation as a measure, there is less variation in the
times for the marathon than for the 100-meter race! ]
Exercises
15.3
Looking Back*
These exercises follow the general outline of the topics presented in
this section and will give you a good overview of the material that
you have just studied.
Find the range, mean, and standard deviation for the following
data sets.
5. 19, 18, 20, 19, 21, 20, 22, 18, 18, 17
6. 89, 72, 100, 87, 65, 98, 77, 92
1. What problem did we point out in Example 1 that can occur when we use the range to measure the spread of a
distribution?
7. 5, 7, 9, 4, 6, 8, 7, 10
2. What do you get if you average the deviations from the mean
without squaring them?
10. 21, 3, 5, 11, 7, 1, 9, 7
3. What is the purpose of the coefficient of variation?
8. 8, 4, 7, 6, 5, 5, 4, 9
9. 2, 12, 3, 11, 14, 5, 8, 9
11. 18, 3, 8, 7, 7, 9, 13, 7
12. 22, 18, 15, 21, 21, 15, 19, 13
4. What was our point in the technology Highlight?
13. 3, 3, 3, 3, 3, 3, 3
Sharpening Your Skills
14. 4, 6, 4, 6, 4, 6, 4, 6
If you want, use a calculator or computer software to solve the following exercises. Some of your answers may differ slightly from
ours due to differences in the way we round off our calculations.
15. The following frequency table shows the number of fires per
month in a city. Complete the table entries to find the mean and
standard deviation for this distribution.
*Before doing these exercises, you may find it useful to review the note How to Succeed at Mathematics
on page xix.
Copyright © 2010 Pearson Education, Inc.
15.3 y Exercises
Number,
x
Frequency,
f
2
1
3
1
4
0
5
2
6
3
7
4
8
4
9
3
10
Deviation2,
( x ⴚ xq )2
Deviation,
( x ⴚ xq )
Product,
x·f
Product,
( x ⴚ xq )2 # f
2
©f =
747
©(x - qx )2 # f =
©x · f =
Table for Exercise 15
16. The following frequency table summarizes the number of job offers made to graduates of a network administrator certification program.
Complete the table entries to find the mean and standard deviation for this distribution.
Number,
x
Frequency,
f
4
2
5
0
6
3
7
5
8
4
9
5
10
Product,
x·f
Deviation2,
( x ⴚ xq )2
Deviation,
( x ⴚ xq )
1
©f =
©(x - qx )2 # f =
©x · f =
19.
Find the mean and standard deviation for the following frequency
distributions.
17.
Product,
( x ⴚ xq )2 # f
20.
x
f
x
f
3
1
12
3
x
f
4
1
13
0
2
8
3
5
0
14
5
3
4
9
2
6
2
15
0
4
2
10
1
7
5
16
1
5
0
11
2
8
3
17
2
6
4
12
3
9
3
18
3
x
f
2
18.
Use the following graphs to solve Exercises 21 and 22.
8
7
6
5
4
3
2
1
0
4
5
(a)
6
2
3
4
5
(b)
6
7
8
2
3
4
5
(c)
Copyright © 2010 Pearson Education, Inc.
6
7
8
3
4
5
(d)
6
7
CHAPTER 15 y Descriptive Statistics
748
21. a. Which data set has the smallest standard deviation?
b. The largest?
22. Rank the data sets in order from smallest standard deviation to
largest. You can do this just by looking at the graphs and without doing any calculations.
26. Summarizing vital statistics. The following is the 2008 roster
of the Houston Comets in the WNBA. Find the mean and the
population standard deviation of these data.
Player
Weight (lbs)
Matee Ajavon
160
Applying What You’ve Learned
Latasha Byears
206
23. Summarizing test score data. The following are the scores of
20 people who took a paramedics licensing test. Find the mean
and standard deviation for these data.
Tamecka Dixon
148
Sequoia Holmes
155
Shannon Johnson
152
Crystal Kelly
190
Sancho Lyttle
175
Mwadi Mabika
165
Hamchétou Maïga-Ba
160
Ashley Shields
155
Michelle Snow
158
Tina Thompson
178
Marcedes Walker
253
Erica White
135
Mistie Williams
184
Score
Frequency
72
73
78
84
86
93
7
1
3
6
2
1
24. Summarizing age data. The following are the ages of 16 World
War II veterans who are attending a reunion commemorating
the Normandy invasion. Find the mean and the standard deviation for these data.
Source: www.wnba.com
27. Summarizing customer data. The manager at the local
Starbucks counted the customers once each hour over a busy
weekend. We summarize the results she obtained in the bar
graph below. Find the mean and standard deviation of
these data.
14
Frequency
12
Age
Frequency
82
83
84
86
93
95
4
2
4
2
3
1
10
8
6
4
2
0
0
25. Summarizing tax data. The following table lists the state
income tax for a person earning $50,000 per year for several
states. Find the mean and sample standard deviation for these
data. (Source: The World Almanac, 2008)
1
2
3
4 5 6 7 8 9
Number of Customers
10 11 12
28. Find the mean and sample standard deviation for the distribution
given by the following bar graph.
5
Income Tax (%)
Frequency
4
State
3
Arizona
3.36
Colorado
4.63
Hawaii
8.25
1
Kansas
6.45
0
Massachusetts
5.3
New York
6.85
Pennsylvania
3.07
Virginia
5.75
2
1
2
3 4
Value
5
6
29. Family incomes. The following table gives the annual incomes
for eight families, in thousands of dollars. Find the number of
standard deviations family H’s income is from the mean.
Copyright © 2010 Pearson Education, Inc.
15.3 y Exercises
Family
A
B
C
D
E
F
G
H
Annual Income
(in $thousands)
47
48
50
49
51
47
49
51
Table for Exercise 29
30. Family incomes. The following table gives the annual incomes
for eight families, in thousands of dollars. Find the number of
standard deviations family A’s income is from the mean.
with a standard deviation of 1.2, did WebRanger have more or
less volatility than the DJIA during that week?
37. Price stability. In the following table, we list the monthly
price (in dollars) of a pound of coffee and a gallon of unleaded
gasoline for 2007. Find the population standard deviation of
each set of data. (Source: Bureau of Labor Statistics)
2007
Coffee Prices
$/lb
Gasoline Prices
$/gal
Jan.
3.288
2.274
Feb.
3.456
2.285
Mar.
3.475
2.592
In Exercises 31 and 32, you are given a distribution of averages in a
literature class. The professor in this class assigns grades as follows:*
Apr.
3.437
2.860
May
3.308
3.130
A: averages that are at least 1.5 standard deviations above the mean
Jun.
3.407
3.052
B: averages that are between 0.5 standard deviation above the
mean, and 1.5 standard deviations above the mean
Jul.
3.529
2.961
Aug.
3.497
2.782
C: averages that are between 0.5 standard deviation below the
mean, and 0.5 standard deviations above the mean
Sep.
3.537
2.789
D: averages that are between 1.5 standard deviations below the
mean, and 0.5 standard deviation below the mean
Oct.
3.577
2.793
Nov.
3.607
3.069
Dec.
3.685
3.020
Family
A
B
C
D
E
F
G
H
Annual Income
(in $thousands)
49
51
52
51
51
50
52
52
749
F: averages that are more than 1.5 standard deviations below
the mean
Source: Bureau of Labor Statistics, U.S. Department of Energy
31. Assigning grades. The averages in the class are 80, 76, 81, 84,
79, 80, 90, 75, 75, and 80. What grade does the person earning
the 76 get?
38. Comparing price stability. Consider the table given in Exercise 37. Use the coefficient of variation to determine whether
coffee prices or gasoline prices were more stable in 2007.
32. Assigning grades. The grades in the class are 72, 71, 73, 70,
71, 79, 65, 73, 74, and 72. What grade does the person earning
the 74 get?
39. Comparing family incomes. Consider the tables given in
Exercises 29 and 30, respectively. Suppose that the eight families are the same in both tables and the incomes are for two
different years that are 2 years apart. Which of the families had
the greatest improvement in annual income over the 2 year
period with respect to the other families? Explain how you
arrived at your conclusion.
In Exercises 33–36, we present information on the performance of
various stocks during a given month. Stocks with greater coefficients of variation are considered more volatile.
33. Comparing stocks. Suppose for a given month that the mean
daily closing price for Apple Computer common stock was
123.76 and the standard deviation was 12.3. For Dell stock, the
mean daily closing price was 78.6 with a standard deviation
of 7.2. Which stock was more volatile?
34. Comparing stocks. Suppose for a given month that the mean
daily closing price for Netflix was 37.4 and the standard deviation was 4.1. For Blockbuster, the mean daily closing price was
18.6 with a standard deviation of 3.2. Which stock was more
volatile?
35. Comparing stocks. The Dow Jones Industrial Average (DJIA)
measures the stock prices of a large group of stocks. If, for a
given week, the DJIA had a mean daily closing price of
11,261.12 with a standard deviation of 72.17 and WebMaster
stock had a mean closing price of 37.6 with a standard deviation of 1.7, did WebMaster have more or less volatility than the
DJIA during that week?
36. Comparing stocks. If, for a given week, the DJIA had a mean
daily closing price of 10,834.8 with a standard deviation of
144.5 and WebRanger stock had a mean closing price of 123.6
40. Comparing family incomes. Suppose that the mean family
income in the United States was $48,000 with a standard deviation of $1,000 in year X. Also, the mean family income was
$51,000 with a standard deviation of $2,250 three years later,
in year X + 3. If a family earned $50,000 during year X and
$54,000 three years later, then in which of these 2 years did the
family have a better annual income in relation to the rest of the
population? Explain how you arrived at your answer.
Communicating Mathematics
41. What do we do differently when we calculate the standard
deviation of a sample versus a population?
Try to answer each of the following as true or false without doing
any computations. Explain your answers in words or give appropriate examples or counterexamples to support your answers.
42. The standard deviation of the set of numbers -2, 2, -2, 2, -2, 2,
-2, 2 is zero.
43. If the standard deviation of a set of data is zero, then all the
numbers in the data set are the same.
*If an average falls on the border of two grades, the professor will assign the higher grade.
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750
CHAPTER 15 y Descriptive Statistics
44. The more numbers there are in a distribution, the larger the
standard deviation.
Using Technology to Investigate
Mathematics
d. What conclusion can you make about changes in the mean
and the standard deviation when the same number is added
to or subtracted from each score in a data set?
e. Use the conclusion reached in part (d) to simplify the calculation of the mean and standard deviation for the distribution 598, 597, 599, 596, 600, 601, 602, 603.
45. Search the Internet for “statistical calculators.” Duplicate some
of the examples in Sections 15.2 and 15.3. Report on your
findings.
49. a. Pick any five numbers and compute the mean and standard
deviation for this data set.
46. See your instructor for tutorials on using a graphing calculator
to do statistical calculations. Use a graphing calculator to redo
some of the examples and exercises in this section.
b. Multiply each number in the data set you created in part (a)
by 4, and compute the mean and standard deviation for this
new data set.
47. Data on the Internet. Do a search on the Internet to locate
data in an area that is of interest to you. For example, you can
go to the New York Times Web site and find the number of
weeks that the top 10 fiction books have been on the bestseller
list. Or, you can locate statistics from the Web site of your
favorite sports team. Other good sources of data are the Bureau
of Labor Statistics and the U.S. Department of Transportation
Web sites. After you have located a site, choose a data set and
find its mean and standard deviation.
c. Multiply each number in the data set you created in part (a)
by 9, and compute the mean and standard deviation for this
new data set.
For Extra Credit
48. a. Pick a data set consisting of any five numbers. Compute the
mean and the standard deviation of this data set. (Consider
the data set to be a sample.)
b. Add 20 to each number in the data set you created in part (a),
and compute the mean and standard deviation for this new
data set.
c. Subtract 5 from each number in the data set you created in
part (a), and compute the mean and standard deviation for
this new data set.
d. What conclusion can you make about changes in the mean
and standard deviation when each score in a data set is multiplied by the same number?
e. The mean is 4 and the standard deviation is 2.2 for the data
set 3, 4, 7, 1, 5. Consider your conclusion from part (d), and
compute the mean and standard deviation for the data set
15, 20, 35, 5, 25.
50. Consider the two distributions given by the bar graphs in Exercises 27 and 28. Can you simply look at the graphs to decide
which of the two distributions has the greater standard deviation? If so, describe what you look for in the graphs on which
to base your decision.
51. Comparative cost of living. According to an Internet service that
compares the cost of living in one location with the cost of living
in another, a salary of $100,000 in Manhattan is comparable
to $38,000 in Allentown, Pennsylvania. What statistical methods
do you think this service uses in determining this information?
Objectives
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