Chapter 1.

144

108

145

109



Chapter 3 Describing Data Numerically

3.1 Measures of Center

Chapter 3 Describing Data Numerically

3

Describing Data Numerically

Introduction

In Chapter 3, students develop numerical summaries to help them discover important characteristics about a data set. They also become acquainted with some powerful and widespread methodologies for applying the tools of descriptive statistics.

Section 3.1 introduces measures of center—the mean, the median, and the mode. Section 3.2 introduces measures of variability—the range, the variance, and the standard deviation, as well as their applications: the Empirical Rule and Chebyshev’s Rule. Section 3.3 discusses how to work with grouped data. Section 3.4 introduces us to measures of position, including z-scores, percentiles, percentile ranks, and quartiles, and how to use z-scores to detect outliers. Section 3.5 discusses the five-number summary, boxplots, and how to use the IQR method to detect outliers.

From the Author

The Chapter 3 Case Study (Can the Financial Experts Beat the Darts?) has been extended throughout the chapter.

Section 3.1 Measures of Center

● Stress the notion of the mean representing the “balance point” of the data, so that students may check their calculations throughout the remainder of the course.

● Early in Section 3.1, you may wish to review the definitions of population and sample.

● The What if scenario, page 115. Usually, this feature is structured in such a way that a calculator will not help. Instead, students need to think about how a change in one aspect of the problem will affect other aspects of the situation.

● Construct Your Own Data Sets, page 125 and page 148. This is a good way for students to apply their understanding of the concepts, by making up their own list of numbers that satisfies a particular set of conditions.

Section 3.2 Measures of Variability

● While many (most?) students now learn mean, median, and mode (Section 3.1) in elementary school, not so many learn about the standard deviation or the variance (Section 3.2). So, for most students, most of the material in this section (and subsequent sections) will be new.

● Discovering Statistics stresses what the statistics mean. This can be helpful when checking calculations, such as the standard deviation. If the student understands what a deviation means, and understands that the standard deviation represents a typical deviation, then the student may catch a calculation error.

Section 3.3 Working with Grouped Data

● Some instructors find that they do not have time to cover Section 3.3. If you choose to omit this section, you may wish to cover Objective 1, The Weighted Mean, using the grading policy in your syllabus as an example.

Section 3.4 Measures of Relative Position and Outliers

● Example 21 has been provided to underscore the fact that z-scores do not have to follow a bell-shaped distribution.

● The dance score data set, which was not real, has been replaced by an exports data set, which represents real data. This data is used for several examples in Sections 3.4 and 3.5.

Section 3.5 Five-Number Summary and Boxplots

● We have moved the section on five-number summary and boxplots ahead of the section on the Empirical Rule and Chebyshev’s Rule. This is because the boxplot uses the quartiles and the IQR, which were learned in the previous section.

Teaching Tips

Students may experience a steeper learning curve beginning at Section 3.2. The material in Section 3.1—mean, median, and mode—is often covered in high school or earlier. The material in Section 3.2 on variability is not usually covered in high school and is very important to an understanding of statistics. Stress the concept of spread—how spread out a data set is. The more spread out the data, the larger the measure of spread will be, whether it is the range, variance, standard deviation, or interquartile range.

In-Class Activities

 1. Access the data set for Old Faithful at the following Web site: www.stat.cmu.edu/~larry/all-of-statistics/=data/faithful.dat.

The data set consists of 225 values in an Excel file for the variables duration and interruption time. The Yellowstone National Park Web site states that “Old Faithful erupts every 35–120 minutes for 1.5–5 minutes” (www.yellowstone.net/geysers/old-faithful). Use the data to construct appropriate graphs for the duration times and interruption times for Old Faithful. Ask students, “What can you say about the distribution of these variables?” Ask them to compute numerical summaries for these two variables. Ask which data set has more variability.

Measures of Center

 2. What is your guess of the typical height of all students in your class?

 3. Make a dotplot of the heights of the students in your class.

 4. Discuss where to place the center of this distribution of student heights. Without crunching any numbers, form a consensus on the location of the center.

 5. Calculate the mean, median, and mode of the student heights.

 6. Which measure (mean, median, or mode) comes closest to the consensus of where the center is located in (4)?

 7. What is the relation between these measures and your guess of the typical height in (2)?

 8. Which measure (mean, median, mode, class consensus, your guess) do you think is the best measure of the center of student heights?

Measures of Spread

 9. Do you think that the distribution of the heights of all students in your class is more spread out or less spread out than the distribution of the heights of only the females in your class?

10. Would the values of our measures of spread (range, standard deviation) be larger for the entire class or for only the females?

11. Make a dotplot of the heights of only the females in the class. Make sure it uses the same scale as the dotplot for the heights of all the students in the class.

12. Use the two dotplots to assess which group has greater variability.

13. Back up your intuition by calculating and comparing our measures of spread (range, standard deviation) for the two groups.

Supplements

● StatTutor 2.1–2.10

● EESEE case studies for describing data numerically

● Weighing Trucks in Motion (Question 2 on mean, median, and standard deviation)

● Acorn Size and Oak Tree Range (Question 7 on boxplots, Question 2 on mean and standard deviation, Question 3 on range and standard deviation)

● Faculty Salary Comparison (Question 1 on boxplots, Question 3 on weighted averages, Question 4 on ranking and means)

Applets

The Mean and Median applet is referenced in Chapter 3 to compute values for the mean and median and for Exercises 104 and 105 in Section 3.1.

Activities and applets that relate to measures of center, spread, and boxplots can be found at http://mathforum.org/mathtools/tool/12489/.

The site Online Statistics: An Interactive Multimedia Course of Study has numerous applets and activities: http://onlinestatbook.com/index.html.

Videos

● Against All Odds: Inside Statistics: www.learner.org/resources/series65.html

● Program 3: Histograms

Web Sites

● CAUSEweb provides resources for statistics education: https://www.causeweb.org/resources/.

● The following Web site has a collection of 20 class projects: www.amstat.org/publications/jse/v6n3/smith.html.

● This Texas Instruments Web site has a host of TI-83/84 statistics activities: http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_83_statistics.html.

● This Web site has a host of activities, simulations, and so on, which relate to elementary statistics: http://davidmlane.com/hyperstat/ch2_contents.html.

● This Web site lists other sites that do statistical calculations: http://statpages.org/.

3

Describing Data Numerically

OVERVIEW

3.1 Measures of Center

3.2 Measures of Variability

3.3 Working with Grouped Data

3.4 Measures of Relative Position and Outliers

3.5 Five-Number Summary and Boxplots

Chapter 3 Formulas and Vocabulary

Chapter 3 Review Exercises

Chapter 3 Quiz

Mark Hooper/Getty Images

Can the Financial Experts Beat the Darts?

Have you ever wondered whether a bunch of monkeys throwing darts to choose stocks could select a portfolio that performed as well as the stocks carefully chosen by Wall Street experts? The Wall Street Journal (www.wsj.com) apparently believed that the comparison was worth a look. The Journal ran a contest between stocks chosen randomly by Journal staff members (instead of monkeys) throwing darts at the Journal stock pages (mounted on a board) and stocks chosen by a team of four professional financial experts. At the end of six months, the Journal compared the percentage change in the price of the experts’ stocks and the dartboard’s stocks and compared both to the Dow Jones Industrial Average as well. So, who do you think did better? Did the six-figure-salary financial experts put the random dart selections to shame?

●●

In Section 3.1, we do some graphical exploration with the data, comparing the balance points (means) of each group using comparative dotplots. We then determine whether the student’s intuition of the location of the means is confirmed by the statistics.

●●

In Section 3.2, we compare the variability of the three groups and find that different measures of spread can disagree about which data set has more variability.

●●

In the Section 3.2 exercises, we calculate the coefficient of skewness for each group.

●●

In the Section 3.3 exercises, we examine how close the estimated mean, variance, and standard deviation for grouped data are to their true values.

●●

In the Section 3.4 exercises, we use the case study data to examine measures of relative position such as z-scores and percentiles.

●●

Finally, in the Section 3.5 exercises, we construct boxplots and identify outliers for each group in the case study data set.

THE BIG PICTURE

Where we are coming from and where we are headed . . .

●●

Chapter 2 showed us graphical and tabular summaries of data.

●●

Here, in Chapter 3, we “crunch the numbers,” that is, we develop numerical summaries of data. We examine measures of center, measures of variability, and measures of relative position.

●●

In Chapter 4, we will learn how to summarize the relationship between two quantitative variables.

The Mean

The most well-known and widely used measure of center is the mean. In everyday usage, the word average is often used to denote the mean.

The Web site CNET.com provides reviews and prices for gadgets and electronics, including cell phones. In Table 1, you will find all eight of the cell phones in CNET’s “Editors’ Picks” for June 27, 2014. Recall from Chapter 1 that a population is the collection of all elements of interest in a particular study. Thus, the data in Table 1 represents a population. Find the mean price of all the cell phones.

Solution

To find the mean, we add up the prices of all eight cell phones and divide by the number of phones:

The population mean price for all eight cell phones is $343.75.

Table 2 contains the number of tropical storms reported by the National Oceanic and Atmospheric Administration for 2006–2013. All years in this period are represented, so this can be considered a population. Find the population mean number of tropical storms.

(The solution is shown in Appendix A.)

Before we proceed, we need to learn some notation.

Notation

Statisticians like to use specialized notation. It is worth learning because it saves a lot of writing, and certain concepts can best be understood by using this special notation.

● The population size, the number of observations in your population, is always denoted as N. We have a population with eight observations in Example 1, so N  8.

● The sample size, which refers to how many observations you have in your sample data set, is always denoted as n.

● The shorthand notation for “the sum of all the data” is x, where x refers to the data, and  (capital sigma), which is the Greek letter for “S,” stands for “Summation.” Note in Example 1 that we added up the prices of all the cell phones. This summing is denoted as x.

● The population mean is denoted as m (pronounced “mew”), which is the Greek letter for m. As we saw in Example 1, to calculate the population mean, we add up all the data and divide by the population size, N. Thus, the formula for the population mean is:

● For Example 1, we therefore have:

● The sample mean is denoted as x_ (pronounced “x-bar”). You should try to commit this to long-term memory because x_ may be the most important symbol used in this book and will return again and again in nearly every chapter. The sample mean is calculated just like the population mean, except that we divide by the sample size n instead of the population size N. Thus, the formula for the sample mean is:

Suppose the cell phones in Table 3 represent a random sample of size four from the population in Table 1. Calculate the sample mean price of this sample of cell phones.

Solution

The sample mean price of this sample of four cell phones is calculated like this:

The sample mean cell phone price for this particular sample is $337.50. Of course, a different sample would have yielded a different value for x_.

Suppose we took a sample of size three instead and obtained the same sample as in Table 3, except that the Sony Xperia Z2 was not included.

a. Would you expect that the sample mean price would be higher or lower than $337.50? Explain.

b. Calculate the sample mean price for the sample of three cell phones. Was your intuition in (a) confirmed?

(The solutions are shown in Appendix A.)

Table 4 contains a sample of six home sales prices for Broward County, Florida, for June 27, 2014. We want to get an idea of the typical home sales price in Broward County.

a. Find the mean sales price of the homes in Table 4.

b. Suppose we add a seventh home in Hillsborough Beach, selling for $6 million. Calculate the mean sales price of all seven homes. Comment on how the extreme value affected the mean sales price.

Solution

a. The mean sales price of the homes in Table 4 is:

5 $422,500

b. Now, suppose that we append a seventh home to our sample: a home in Hillsborough Beach listed for $6 million, which is much more expensive than any of the other homes in the sample. Recalculating the mean, we get

Note that the mean sales price nearly tripled from $422,500 to $1,220,000 when we added this extreme value. Also, this new mean is much higher than every price in the original sample. Thus, it is highly unlikely that this new mean of about $1.2 million is representative of the typical sales price of homes in Broward County. This example shows how the mean is sensitive to the presence of extreme values. For situations like this, we prefer a measure of center that is not so sensitive to extreme values. Fortunately, the median is just such a measure.

The Median

Recall that the median strip on a highway is the slice of land in the middle of the two lanes of the highway. In statistics, the median of a data set is the middle data value when the data are put into ascending order. There are two cases, depending on whether the sample size is odd or even.

The case when the sample size is even is clear if you hold up four fingers on one hand. Notice that there is no unique finger in the middle. No middle value exists when the sample size is even, so we take the two data values in the middle and split the difference.

EXAMPLE 4 Median is not sensitive to extreme values

Show that the median is not sensitive to extreme values by doing the following:

a. Find the median sales price of the homes in Table 4.

b. Add the seventh home in Hillsborough Beach, selling for $6 million. Calculate the median sales price of all seven homes.

Solution

a. Fortunately, the data are already presented in ascending order in the table. Because n = 6 is even, the median is the mean of the two data values that lie on either side of the 5 3.5th position. That is, the median is the mean of the 3rd and 4th data values, $360,000 and $425,000. Splitting the difference between these two, we get

We note that, in Table 4, there are exactly as many homes with prices lower than $392,500 as homes with prices higher than $392,500.

b. Now, what happens to the median when we add in the $6 million home from Hillsborough Beach? Because n = 7 is odd, the median is the unique 5 4th observation, given by the home in Miramar for $425,000. The extreme value increased the median only from $392,500 to $425,000. In Example 3, we showed that the value of the mean price nearly tripled when the expensive home was added. Thus, the median home sales price is a better measure of center because it more accurately reflects the typical sales prices of homes in Broward County.

EXAMPLE 5 Using technology to find the mean and median

Find the mean and median of the home sales prices in Table 4, using (a) the TI-83/84, (b) Excel, (c) Minitab, and (d) JMP.

Solution

Using the instructions in the Step-by-Step Technology Guide on page 117, we get the following output:

a. The first TI-83/84 screen shows x 5 422,500 and n 5 6. The second screen shows the median, Med 5 392,500.

  

b. The mean and median are shown in the Excel output.

c. The mean and median are shown in the Minitab output.

d. The mean and median are shown in the JMP output.

3

The Mode

Sometimes the mode does not indicate the center of a data set. For example, suppose we have the following set of biology lab scores: 60, 80, 100, 100. The mode is 100, but it is not near the center of the data.

A third measure of center is called the mode. French speakers will recognize that the term mode in French refers to fashion. The popularity of clothing, cosmetics, music, and even basketball shoes often depends on just which style is in fashion. In a data set, the value that is most “in fashion” is the value that occurs the most.

The mode of a data set is the data value that occurs with the greatest frequency.

EXAMPLE 6 Finding the mean, median, and mode: Music videos

The Web site MTV.com contains music videos for many performers. Table 5 provides the number of music videos available for download for four performers, as of May 21, 2012. Find the (a) mean, (b) median, and (c) mode number of music videos.

Table 5 Music videos for four performers

Performer

Music Videos

Michael Jackson

31

Taylor Swift

26

Usher

26

Katy Perry

15

Solution

a. The sample mean number of music videos is

The mean number of music videos is 24.5.

b. Because n 5 4 is even, the median is the mean of the two middle data values:

Median 5 5 26 music videos.

c. The mode is the data value that occurs with the greatest frequency. Two performers have 26 music videos: Taylor Swift and Usher. No other data value occurs more than once. Therefore, the mode is 26 music videos, as shown in Figure 3.

FIGURE 3 Dotplot of music videos, showing 26 as the mode.

One of the strengths of the mode is that it can also be used with categorical, or qualitative, data. Suppose you asked your friends to name their favorite flower. Six of them answered “rose,” three answered “lily,” and one answered “daffodil.” Note that these data are categorical, not numerical. The most frequently occurring flower is “rose”; therefore, the rose represents the mode of the variable favorite flower. Unfortunately, we cannot use arithmetic with categorical variables, and thus the mean or median for this variable cannot be found.

It may happen that no value occurs more than once, in which case we say there is no mode. On the other hand, more than one data value could occur with the greatest frequency, in which case we would say there is more than one mode. Data sets with one mode are unimodal; data sets with more than one mode are multimodal.

Figure 1.10: WHAT IF?

What If Scenario

Consider Example 6 once again. Now imagine: what if there was an incorrect data entry, such as a typo, and the number of Michael Jackson’s videos was greater than 31 by some unspecified amount?

Describe how and why this change would have affected the following, if at all:

a. The mean number of music videos

b. The median number of music videos

c. The mode number of music videos

Solution

a. Consider Figure 4, a dotplot of the number of music videos, with the triangle indicating the mean, or balance point, at 24.5. Recall that this represents the balance point of the data. As the number of Michael Jackson’s videos increases (arrow), the point at which the data balance (the mean) also moves somewhat to the right. Thus, the mean number of followers will increase.

b. Recall from Example 6 that the median is the mean of the middle two data values. In other words, the mean ignores most of the data values, including the largest value, which is the only one that has increased. Therefore, the median will remain unchanged.

c. The mode also remains unchanged, because the only data value that occurs more than once is the original mode—26 music videos—and this remains unchanged.

FIGURE 4 As the number of Michael Jackson’s videos increases, so does the mean, but not the median or mode.

4

Skewness and Measures of Center

The skewness of a distribution can often tell us something about the relative values of the mean, median, and mode (see Figure 5).

FIGURE 5 How skewness affects the mean and median.

EXAMPLE 7 Mean, median, and skewness

The histogram of the average size of households in the 50 states and the District of Columbia from Example 21 of Chapter 2 (page 74) is reproduced here as Figure 6.

a. Based on the skewness of the distribution, state the relative values of the mean, median, and mode.

b. Use Minitab to verify your claim in (a).

Solution

a. The distribution of average household size is somewhat right-skewed. Thus, from Figure 6, we would expect the mean to be greater than the median, which is greater than the mode.

b. The Minitab descriptive statistics are shown here. Note that the mean is greater than the median, which is greater than the mode.

Can the Financial Experts Beat the Darts?

Recall the contest held by the Wall Street Journal to compare the performance of stock portfolios chosen by financial experts and stocks chosen at random by throwing darts at the Journal stock pages. We will examine the results of 100 such contests in various ways, using the methods we have learned thus far, and will return to examine them further as we acquire more analysis tools. Let’s start by reporting the raw result data. The percentage increase or decrease in stock prices was calculated for the portfolios chosen by the professional financial advisers and by the randomly thrown darts, and was compared with the percentage net change in the Dow Jones Industrial Average (DJIA).

Exploratory Data Analysis

Figure 7 shows comparative dotplots of the percentage net change in price for the professionally selected portfolio, the randomly selected darts portfolio, and the DJIA, over the course of the 100 contests. First, estimate the mean of each distribution by choosing the balance point of the data. This balance spot is the mean. For fun, write down your guess for the mean for the professionals so you can see how close you were when we provide the descriptive statistics later. Now compare this with where you would find the balance spot (mean) for the darts dotplot. Which numerical value is larger: the balance spot for the pros or the darts? Just think: you are comparing the mean portfolio performances for the professionals and the darts without using a formula or a calculator. This is exploratory data analysis. You are using graphical methods to compare numerical statistics.

FIGURE 7 Dotplot of the percentage net price change for the professionally selected portfolio, the randomly selected darts portfolio, and the DJIA.

Hopefully, you discovered that the estimated mean for the pros is greater than the estimated mean for the darts. This is not particularly surprising, is it? Next, find the balance point for the DJIA dotplot. Compare the numerical value for the DJIA balance spot with the mean you found for the dotplot for the pros. Write down your estimate of the means for the DJIA and darts dotplots, so you can see how close you were later. Again, hopefully, you found that the estimated professionals’ mean was higher than that of the DJIA. Now, a tougher comparison is to compare the estimated DJIA mean with that of the darts. Which of these two do you think is higher?

Finally, Minitab provides us with the mean percentage net price changes, as shown in Figure 8. Over the course of 100 contests, the mean price for the portfolios chosen by the professional financial advisers increased by 10.95%, by 6.793% for the DJIA, and by 4.52% for the random darts portfolio.

This is evidence in support of the view that financial experts can consistently outperform the market.

STEP-BY-STEP TECHNOLOGY GUIDE: Descriptive Statistics

TI-83/84

Step 1 Press STAT > 1: Edit. Enter the data in L1 using the instructions found in the Step-by-Step Technology Guide in Section 2.2.

Step 2 Press STAT. Use the right arrow button to move the cursor so that CALC is highlighted.

Step 3 Select 1-Var Stats, and press ENTER.

Step 4 On the home screen, the command 1-Var Statistics is shown. Press 2nd, then L1 (above the 1 key), and press ENTER.

EXCEL

Step 1 Enter the data in column A.

Step 2 Select Data > Data Analysis.

Step 3 Select Descriptive Statistics, and click OK.

Step 4 For the Input Range, click and drag to select the data in column A. If the variable name is at the top of the column, click Labels in the First Row.

Step 5 Check Summary Statistics, and click OK.

MINITAB

Step 1 Enter the data in column C1.

Step 2 Select Stat > Basic Statistics > Display Descriptive Statistics…

Step 3 The variable selection dialog box appears. Select the variable you want to summarize by double-clicking on it until it appears in the Variables box.

Step 4 Click Statistics…

Step 5 Select the desired statistics, and click OK. Then click OK.

SPSS

Step 1 Enter the data in the first column.

Step 2 Click Analyze > Descriptive Statistics > Frequencies…

Step 3 Click the variable name, then click the arrow to move it to the Variable(s) box.

Step 4 Click Statistics… and choose the desired statistics. Click Continue, and then OK.

JMP

Step 1 Click File > New > DataTable. Enter the data in Column 1.

Step 2 Click Tables > Summary.

Step 3 Select the column, and then select the desired statistics from the Statistics drop-down menu one by one. Click OK.

CRUNCHIT!

We will use the data from Example 3 (page 111).

Step 1 Click File, highlight Load from Larose, Discostat3e > Chapter 3, and click on Example 01_03.

Step 2 Click Statistics and select Descriptive Statistics. For Data, select Price, and then click Calculate.

Section 3.1 Summary

1. Measures of center are introduced in Section 3.1. The sample mean (x) represents the sum of the data values in the sample divided by the sample size (n). The population mean (m) represents the sum of the data values in the population divided by the population size (N). The mean is sensitive to the presence of extreme values.

2. The median occupies the middle position when the data are put in ascending order and is not sensitive to extreme values.

3. The mode is the data value that occurs with the greatest frequency. Modes can be applied to categorical data as well as numerical data but are not always reliable as measures of center.

4. The skewness of a distribution can often tell us something about the relative values of the mean and the median.

Section 3.1 Exercises

CLARIFYING THE CONCEPTS

 1. Explain what a measure of center is. (p. 108)

 2. Which measure may be used as the balance point of the data set? Explain how this works. (p. 110)

 3. Explain what we mean when we say that the mean is sensitive to the presence of extreme values. Explain whether the median is sensitive to extreme values. (pp. 111–112)

 4. What are the three measures of center that we learned about in this section? (p. 108)

For Exercises 5–12, either state what is being described or provide the notation.

 5. The number of observations in your sample data set (p. 109)

 6. The number of observations in your population data set (p. 109)

 7. Notation denoting “sum all the data” (p. 109)

 8. Notation for what we get when we add up all the data values in the population, and divide by how many observations there are in the population (p. 109)

 9. Notation for what we get when we add up all the data values in the sample, and divide by how many observations there are in the sample (p. 109)

10.

The middle data value when the data are put in ascending order (p. 112)

11.

The data value that occurs with the greatest frequency (p. 114)

12.

The sample mean (p. 109)

PRACTICING THE TECHNIQUES

Figure 1.15:

CHECK IT OUT!

To do

Check out

Topic

Exercises 13–18

Example 1

Population mean

Exercises 19–24

Example 2

Sample mean

Exercises 25–30

Example 3

Sensitivity of mean

Exercises 31–36

Example 4

Median

Exercises 37–40

Example 6

Mode

Exercises 41–44

Example 7

Mean, median, and skewness

For the data in Exercises 13–18:

a. Find the population size N.

b. Calculate the population mean m.

13.

State exports to other countries are shown in the table for the population of all New England states, for the month of June 2014, expressed in billions of dollars.

State

Exports

State

Exports

Connecticut

1.4

New Hampshire

0.4

Maine

0.3

Rhode Island

0.2

Massachusetts

2.4

Vermont

0.3

Source: U.S. Census Bureau.

14.

The number of wins for each baseball team in the population of the American League West division for 2013 is shown in the table.

Team

Wins

Team

Wins

Oakland Athletics

96

Seattle Mariners

71

Texas Rangers

91

Houston Astros

51

Los Angeles Angels

78

Source: MLB.mlb.com.

15.

The table provides the motor vehicle theft rate for the population of the top 10 countries in the world for motor vehicle theft, for 2012. The theft rate equals the number of motor vehicles stolen in 2012 per 100,000 residents.

Country

Theft rate

Country

Theft rate

Italy

208.0

Greece

100.2

France

174.1

Norway

94.1

USA

167.8

Netherlands

75.2

Sweden

117.2

Spain

75.1

Belgium

106.0

Cyprus

66.0

Source: United Nations Office on Drugs and Crime.

16.

The National Center for Education Statistics sponsors the Trends in International Mathematics and Science Study (TIMSS). The table contains the mean science scores for the eighth-grade science test for the populations of all Asian-Pacific countries that took the exam.

Country

Science score

Country

Science score

Singapore

578

Australia

527

Taiwan

571

New Zealand

520

South Korea

558

Malaysia

510

Hong Kong

556

Indonesia

420

Japan

552

Philippines

377

17.

The table contains the number of petit larceny cases for the population of all police precincts in South Manhattan in 2013.

Precinct

Petit larcenies

Precinct

Petit larcenies

1

2014

10

995

5

1288

13

2094

6

1555

14

4551

7

584

17

823

9

1607

18

2071

Source: New York City Police Department.

18.

The table contains the number of criminal trespass cases for the population of all police precincts in South Manhattan in 2013.

Precinct

Criminal trespasses

Precinct

Criminal trespasses

1

108

10

207

5

105

13

135

6

113

14

340

7

233

17

74

9

219

18

120

Source: New York City Police Department.

For the data in Exercises 19–24:

a. Find the sample size n.

b. Calculate the sample mean x.

19.

A sample of the state export data from Exercise 13 is provided in the table.

State

Exports

Connecticut

1.4

Massachusetts

2.4

Rhode Island

0.2

20.

A sample from the baseball data in Exercise 14 is shown here.

Team

Wins

Texas Rangers

91

Los Angeles Angels

78

Seattle Mariners

71

21.

A sample from the motor vehicle theft data in Exercise 15 is as follows.

Country

Theft rate

Italy

208.0

USA

167.8

Greece

100.2

22.

A sample from the science score data in Exercise 16 is given here.

Country

Science score

South Korea

558

Hong Kong

556

Japan

552

Australia

527

23.

The following sample is taken from the petit larceny data in Exercise 17.

Precinct

Petit larcenies

1

2014

6

1555

9

1607

14

4551

17

 823

24.

A sample taken from the criminal trespass data in Exercise 18 is as follows.

Precinct

Criminal trespasses

1

108

7

233

14

340

18

120

For Exercises 25–30, use the data from the indicated exercise, along with the indicated extreme, to show that the mean is more sensitive to extreme values. For each exercise, find the sample mean including the extreme value. Compare your answer to the mean calculated without the extreme value from the earlier exercise.

25.

Data from Exercise 19. Extreme value 5 10

26.

Data from Exercise 20. Extreme value 5 20

27.

Data from Exercise 21. Extreme value 5 1000

28.

Data from Exercise 22. Extreme value 5 0

29.

Data from Exercise 23. Extreme value 5 20,000

30.

Data from Exercise 24. Extreme value 5 1500

For Exercises 31–36, use the data from the indicated exercise, along with the indicated extreme, to show that the mean is more sensitive to extreme values than the median is. Do the following:

a. Calculate the median of the data without the extreme value.

b. Find the median of the data including the extreme value. Compare your answers from (a) and (b). Note that the median did not change as much as the mean did in Exercises 25–30.

31.

Data from Exercise 19. Extreme value 5 10

32.

Data from Exercise 20. Extreme value 5 20

33.

Data from Exercise 21. Extreme value 5 1000

34.

Data from Exercise 22. Extreme value 5 0

35.

Data from Exercise 23. Extreme value 5 20,000

36.

Data from Exercise 24. Extreme value 5 1500

For the data in Exercises 37–40, find the mode.

37.

The table contains the number of dangerous weapons cases for four police precincts in Manhattan.

Precinct

Dangerous weapons cases

1

19

5

24

20

24

22

9

38.

The Recording Industry Association of America (RIAA) awards multi-platinum status for any musical recording that sells more than 2 million copies. The table contains a random sample of 10 of the musical artists with the most multi-platinum singles.

Artist

Multi-platinums

Artist

Multi-platinums

Beyoncé

4

Linkin Park

2

Bruno Mars

4

The Beatles

4

Jay-Z

4

Michael Jackson

1

Katy Perry

8

Taylor Swift

8

Lady Gaga

6

Tim McGraw

2

Source: RIAA.

39.

The table contains the unemployment rates in August 2014 for 10 countries.

Country

Unemployment rate

Country

Unemployment rate

Britain

6.4

Japan

3.7

Canada

7.0

Mexico

4.8

China

4.1

Pakistan

6.2

India

8.8

South Korea

3.4

Italy

12.3

United States

6.2

Source: The Economist, www.economist.com/node/21604509.

40.

The table contains the top 10 most downloaded free apps for the IOS platform, as reported by Apple.com, along with the app type, for June 2014. Find the mode of App Type.

Rank

App

App type

Rank

App

App type

1

Two Dots

Games

6

Snap Chat

Photo and video

2

The Line

Games

7

Instagram

Photo and video

3

Traffic Racer

Games

8

The Test

Games

4

Rival Knights

Games

9

Republique

Games

5

Piano Tiles

Games

10

YouTube

Photo and video

For Exercises 41–44, consider the accompanying distributions. What can we say about the values of the mean, median, and mode in relation to one another for the given histograms?

A

B

C

41.

The distribution in A

42.

The distribution in B

43.

The distribution in C

44.

The distribution in D

APPLYING THE CONCEPTS

45.

NFL Football, Southern Style. The table contains the population of all the teams in the National Football Conference South Division, along with the number of wins in the 2013 season.

a. What is the population size, N, where the population is the NFC South Division?

b. What is the population mean number of wins, m?

NFC South team

Wins

Carolina Panthers

12

New Orleans Saints

11

Atlanta Falcons

4

Tampa Bay Buccaneers

4

46.

New England Electoral Votes. The table contains the population of all the New England states, along with their electoral votes.

a. What is the population size, N?

b. Calculate the population mean number of electoral votes, m.

Electoral votes

Connecticut

7

Maine

4

Massachusetts

11

New Hampshire

4

Rhode Island

4

Vermont

3

47.

NFL Football, Southern Style. Refer to the population data in Exercise 45. Suppose we take a sample from the population, and we get the Carolina Panthers and the Atlanta Falcons.

a. What is the sample size n?

b. Calculate the sample mean number of wins, x.

48.

New England Electoral Votes. Refer to the population data in Exercise 46. Suppose we take a sample from the population, and get Massachusetts, Rhode Island, and Vermont.

a. What is the sample size n?

b. Calculate the sample mean number of electoral votes, x.

Video Game Sales. The Chapter 1 Case Study looked at video game sales for the top 30 video games. The following table contains the total sales (in game units) and weeks on the top 30 list for a sample of five randomly selected video games. Use this information for Exercises 49 and 50.

Video game

Total sales in millions of units

Weeks

Super Mario Bros. U for WiiU

1.7

78

NBA 2K14 for PS4

0.6

27

Battlefield 4 for PS3

0.9

29

Titanfall for XBoxOne

1.2

10

Yoshi’s New Island for 3DS

0.2

10

Source: www.vgchartz.com.

Figure 1.20: videogamereg

49.

Find the following measures of center for total sales.

a. Mean

b. Median

50.

Calculate the following measures of center for weeks.

a. Mean

b. Median

Figure 1.21: CASE STUDY

Darts and the Dow Jones. The following table contains a random sample of eight days from the Chapter 3 Case Study data set, indicating the stock market gain or loss for the portfolio chosen by the random darts, as well as the Dow Jones Industrial Average gain or loss for that day. Use this information for Exercises 51 and 52.

51.

Find the following measures of center for the darts stock returns.

a. Mean

b. Median

52.

Find the following measures of center for the DJIA.

a. Mean

b. Median

Darts

DJIA

–27.4

–12.8

18.7

9.3

42.2

8

–16.3

–8.5

11.2

15.8

28.5

10.6

1.8

11.5

16.9

–5.3

Source: Wall Street Journal.

Figure 1.22: dartsdjia

Age and Height. The following table provides a random sample from the Chapter 4 Case Study data set body_females, showing the age and height of the eight women. Use this information for Exercises 53 and 54.

Age

Height

40

63.5

28

63

25

64.4

34

63

26

63.8

21

68

19

61.8

24

69

Source: Journal of Statistics Education.

53.

Find the following measures of center for the women’s ages.

a. Mean

b. Median

54.

Find the following measures of center for the women’s heights.

a. Mean

b. Median

Figure 1.23: ageheight

Saturated Fat and Calories. The table contains the calories and saturated fat in a sample of ten food items. Use this information for Exercises 55 and 56.

55.

Find the following measures of center for calories.

a. Mean

b. Median

56.

Find the following measures of center for the grams of saturated fat.

a. Mean

b. Median

Figure 1.24: satfatcorr

Food item

Calories

Grams of saturated fat

Chocolate bar (1.45 ounces)

216

7.0

Meat & veggie pizza (large slice)

364

5.6

New England clam chowder (1 cup)

149

1.9

Baked chicken drumstick (no skin, medium size)

75

0.6

Curly fries, deep-fried (4 ounces)

276

3.2

Wheat bagel (large)

375

0.3

Chicken curry (1 cup)

146

1.6

Cake doughnut hole (one)

59

0.5

Rye bread (1 slice)

67

0.2

Raisin bran cereal (1 cup)

195

0.3

Source: Food-a-Pedia.

Table 6 contains the trade balance currently maintained by the United States with a sample of 9 countries, for the month of June 2014. Use this data for Exercises 57–60.

TABLE 6 Trade balance

Country

Trade balance ($ billions)

Brazil

1

France

–1.2

Germany

–5.6

India

–1.3

Italy

–2.4

Japan

–5.6

South Korea

–1.8

Saudi Arabia

–1.8

United Kingdom

0

Source: Foreign Trade Division, U.S. Census Bureau.

57.

Find the sample size, n.

58.

Calculate the sample mean trade balance, x.

59.

Find the median.

60.

Find the modes.

Table 7 contains the number of cylinders, the engine size (in liters), the fuel economy (miles per gallon [mpg], city driving), and the country of manufacture for six 2011 automobiles. Use this information for Exercises 61–65.  

Figure 1.25: cylinderengine

TABLE 7 Cylinders, engine size, and fuel economy for six cars

Vehicle

Cylinders

Engine size

City mpg

Country of manufacture

Cadillac CTS

6

3.0

18

USA

Ford Fusion

Hybrid

4

2.5

41

USA

Ford Taurus

6

3.5

18

USA

Honda Civic

4

1.8

25

Japan

Rolls Royce

12

6.7

11

UK

Toyota Camry

Hybrid

4

2.4

31

Japan

Source: www.fueleconomy.gov.

61.

Find the following for the number of cylinders:

a. Mean b. Median c. Mode

62.

Refer to your work in Exercise 61. Which measure of center do you think is most representative of the typical number of cylinders? Explain.

63.

Find the following for the engine size:

a. Mean b. Median c. Mode

64.

Find the following for the city mpg:

a. Mean b. Median c. Mode

65.

Find the mode for country of manufacture.

Use the information in Table 8 to answer Exercises 66−68, which gives the number of wins for the top 10 NASCAR racing drivers in various categories. 

Figure 1.26: nascar

TABLE 8 Top 10 NASCAR winners in the modern era

Rank

Driver

Total

Super speedways

Short tracks

 1

Darrell Waltrip

84

18

47

 2

Dale Earnhardt

76

29

27

 3

Jeff Gordon

75

15

15

 4

Cale Yarborough

69

15

29

 5

Richard Petty

60

19

23

 6

Bobby Allison

55

24

12

 7

Rusty Wallace

55

 5

25

 8

David Pearson

45

20

 1

 9

Bill Elliott

44

16

 2

10

Mark Martin

35

 5

 7

Source: www.nascar.com.

66.

Refer to the super speedways data. Find the following:

a. Mean b. Median c. Mode

67.

Refer to the short tracks data. Find the following:

a. Mean b. Median c. Mode

68.

Refer to the totals data. Find the following:

a. Mean b. Median c. Mode

For Exercises 69–73, refer to Table 9, which lists the top five mass market paperback fiction books for the week of July 1, 2014, as reported by the New York Times.

TABLE 9 Top five best-sellers in paperback trade fiction

Rank

Title

Author

Price

1

A Game of Thrones

George R. R. Martin

$7.83

2

Takedown Twenty

Janet Evanovich

$7.64

3

Inferno

Dan Brown

$8.48

4

A Dance with Dragons

George R. R. Martin

$6.71

5

The 9th Girl

Tami Hoag

$8.47

69.

Find the mean, median, and mode for the price of these five books on the best-seller list. Suppose a salesperson claimed that the price of a typical book on the best-seller list is less than $14. How would you use these statistics to respond to this claim?

70.

Linear Transformations. Add $10 to the price of each book.

a. Now find the mean of these new prices.

b. How does this new mean relate to the original mean?

c. Construct a rule to describe this situation in general.

71.

Linear Transformations. Multiply the price of each book by 5.

a. Now find the mean of these new prices.

b. How does this new mean relate to the original mean?

c. Construct a rule to describe this situation in general.

72.

Find the mode for the following variables:

a. Price

b. Author

73.

Explain whether it makes sense to find the mean or median of the variable author.

Mode of Categorical Data. The New York City Police Department tracks the number and type of traffic violations. The table contains a random sample of 12 traffic violations and the borough in which they occurred (Manhattan or Brooklyn). Use the data for Exercises 74–76.

Violation type

Borough

Violation type

Borough

Cell phone

Brooklyn

Disobey sign

Manhattan

Safety belt

Manhattan

Speeding

Brooklyn

Cell phone

Brooklyn

Safety belt

Manhattan

Cell phone

Manhattan

Disobey sign

Manhattan

Speeding

Brooklyn

Disobey sign

Brooklyn

Safety belt

Manhattan

Cell phone

Manhattan

74.

Find the mode for violation type. Does this mean that most violations are of this type?

75.

Calculate the mode for borough.

76.

Does the idea of the mean or median of these two variables make any sense? Explain clearly why not.

Car Model Years. The dotplot in Figure 9 represents the model year for a sample of cars in a used car lot. Refer to the dotplot for Exercises 77−79.

FIGURE 9 Dotplot of model year.

77.

What are the mean, median, and mode of the model year?

78.

Calculate a new statistic “age of the car in 2015” as follows: take the model year and subtract it from 2015.

a. Find the mode of the car ages.

b. Find the mean and median of the car ages.

79.

What will be the mean, median, and mode of the car ages in 2025?

80.

Five friends have just had dinner at the local pizza joint. The total bill came to $30.60. What is the mean cost of each person’s meal?

81.

Lindsay just bought four shirts at the boutique in the mall, costing a total of $84.28. What was the mean cost of each shirt?

Dealing with Missing Data. Exercises 82−85 ask you to calculate measures of center when one of the values is missing.

82.

The mean cost of a sample of five items is $20. The cost of four of the items is as follows: $25, $15, $15, $20. What is the cost of the 5th item?

83.

The mean size of four downloaded music files is 3 Mb (megabytes). The size of three of the files is as follows: 5 Mb, 2 Mb, 3 Mb. What is the length of the 4th music file?

84.

The median number of students in a sample of seven statistics classes is 25. The ordered values are: 20, 22, 24, __, 27, 27, 28. What is the missing value?

85.

The median number of academic credits taken in a sample of six students is 15. The ordered values are: 12, 12, 14, __, 17, 17. What is the missing value?

Nutrition Ratings of Breakfast Cereals. Refer to the following information for Exercises 86−89. (Note that Minitab denotes both the sample size and the population size as N.) The data represent the nutrition rate of 59 cereals based on sugar content, vitamin content, and so on.

86.

Find the following sample statistics.

a. The sample size

b. The sample mean

c. The sample median

d. The highest and lowest ratings in the sample

87.

What do these statistics tell us about the skewness of the distribution?

88.

Linear Transformations. If we take each cereal rating and subtract 5 from it, how would that affect the mean, median, and mode? Would it affect each of the measures equally?

89.

Linear Transformations. If we cut each of the cereals’ ratings in half, how would that affect the mean, median, and mode? Would it affect each of the measures equally?

BRINGING IT ALL TOGETHER

Pulse Rates for Men and Women. To answer Exercises 90−93, refer to Figure 10, which includes comparative dotplots of the pulse rates for males and females.2

FIGURE 10 Comparative dotplots of pulse rates, by gender.

90.

Examine Figure 10.

a. Without doing any calculations, what is your impression of which gender, if any, has the higher overall pulse rate?

b. Find the mean pulse rate for the males by estimating the location of the balance point.

c. Find the mean pulse rate for the females by estimating the location of the balance point.

d. Based on (b) and (c), which gender has the higher mean pulse rate? Does this agree with your earlier impression?

91. Find the following medians:

a. The median pulse rate for the males

b. The median pulse rate for the females

c. Which gender has the higher median pulse rate? Does this agree with your findings for the mean earlier?

92. Find the following modes:

a. The mode pulse rate for the males

b. The mode pulse rate for the females

c.

Figure 1.30: WHAT IF?

Which gender has the higher mode pulse rate? Does this agree with your findings for the mean earlier?

93. What if the fastest pulse rate for the men was a typo and should have been an unspecified lower pulse rate? Describe how and why this change would have affected the following, if at all. Would they increase, decrease, or remain unchanged? Or is there insufficient information to tell what would happen? Explain your answers.

a. The mean men’s pulse rate

b. The median men’s pulse rate

c. The mode men’s pulse rate

 94. Trimmed Mean. Because the mean is sensitive to extreme values, the trimmed mean was developed as another measure of center. To find the 10% trimmed mean for a data set, omit the largest 10% of the data values and the smallest 10% of the data values, and calculate the mean of the remaining values. Because the most extreme values are omitted, the trimmed mean is less sensitive, or more robust (resistant), than the mean as a measure of center. For the data in the table, calculate the following:

a. The mean

b. The 10% trimmed mean

c. The 20% trimmed mean

The data represent the number of business establishments in a sample of states.

Figure 1.31: statebusinesses

State

Businesses (1000s)

State

Businesses (1000s)

Alabama

3.8

Michigan

7.5

Arizona

7.9

Minnesota

6.1

Colorado

8.9

Missouri

5.9

Connecticut

3.1

Ohio

9.5

Georgia

10.3

Oklahoma

3.8

Illinois

11.9

Oregon

5.4

Indiana

5.6

South Carolina

4.6

Iowa

2.7

Tennessee

5.4

Maryland

5.7

Virginia

8.6

Massachusetts

6.3

Washington

9.3

Source: U.S. Census Bureau.

 95. Challenge Exercise. In general, would you expect the trimmed mean to be larger, smaller, or about the same as the mean for data sets with the following shapes?

a. Right-skewed data

b. Left-skewed data

c. Symmetric data

 96. Midrange. Another measure of center is the midrange.

Because the midrange is based on the maximum and minimum values in the data set, it is not a robust statistic, but it is sensitive to extreme values. Calculate the midrange for the following data:

a. The price data from Table 9 on page 123.

b. The car model year data from Figure 9 on page 124.

 97. Harmonic Mean. The harmonic mean is a measure of center most appropriately used when dealing with rates, such as miles per hour (mph). The harmonic mean is calculated as

where n is the sample size and the x’s represent rates, such as the speeds in mph. Emily walked five miles today, but her walking speed slowed as she walked farther. Her walking speed was 5 mph for the first mile, 4 mph for the second mile, 3 mph for the third mile, 2 mph for the fourth mile, and 1 mph for the fifth mile. Calculate her harmonic mean walking speed over the entire five miles.

 98. Challenge Exercise. The (arithmetic) mean for Emily’s five-mile walk in Exercise 97 is 3 mph. Explain clearly why the value you calculated for the harmonic mean in Exercise 97 makes more sense than this arithmetic mean of 3 mph. (Hint: Consider time.)

 99. Geometric Mean. The geometric mean is a measure of center used to calculate growth rates. Suppose that we have n positive values; then the geometric mean is the nth root of the product of the n values. Jamal has been saving money in an account that has had 4% growth, 6% growth, and 10% growth over the last three years. Calculate the average growth rate over these three years. (Hint: Find the geometric mean of 1.04, 1.06, and 1.10 and subtract 1.)

CONSTRUCT YOUR OWN DATA SETS

100.

Construct your own data set with n 5 10, where the mean, the median, and the mode are all the same. Yes, just make up your own list of numbers, as long as the mean, median, and mode are all the same. Draw a dotplot. Comment on the skewness of the distribution.

101.

Construct your own data set with n 5 10, where the mean is greater than the median, which is greater than the mode. Draw a dotplot. Comment on the skewness of the distribution.

102.

Construct your own data set with n 5 10, where the mode is greater than the median, which is greater than the mean. Draw a dotplot. Comment on the skewness of the distribution.

103.

Construct your own data set with n 5 3. Let the mean and median be equal. Now, alter the three data values so that the mean of the altered data set has increased, while the median of the altered data set has decreased.

Use the Mean and Median applet for Exercises 104 and 105.

104.

Insert three points on the line by clicking just below it: two near the left side and one near the middle.

a. Click and drag the rightmost point to the right.

b. Describe what happens to the mean when you do this.

c. Describe what happens to the median when you do this.

105.

Explain why each of the measures behaves the way it does in the previous exercise.

WORKING WITH LARGE DATA SETS

Open the VideoGameSales data set from the Chapter 1 Case Study. The data set represents a sample. Use technology to do the following.

106.

Find the mean and median weekly sales.

107.

Suppose we remove the biggest seller for the week, Minecraft for PS3, from the data. Given what you have learned about the sensitivity of the mean to the presence of extreme values, which measure do you expect will change the most, the mean or the median?

108.

Recalculate the mean and the median of weekly sales, this time omitting Minecraft for PS3. Was your intuition in Exercise 107 confirmed?

109.

Compute the mean and median total sales for the 30 games.

110.

Identify the video game with the largest total sales. Omit this video game, and recompute the mean and median total sales. Which measure of center was more sensitive to the removal of the extreme value?

111.

Find the mode for each of the following variables:

a. Platform

b. Studio

c. Game type

112.

Compute the mean, median, and mode for the variable weeks on list.

113. What if we add a certain unknown amount x to each value in the variable weeks on list? Describe what will happen to the following measures of center.

a. Mean

b. Median

c. Mode

1

The Range

In Section 3.1, we learned how to find the center of a data set. Is that all there is to know about a data set? Definitely not! Two data sets can have exactly the same mean, median, and mode and yet be quite different. We need measures that summarize the data set in a different way, namely, the variation or variability of the data. In Section 3.2, we will learn measures of variability that will help us answer the question: “How spread out is the data set?”

Table 10 contains the heights (in inches) of the players on two volleyball teams.

Table 10 Women’s volleyball team heights (in inches)

Western Massachusetts University

Northern Connecticut University

60

66

70

67

70

70

70

70

75

72

a. Describe in words and graphs the variability of the heights of the two teams.

b. Verify that the means, medians, and modes for the two teams are equal.

Solution

a. There are some distinct differences between the teams. The Western Massachusetts (WMU) team has a player who is relatively short (60 inches: 5 feet tall) and a player who is very tall (75 inches: 6 feet, 3 inches tall). The Northern Connecticut (NCU) team has players whose heights are all within 6 inches of each other.

b. But despite the differences in (a), the mean, median, and mode of the heights for the two teams are precisely the same. As illustrated in Figure 11, the mean height (red triangle) for each team is 69 inches, the median height (green triangle) for each team is 70 inches, and the mode height (yellow triangle) for each team is 70 inches.

Clearly, these measures of location do not give us the whole picture. We need measures of variability (or measures of spread or measures of dispersion) that will describe how spread out the data values are. Figure 11 illustrates that the heights of the WMU team are more spread out than the heights of the NCU team.

FIGURE 11 Comparative dotplots of the heights of two volleyball teams.

Just as there were several measures of the center of a data set, there are also a variety of ways to measure how spread out a data set is. The simplest measure of variability is the range.

A larger range is an indication of greater variability, or greater spread, in the data set.

Calculate the range of player heights for each of the WMU and NCU teams.

Solution

rangeWMU 5 largest value 2 smallest value 5 75 2 60 5 15 inches

rangeNCU 5 largest value 2 smallest value 5 72 2 66 5 6 inches

As we expected, the range for WMU players is indeed larger than the range for NCU players, reflecting WMU’s players’ greater variability in height.

Table 11 contains a sample from the data set for the Chapter 3 Case Study. The percent increase or decrease in stock portfolio is recorded for the set of stocks chosen by throwing darts at the stock pages, along with the Dow Jones Industrial Average (DJIA) for the same day.

Table 11 Sample set of stock market returns

Darts

DJIA

11.2

15.8

72.9

16.2

16.6

17.3

28.7

17.7

1. Construct a comparison dotplot of the darts returns and the DJIA returns.

2. Using the dotplot, which group would you say has the larger range?

3. Calculate the range for each group. Is your intuition from (2) confirmed?

(The solutions are shown in Appendix A.)

The range is quite simple to calculate; however, it does have its drawbacks. For example, the range is quite sensitive to extreme values, because it is calculated from the difference of the two most extreme values in the data set. It completely ignores all the other data values in the data set. We would prefer our measure of variability to quantify spread with respect to the center, as well as to actually use all the available data values. Two such measures are the variance and the standard deviation.

2

Population Variance and Population Standard Deviation

Before we learn about the variance and the standard deviation, we need to get a firm understanding of what a deviation means, in the statistical sense.

EXAMPLE 10 Calculating deviations

Ashley and Brandon are certified public accountants who work for a large accounting firm, preparing tax returns for small business clients. Because tax returns are often filed close to the deadline, it is important that the returns be prepared in a timely fashion, with not a lot of variability in the length of time it takes to prepare a return. The chief accountant kept careful track of the amount of time (in hours, Table 12) for all the tax returns prepared by Ashley and Brandon during the last week of March.

a. Find the mean preparation time for each accountant.

b. Use comparative dotplots to compare the variability of Ashley and Brandon’s tax preparation times.

c. Calculate the deviations for each of Ashley and Brandon’s tax preparation times.

Table 12 Preparation times (in hours) for Ashley and Brandon

Ashley

5

7

8

9

11

Brandon

3

5

7

11

14

Solution

Because the data represent all the tax returns for the indicated period, they may be considered a population.

a. For Ashley:

For Brandon:

So the two accountants spent the same mean amount of time in tax preparation.

b. Figure 12 contains comparative dotplots of Ashley and Brandon’s tax preparation times. Note that Brandon’s preparation times vary more than Ashley’s. Compared to Ashley, we can say that Brandon’s tax preparation times

● are more spread out,

● show greater variability,

● have more variation, and

● are more dispersed.

The chief accountant probably prefers a more consistent tax preparation time, with less variability.

FIGURE 12  Brandon’s tax preparation times are more spread out.

c. Here we find the deviations, x 2 m.

● Ashley’s mean preparation time is m 5 8 hours. Her first tax return took x 5 5 hours, so the deviation for this first tax return is x 2m 5 5 2 8 5 23. Note that, when x , m, the deviation is negative.

● Ashley’s last tax return took 11 hours, so the deviation for this last return is x 2 m 5 11 2 8 5 3. Note that, when x . m, the deviation is positive.

● Continuing in this way, we find the deviations for all of Ashley’s and Brandon’s tax preparation times, as recorded in Table 13.

Table 13 Tax preparation times and their deviations

Ashley’s times

5

7

8

9

11

Ashley’s deviations

5 2 8 5 23

7 2 8 5 21

8 2 8 5 0

9 2 8 5 1

11 2 8 5 3

Brandon’s times

3

5

7

11

14

Brandon’s deviations

3 2 8 5 25

5 2 8 5 23

7 2 8 5 21

11 2 8 5 3

14 2 8 5 6

These deviations are used for the most widespread measures of spread: the variance and the standard deviation. However, we cannot use the mean deviation, because the mean deviation always equals zero. For example,

● Ashley’s mean deviation:

● Brandon’s mean deviation:

The mean deviation always equals zero for any data set because the positive and negative deviations cancel each other out. Thus, the mean deviation is not a useful measure of spread. To avoid this problem, we will work with the squared deviations.

Table 14 shows the squared deviations for Ashley and Brandon. Note that Brandon’s squared deviations are, on average, larger than Ashley’s, reflecting the greater spread in Brandon’s preparation times. It is therefore logical to build our measure of spread using the mean squared deviation.

Table 14 Squared deviations of tax preparation times

Ashley’s deviations

–3

–1

0

1

3

Ashley’s squared deviations

9

1

0

1

9

Brandon’s deviations

–5

–3

–1

3

6

Brandon’s squared deviations

25

9

1

9

36

The Population Variance, s2

For populations, the mean squared deviation is called the population variance and is symbolized by s2. This is the lowercase Greek letter sigma, not to be confused with the uppercase sigma () used for summation.

Notice that the numerator in s2 is a sum of squares. Squared numbers can never be negative, so a sum of squares also can never be negative. The denominator, N, which is the population size, also can never be negative. Thus, s2 can never be negative. The only time s2 5 0 is when all the population data values are equal.

Calculate the population variances of the tax preparation times for Ashley and Brandon.

Solution

Using the squared deviations from Table 14, we have

for Ashley, and

for Brandon. The population variance of the tax preparation times for Brandon is greater than the variance for Ashley, thus indicating that Brandon’s tax preparation times are more variable than Ashley’s.

Table 15 contains the funding provided by the Centers for Disease Control (CDC) to all the states in New England, in order to fight HIV/AIDS.3 This includes all the states in New England, so we may consider this a population.

1. Find the population mean funding, m.

2. Calculate the population variance of the funding, s2.

(The solution is shown in Appendix A.)

Table 15 CDC funding to fight HIV/AIDS for New England states

State

Funding (in millions)

Connecticut

7.8

Maine

1.9

Massachusetts

14.9

New Hampshire

1.5

Rhode Island

2.7

Vermont

1.6

However, what is the meaning of the values we obtained for s 2, 4, and 16, apart from their comparative value? The problem is that the units of these values represent hours squared, which is not a useful measure. Unfortunately, the intuitive meaning of the population variance is not self-evident.

The Population Standard Deviation, s

In practice, the standard deviation is easier to interpret than the variance. The standard deviation is simply the square root of the variance, and by taking the square root, we return the units of measure back to the original data unit (for example, “hours” instead of “hours squared”). The symbol for the population standard deviation is s. Conveniently,

The population standard deviation, s, is the positive square root of the population variance and is found by

Calculate the population standard deviations of the tax preparation times for Ashley and Brandon.

Solution

Brandon’s population variance of 16 is larger than Ashley’s population variance of 4, so Brandon’s population standard deviation will also be larger because we are simply taking the square root. We have

for Ashley and

for Brandon.

The population standard deviation of Brandon’s tax preparation times is 4 hours, which is larger than Ashley’s 2 hours. As expected, the greater variability in Brandon’s preparation times leads to a larger value for his population standard deviation, s.

Compute the Sample Variance and Sample Standard Deviation

The Sample Variance, s2, and the Sample Standard Deviation s

In the real world, we usually cannot determine the exact value of the population mean or the population standard deviation. Instead, we use the sample mean and sample standard deviation to estimate the population parameters. The sample variance also depends on the concept of the mean squared deviation. If the sample mean is x, and the sample size is n, then we would expect the formula for the sample variance to resemble the formula for the population variance, namely

However, this formula has been found to underestimate the population variance, so that we need to replace the n in the denominator with n 2 1. We therefore have the following.

The sample variance, s2, is approximately the mean of the squared deviations in the sample and is found by

The sample standard deviation is perhaps the second most important statistic you will encounter in this book (after the sample mean, x). It is the most commonly used measure of spread. The sample standard deviation is simply the square root of the sample variance and takes as its symbol the letter s, which is the Roman letter for the Greek s. Again, .

Suppose we obtain a sample of size n 5 3 from Ashley’s population of tax preparation times, as follows: 5 hours, 8 hours, 11 hours, as shown.

Ashley’s Population

5

7

8

9

11

Ashley’s Sample

5

8

11

a. Calculate the sample variance of the tax preparation times.

b. Compute the sample standard deviation of the tax preparation times.

c. Interpret the sample standard deviation.

Solution

a. We first find the sample mean, . It so happens that the value for this sample mean equals the population mean m 5 8, but this is only a coincidence.

 Then the sample variance is

The sample variance is s2 5 9 hours squared.

b. Then the sample standard deviation is

c. For this sample of Ashley’s tax returns, the typical difference between a tax preparation time and the mean preparation time is 3 hours.

In the exercises, you will find alternative computational formulas for the variance and standard deviation.

Find the sample standard deviation and the sample variance of the city gas mileage for the 2015 cars shown in the following table. Use (a) the TI-83/84, (b) Excel, (c) Minitab, (d) JMP, and (e) SPSS.

Vehicle

City mpg

Subaru Forester

22

Lexus RX 350

18

Ford Taurus

19

Mini Cooper

25

Cadillac Escalade

14

Mazda MX-5

21

Source: www.fueleconomy.gov.

Solution

Using the instructions in the Step-by-Step Technology Guide on page 117, we obtain the following output:

a. The TI-83/84 output is shown in Figure 13. The sample standard deviation, s, is given as Sx 5 3.763863264. The sample variance is s2 5 (3.763863264)2 5 14.16667.

b. The Excel output is provided in Figure 14. The sample standard deviation and sample variance are highlighted.

c. The Minitab output is provided in Figure 15. Note that Minitab rounds s to two decimal places.

d. The JMP output is shown in Figure 16.

e. The SPSS results are provided in Figure 17.

Next, we turn to methods for applying the standard deviation.

4

The Empirical Rule

If the data distribution is bell-shaped, we may apply the Empirical Rule to find the approximate percentage of data that lies within k standard deviations of the mean, for k 5 1, 2, or 3.

EXAMPLE 15 Using the Empirical Rule to find percentages

The College Board reports that the population mean Math SAT score for 2014 is m 5 514, with a population standard deviation of s 5 118. Assume the distribution of Math SAT scores is bell-shaped.

a. Find the percentage of Math SAT scores between 396 and 632.

b. Compute the percentage of Math SAT scores that are above 750.

Solution

a. We see that a Math SAT score of 396 represents 1 standard deviation below the mean, because

m – 1s 5 514 2 1(118) 5 396.

Similarly, a Math SAT score of 632 represents 1 standard deviation above the mean, because

m 1 1s 5 514 1 1(118) 5 632.

Thus, “Math SAT scores between 396 and 632” represents between m – 1s and m 11s, that is, within 1 standard deviation of the mean. The data distribution is bell-shaped, so we may use the Empirical Rule. Therefore, about 68% of the Math SAT scores lie between 396 and 632, as shown in Figure 19.

b. We note that a Math SAT score of 750 represents 2 standard deviations above the mean, because

m 1 2s 5 514 1 2(118) 5 750.

We know from the Empirical Rule that about 95% of the Math SAT scores lie within 2 standard deviations of the mean, so that about 95% of the Math SAT scores lie between 278 and 750. The left-over area of about 5% in the two tails in Figure 19 is the percentage of Math SAT scores above 750 or below 278. Because the bell-shaped curve is symmetric, the two tail areas are equal in area, which means that about 2.5% of the Math SAT scores lie above 750 (Figure 19).

5

Chebyshev’s Rule

P. L. Chebyshev (1821–1894, Russia) derived a result, called Chebyshev’s Rule, that can be applied to any continuous data set.

Because of the phrase “at least,” we say that Chebyshev’s Rule provides minimum percentages, instead of the approximate percentages provided by the Empirical Rule. The actual percentage may be much greater than the minimum percentage provided by Chebyshev’s Rule.

EXAMPLE 16 Using Chebyshev’s Rule to find minimum percentages

The College Board reports that the population mean SAT Writing exam score for 2014 is m 5 488, with a population standard deviation of s 5 114. However, assume we do not know the data distribution. Find the minimum percentage of exam scores that is

a. between 260 and 716.

b. between 317 and 659.

c. between 374 and 602.

Solution

The data distribution is unknown, so we cannot apply the Empirical Rule.

a. Because 260 lies 2 standard deviations below the mean

m 2 2s 5 488 2 2(114) 5 260

and 716 lies 2 standard deviations above the mean

m 1 2s 5 488 1 2(114) 5 716,

this question is really asking what is the minimum percentage within k 5 2 standard deviations of the mean. From Chebyshev’s Rule, the minimum percentage is

Thus, at least 75% of the SAT Writing exam scores will lie between 260 and 716.

b. The exam scores 317 and 659 lie k 5 1.5 standard deviations below and above the mean, respectively. Therefore, at least

of the SAT Writing exam scores will lie between 317 and 659.

c. The scores 374 and 602 lie k 5 1 standard deviation below and above the mean, respectively. Unfortunately, Chebyshev’s Rule is restricted to situations where k . 1. Thus, we cannot answer this question.

If a given data set is bell-shaped, either the Empirical Rule or Chebyshev’s Rule may be applied to it.

Section 3.2 Summary

 1. The simplest measure of variability, or measure of spread, is the range. The range is simply the difference between the maximum and minimum values in a data set, but the range has drawbacks because it relies on the two most extreme data values.

 2. The variance and standard deviation are measures of spread that utilize all available data values. The population variance can be thought of as the mean squared deviation. The standard deviation is the square root of the variance. We interpret the value of the standard deviation as the typical deviation, that is, the typical distance between a data value and the mean.

 3. The variance and standard deviation may also be calculated for a sample. Again, we interpret the value of the standard deviation as the typical deviation, that is, the typical distance between a data value and the mean.

 4. For bell-shaped distributions, the Empirical Rule may be applied. The Empirical Rule states that, for bell-shaped distributions, about 68%, 95%, and 99.7% of the data values will fall within 1, 2, and 3 standard deviations of the mean, respectively.

 5. Chebyshev’s Rule allows us to find the minimum percentage of data values that lie within a certain interval. Chebyshev’s Rule states that the proportion of values from a data set that will fall within k standard deviations of the mean will be at least [1 2 1/(k)2 ]100%, where k . 1.

Section 3.2 Exercises

Unless a data set is identified as a population, you can assume that it is a sample.

CLARIFYING THE CONCEPTS

 1. Explain what a deviation is. (p. 128)

 2. What is the interpretation of the value of the standard deviation? (p. 132)

 3. State one benefit and one drawback of using the range as a measure of spread. (p. 128)

 4. True or false: If two data sets have the same mean, median, and mode, then they are identical. (p. 127)

 5. What is one benefit of using the standard deviation instead of the range as a measure of spread? What is one drawback? (p. 128)

 6. Which measure of spread represents the mean squared deviation for the population? (p. 130)

 7. True or false: Chebyshev’s Rule provides exact percentages. (p. 138)

 8. When can the sample standard deviation, s, be negative? (p. 133)

 9. When does the sample standard deviation, s, equal zero? (p. 133)

10.

When may the Empirical Rule be used? (p. 135)

PRACTICING THE TECHNIQUES

Figure 1.37:

CHECK IT OUT!

To do

Check out

Topic

Exercises 11a–16a

Example 9

Range

Exercises 11c–16c

Example 10

Calculating deviations

Exercises 11d–16d

Example 11

Population variance

Exercises 11e–16e

Example 12

Population standard deviation

Exercises 17–22

Example 13

Sample variance and sample standard deviation

Exercises 23–30

Example 15

Empirical Rule

Exercises 31–38

Example 16

Chebyshev’s Rule

For the population data in Exercises 11–16, do the following:

a. Compute the range.

b. Find the population mean, m.

c. Calculate the deviations, x 2 m.

d. Compute the population variance, s2.

e. Find the population standard deviation, s.

11.

State exports to other countries are shown in the table for the population of all New England states, for the month of June 2014, expressed in billions of dollars.

State

Exports

State

Exports

Connecticut

1.4

New Hampshire

0.4

Maine

0.3

Rhode Island

0.2

Massachusetts

2.4

Vermont

0.3

Source: U.S. Census Bureau.

12.

The number of wins for each baseball team in the population of the American League West division for 2013 is shown in the table.

Team

Wins

Team

Wins

Oakland Athletics

96

Seattle Mariners

71

Texas Rangers

91

Houston Astros

51

Los Angeles Angels

78

Source: MLB.mlb.com.

13.

The table provides the motor vehicle theft rate for the population of the top 10 countries in the world for motor vehicle theft, for 2012. The theft rate equals the number of motor vehicles stolen in 2012 per 100,000 residents.

Country

Theft rate

Country

Theft rate

Italy

208.0

Greece

100.2

France

174.1

Norway

94.1

USA

167.8

Netherlands

75.2

Sweden

117.2

Spain

75.1

Belgium

106.0

Cyprus

66.0

Source: United Nations Office on Drugs and Crime.

14.

The National Center for Education Statistics sponsors the Trends in International Mathematics and Science Study (TIMSS). The table contains the mean science scores for the eighth-grade science test for the population of all Asian-Pacific countries that took the exam.

Country

Science Score

Country

Science score

Singapore

578

Australia

527

Taiwan

571

New Zealand

520

South Korea

558

Malaysia

510

Hong Kong

556

Indonesia

420

Japan

552

Philippines

377

15.

The table contains the number of petit larceny cases for the population of all police precincts in South Manhattan in 2013.

Precinct

Petit larcenies

Precinct

Petit larcenies

1

2014

10

995

5

1288

13

2094

6

1555

14

4551

7

584

17

823

9

1607

18

2071

Source: New York City Police Department.

16.

The table contains the number of criminal trespass cases for the population of all police precincts in South Manhattan in 2013.

Precinct

Criminal trespass

Precinct

Criminal trespass

1

108

10

207

5

105

13

135

6

113

14

340

7

233

17

74

9

219

18

120

Source: New York City Police Department.

For the sample data in Exercises 17–22, do the following:

a. Calculate the sample variance.

b. Compute the sample standard deviation.

c. Interpret the sample standard deviation.

17.

A sample of the state export data from Exercise 11 is provided in the table.

State

Exports

Connecticut

1.4

Massachusetts

2.4

Rhode Island

0.2

18.

A sample from the baseball data in Exercise 12 is shown here.

Team

Wins

Texas Rangers

91

Los Angeles Angels

78

Seattle Mariners

71

19.

A sample from the motor vehicle theft data in Exercise 13 is as follows.

Country

Theft rate

Italy

208.0

USA

167.8

Greece

100.2

20.

A sample from the science score data in Exercise 14 is given here.

Country

Science score

South Korea

558

Hong Kong

556

Japan

552

Australia

527

21.

The following sample is taken from the petit larceny data in Exercise 15.

Precinct

Petit larcenies

1

2014

6

1555

9

1607

14

4551

17

823

22.

A sample taken from the criminal trespass data in Exercise 16 is as follows.

Precinct

Criminal trespass

1

108

7

233

14

340

18

120

For Exercises 23–26, use the following information. A data distribution is bell-shaped, with a mean of 50 and a standard deviation of 5. Use the Empirical Rule to approximate the percentage of data.

23.

Between 45 and 55

24.

Between 40 and 60

25.

Between 35 and 65

26.

Less than 45

For Exercises 27–30, use the following information. A data distribution is bell-shaped, with a mean of 0 and a standard deviation of 1. Use the Empirical Rule to approximate the percentage of data.

27.

Between –1 and 1

28.

Greater than 2

29.

Less than –2

30.

Between –2 and 2

For Exercises 31–34, use the following information. A data set has an unknown distribution, with a mean of 20 and a standard deviation of 2. Use Chebyshev’s Rule to estimate the minimum possible percentage of data.

31.

Between 16 and 24

32.

Between 14 and 26

33.

Between 12 and 28

34.

Between 13 and 27

For Exercises 35–38, use the following information. A data set has an unknown distribution, with a mean of 20 and a standard deviation of 5. If possible, use Chebyshev’s Rule to estimate the minimum possible percentage of data.

35.

Between 0 and 40

36.

Between 5 and 35

37.

Between 12.5 and 27.5

38.

Between 15 and 25

APPLYING THE CONCEPTS

39.

Match the histograms in (a)–(d) to the statistics in (i)–(iv).

 i. Mean 5 75, standard deviation 5 20

ii. Mean 5 75, standard deviation 5 10

iii. Mean 5 50, standard deviation 5 20

iv. Mean 5 50, standard deviation 5 10

40.

Match the histograms in (a)–(d) to the statistics in (i)–(iv).

 i. Mean 5 1, standard deviation 5 1

ii. Mean 5 1, standard deviation 5 0.1

iii. Mean 5 0, standard deviation 5 1

iv. Mean 5 0, standard deviation 5 0.1

For the following exercises, make sure to state your answers in the proper units, such as “years” or “years squared.”

Video Game Sales. The Chapter 1 Case Study looked at video game sales for the top 30 video games. The following table contains the total sales (in game units) and weeks on the top 30 list for a sample of five randomly selected video games. Use this information for Exercises 41 and 42.

41.

Find the following measures of spread for total sales:

a. Range

b. Sample variance

c. Sample standard deviation

42.

Calculate the following measures of spread for the number of weeks on the top 30 list:

a. Range

b. Sample variance

c. Sample standard deviation

Video Game

Total sales in millions of units

Weeks on list

Super Mario Bros. U for WiiU

1.7

78

NBA 2K14 for PS4

0.6

27

Battlefield 4 for PS3

0.9

29

Titanfall for XboxOne

1.2

10

Yoshi’s New Island for 3DS

0.2

10

Source: www.vgchartz.com.

Figure 1.40: videogamereg

Figure 1.41: CASE STUDY

Darts and the DJIA. The following table contains a random sample of eight days from the Chapter 3 Case Study data set, indicating the stock market gain or loss for the portfolio chosen by the random darts, as well as the DJIA gain or loss for that day. Use this information for Exercises 43 and 44.

43.

Find the following measures of spread for the darts:

a. Range

b. Sample variance

c. Sample standard deviation

44.

Calculate the following measures of spread for the DJIA:

a. Range

b. Sample variance

c. Sample standard deviation

Figure 1.42: dartsdjia

Darts

DJIA

227.4

212.8

18.7

9.3

42.2

8

216.3

28.5

11.2

15.8

28.5

10.6

1.8

11.5

16.9

25.3

Source: Wall Street Journal.

Age and Height. The following table provides a random sample from the Chapter 4 Case Study data set body_females, showing the age and height of the eight women. Use this information for Exercises 45 and 46.

45.

Find the following measures of spread for age:

a. Range

b. Sample variance

c. Sample standard deviation

46.

Calculate the following measures of spread for height:

a. Range

b. Sample variance

c. Sample standard deviation

Figure 1.43: ageheight

Age

Height

40

63.5

28

63.0

25

64.4

34

63.0

26

63.8

21

68.0

19

61.8

24

69.0

Source: Journal of Statistics Education.

CASE STUDY

107

3.1

Measures of Center

OBJECTIVES By the end of this section, I will be able to . . .

1

Calculate the mean for a given data set.

2

Find the median, and describe why the median is sometimes preferable to the mean.

3

Find the mode of a data set.

4

Describe how skewness and symmetry affect these measures of center.

Do you like to make money? Then you might want to stay in school and finish your Bachelor’s degree. The Pew Research Center reports that the median annual earnings among young people ages 25–32 with a Bachelor’s degree was $45,500, compared with $30,000 for those who did not finish their college degree (Source: Pew Research Center: The Rising Cost of Not Going to College1). The $45,500 is a sample median, which was calculated from the sample taken by the researchers. As such, it summarizes the earnings of over 1000 different young people from all over the country. In Chapter 3, we learn how to do this: to summarize an entire dataset with just a few numbers. In Section 3.1, we will learn about three numerical measures that tell us where the center of the data lies: the mean, the median, and the mode.

1

The mean is often called the arithmetic mean.

To find the mean of the values in a data set, simply add up all the numbers and divide by how many numbers you have.

EXAMPLE 1 Calculating the population mean

Table 1 Prices for a population of cell phones

Samsung Galaxy S5 Standard

$200

Samsung Galaxy S5 Active

$200

Sony Xperia Z2

$600

Nokia Lumia Icon

$200

LG G3

$800

Apple iPhone 5s

$250

HTC One M8

$200

Samsung Galaxy Note 3

$300

Source: www.cnet.com/topics/phones/best-phones.

NOW YOU CAN DO

Exercises 13–18.

YOUR TURN

#1

Table 2 Number of tropical storms

Year

2006

2007

2008

2009

2010

2011

2012

2013

Tropical storms

10

15

16

9

19

19

19

14

EXAMPLE 2 Calculating the sample mean

Table 3 Prices for a sample of cell phones

Samsung Galaxy S5 Active

$200

Sony Xperia Z2

$600

Apple iPhone 5s

$250

Samsung Galaxy Note 3

$300

NOW YOU CAN DO

Exercises 19–24.

YOUR TURN

#2

The Mean as the Balance Point of the Data

Let’s explore our sample cell phone price data a bit further. Consider the dotplot of the cell phone prices in Figure 1. To find out where the mean price lies on this number line, imagine that the dots are little blocks on a ruler or a seesaw and that you must decide where to place the support (like the triangle in Figure 1) so that the ruler balances perfectly. The place where the data set balances perfectly is the location of the mean. Placing the fulcrum too far to the right or left would create an imbalance. This data set balances precisely at the sample mean, x_ 5 $337.50

What Does This Number Mean?

FIGURE 1 The price data balance at the mean.

Checking Your Results Against Experience and Common Sense

When you have found the balance point, you have found the mean. When you calculate the mean, or have a computer or calculator do it for you, don’t just accept whatever value pops out. Make sure the result makes sense. Because the mean always indicates the place where the data values are in balance, the mean is often near the center of the data. If the value you have calculated lies nowhere near the center of the data, then you may want to check your calculations.

For example, suppose we were finding the mean of the cell phone data, and we accidentally entered 6000 instead of 600 for the price of the Sony Xperia Z2. Then, our value for the mean resulting from this incorrect calculation would be

The mean price cannot equal $1687.50 because all the values in the data set are less than $1687.50. The mean can never be larger or smaller than all the values in the data set.

Don’t automatically accept the result you get from a computer or calculator. Remember GIGO: Garbage In Garbage Out. If you enter the wrong data, the calculator or computer will not bail you out. Human error is one reason for the explosion of faulty statistical analyses in the newspapers and on the Internet. Now more than ever, data analysts must use good judgment. When you calculate a mean, always have an idea of what you expect the sample mean to be, that is, at least a ballpark figure.

Developing Your Statistical Sense

For calculating the mean, we will adopt the convention of rounding our final calculation, if necessary, to one more decimal place than that in the original data.

The Mean Is Sensitive to Extreme Values

One drawback of using the mean to measure the center of the data is that the mean is sensitive to the presence of extreme values in the data set. We illustrate this phenomenon with the following example.

EXAMPLE 3 Sensitivity of the mean to extreme values

Table 4 Home sales prices in Broward County, Florida

Location

Price

Pembroke Pines

$300,000

Weston

$350,000

Hallandale

$360,000

Miramar

$425,000

Davie

$500,000

Fort Lauderdale

$600,000

homesales

Source: www.homes.com (prices rounded to nearest $1000).

= $1,220,000

NOW YOU CAN DO

Exercises 25–30.

2

The Median

The median of a data set is the middle data value when the data are put into ascending order. Half of the data values lie below the median, and half lie above.

● If the sample size n is odd, then the median is the middle value and lies at the position when the data are put in ascending order.

● If the sample size n is even, then the median is the mean of the two middle data values that lie on either side of the position.

The Median Is Not Sensitive to Extreme Values

Unlike the mean, the median is not sensitive to extreme values. If the expensive home is included in the sample, the median price should not change much, even though, as we saw in Example 3, the mean sales price nearly tripled. Let’s look at an example of how this would occur.

Phillip Spears/Getty Images

Because the median is not sensitive to extreme values, we say that it is a robust, or resistant, measure of center. The mean is neither robust nor resistant.

NOW YOU CAN DO

Exercises 31–36.

FIGURE 2 The mean (red triangles) is sensitive to extreme values, but the median (green triangles) is not.

The Mean and Median applet allows you to insert your own data values and see how changes in these values affect both the mean and the median.

Note that the formula gives the position, not the value, of the median. For example, the median home sales price for Table 4 is not 5

CAUTION

!

Theron Kirkman/AP Photo

Taylor Swift.

NOW YOU CAN DO

Exercises 37–40.

Take a sample from Table 2 that consists of the number of tropical storms from the even-numbered years. Find the mean, median, and mode number of tropical storms.

(The solutions are shown in Appendix A.)

YOUR TURN

#3

How Skewness Affects the Mean and Median

For a right-skewed distribution, the mean is larger than the median.

For a left-skewed distribution, the median is larger than the mean.

For a symmetric unimodal distribution, the mean, median, and mode are fairly close to each other.

darts

FIGURE 6 Household size is somewhat right-skewed.

NOW YOU CAN DO

Exercises 41–44.

CASE STUDY

Mark Hooper/Getty Images

Remember: It is often helpful to have a “ballpark” estimate of the mean or other statistics as a reality check of your calculations.

Note: In exploratory data analysis, we use graphical methods to compare numerical statistics.

FIGURE 8 Mean percentage net price change for the professionals, darts, and DJIA.

118

Chapter 3 Describing Data Numerically

119

3.1 Measures of Center

120

Chapter 3 Describing Data Numerically

121

3.1 Measures of Center

D

122

Chapter 3 Describing Data Numerically

123

3.1 Measures of Center

124

Chapter 3 Describing Data Numerically

125

3.1 Measures of Center

videogamesales

126

Chapter 3 Describing Data Numerically

WHAT IF

?

3.2

Measures of Variability

OBJECTIVES By the end of this section, I will be able to . . .

1

Find the range of a data set.

2

Calculate the variance and the standard deviation for a population.

3

Compute the variance and the standard deviation for a sample.

4

Use the Empirical Rule to find approximate percentages for a bell-shaped distribution.

5

Apply Chebyshev’s Rule to find minimum percentages.

EXAMPLE 8 Different data sets with the same measures of center

Martin Meissner/AP Photo

volleyball

The range of a data set is the difference between the largest value and the smallest value in the data set:

range 5 largest value 2 smallest value 5 maximum 2 minimum

EXAMPLE 9 Range of the volleyball teams’ heights

What Results Might We Expect?

From Figure 11, it is intuitively clear that the heights of the WMU team are more spread out than the heights of the NCU team. Therefore, we would expect the range of the WMU team to be larger than the range of the NCU team, reflecting its greater variability.

NOW YOU CAN DO

Exercises 11a–16a.

YOUR TURN

#4

CASE STUDY

Deviation

A deviation for a given data value x is the difference between the data value and the mean of the data set. For a sample, the deviation equals x 2 x. For a population, the deviation equals x 2m.

● If the data value is larger than the mean, the deviation will be positive.

● If the data value is smaller than the mean, the deviation will be negative.

● If the data value equals the mean, the deviation will be zero.

The deviation can roughly be thought of as the distance between a data value and the mean, except that the deviation can be negative, whereas distance is always positive.

Catherine Yeulet/Getty Images

Ashley and Brandon, certified public accountants.

NOW YOU CAN DO

Exercises 11c–16c.

The population variance, s2, is the mean of the squared deviations in the population and is given by the formula

EXAMPLE 11  Calculating the population variances for Ashley and Brandon

NOW YOU CAN DO

Exercises 11d–16d.

YOUR TURN

#5

Note: s can never be negative.

CAUTION

!

EXAMPLE 12 Calculating the population standard deviations for Ashley and Brandon

NOW YOU CAN DO

Exercises 11e−16e.

Calculate the population standard deviation of the CDC from Table 15.

(The solution is shown in Appendix A.)

YOUR TURN

#6

The Standard Deviation

So how do we interpret these values for s? One quick thumbnail interpretation of the standard deviation is that it represents a “typical” deviation. That is, the value of s represents a distance from the mean that is representative for that data set. For example, the typical distance from the mean for Ashley’s and Brandon’s tax preparation times is 2 hours and 4 hours, respectively.

What Do These Numbers Mean?

Communicating the Results

As you study statistics, keep in mind that during your career you will likely need to explain your results to others who have never taken a statistics course. Therefore, you should always keep in mind how to interpret your results to the general public. Communication and interpretation of your results can be as important as the results themselves.

Developing Your Statistical Sense

3

Note: In this book, we will work with sample statistics unless the data set is identified as a population.

The sample standard deviation, s, is the positive square root of the sample variance s2:

The value of s may be interpreted as the typical distance between a data value and the sample mean, for a given data set.

Neither s2 nor s can ever be negative. Both the variance and standard deviation are equal to zero only when all the data values in the data set are the same.

EXAMPLE 13  Calculating the sample variance and the sample standard deviation

NOW YOU CAN DO

Exercises 17–22.

Suppose we take as our sample from the CDC funding data set in Table 15 the three northernmost (and least populated) New England states: Maine, New Hampshire, and Vermont.

1. Look at the funding values for the sample states. Would you expect our measures of spread to be larger or smaller than those of all the New England states? Why?

2. Find the variance of this sample. Express it in dollars squared.

3. Use your answer from (2) to calculate the standard deviation. Express it in dollars.

4. Interpret the value of the standard deviation.

(The solutions are shown in Appendix A.)

YOUR TURN

#7

Less Variation Is Better

In most real-world applications, consistency is a great advantage. In statistical data analysis, less variation is often better, even though variability is natural and cannot be eliminated. Throughout the text, you will find that smaller variability will lead to

● more precise estimates and

● higher confidence in conclusions.

Developing Your Statistical Sense

EXAMPLE 14  Using technology to find the sample variance and sample standard deviation

gasmileage

For the TI-83/84, do not confuse Sx, the TI’s notation for the sample standard deviation, with sx, which the TI-83/84 uses to label the population standard deviation.

CAUTION

!

FIGURE 13 TI-83/84 output.

FIGURE 15 Minitab output.

FIGURE 14 Excel output.

FIGURE 16 JMP output.

FIGURE 17 SPSS output.

The Empirical Rule

If the data distribution is bell-shaped:

● About 68% of the data values will fall within 1 standard deviation of the mean.

● For a population, about 68% of the data will lie between m 2 1s and m 1 1s.

● For a sample, about 68% of the data will lie between x 2 1s and x 1 1s.

● About 95% of the data values will fall within 2 standard deviations of the mean.

● For a population, about 95% of the data will lie between m 2 2s and m 1 2s.

● For a sample, about 95% of the data will lie between x 2 2s and x 1 2s.

● About 99.7% of the data values will fall within 3 standard deviations of the mean.

● For a population, about 99.7% of the data will lie between m 2 3s and m 1 3s.

● For a sample, about 99.7% of the data will lie between x 2 3s and x 1 3s.

Figure 18 illustrates these approximate percentages.

Remember: The Empirical Rule may be applied only if the data distribution is bell-shaped.

CAUTION

!

FIGURE 18 Empirical Rule, with approximate percentages.

Remember: The English word “about” is not optional; it is required. The Empirical Rule is an approximation of normal distribution probabilities that we will examine more closely in Chapter 6.

NOW YOU CAN DO

Exercises 23–30.

FIGURE 19 Example of Empirical Rule applied to Math SAT scores.

Suppose vehicle speeds on the local interstate highway are bell-shaped, with a mean of m 5 70 mph and a standard deviation of s 5 5 mph.

1. Find the percentage of vehicle speeds between 65 mph and 75 mph.

2. Compute the percentage of vehicles that are obeying the speed limit of at most 65 mph.

(The solutions are shown in Appendix A.)

YOUR TURN

#8

Chebyshev’s Rule

The proportion of values from a data set that will fall within k standard deviations of the mean will be at least

where k . 1. Chebyshev’s Rule may be applied to either samples or populations. For example:

● When k 5 2, at least 3/4 (or 75%) of the data values will fall within 2 standard deviations of the mean.

● When k 5 3, at least 8/9 (or 88.89%) of the data values will fall within 3 standard deviations of the mean.

State Central Artillery Museum, St. Petersburg, Russia/The Bridgeman Art Library

Portrait of Pafnuty Chebyshev–Yaroslav Sergeyevich (1821–1894).

NOW YOU CAN DO

Exercises 31–38.

Strengths and Weaknesses of the Empirical Rule and Chebyshev’s Rule

Example 16 shows that the lack of knowledge of a bell-shaped distribution can have a cost.

a. For part (a), using the Empirical Rule with k 5 2 would have given us an answer of “about 95%,” which is more precise than “at least 75%.” However, this extra precision comes only if we know the distribution is bell-shaped.

b. For part (b), however, the Empirical Rule does not apply to any values other than 1, 2, or 3, so would have been no help here.

c. Finally, had we been able to apply the Empirical Rule in part (c), then we could have gotten an answer of “about 68%” for k 5 1.

Developing Your Statistical Sense

Suppose systolic blood pressure in a population of senior citizens has a mean of m 5 130 and a standard deviation of s 5 10. Find the minimum percentage of systolic blood pressure readings between 110 and 150.

(The solution is shown in Appendix A.)

YOUR TURN

#9

Can the Financial Experts Beat the Darts?

Recall from the Case Study at the beginning of this chapter, the Wall Street Journal competition between stocks chosen randomly by Journal staff members throwing darts and stocks chosen by a team of four financial experts. Note from Figure 20 that the DJIA exhibits less variability than the other two portfolios. This smaller variability is due to the fact that the DJIA is made up of 30 component stocks, whereas each portfolio is made up of only four stocks. Smaller sample sizes can be associated with increased variability, because an unusual result in one value has a relatively strong effect on the mean when it is not offset by a large sample.

Which of the portfolios, pros or darts, shows greater variability? It is difficult to determine which has the greater standard deviation, just by examining Figure 20. We therefore turn to the Minitab descriptive statistics in Figure 21. The range for the darts, 115.90, is greater than the range for the pros, 112.80. But the standard deviation for the darts (19.39) is less than that of the pros (22.25).

FIGURE 21 Descriptive statistics for the portfolios.

Measures of spread may disagree about which data set has more variability. However, the range takes into account only the two most extreme data values; therefore, the standard deviation is the preferred measure of spread because it uses all the data values. Our conclusion, therefore, is that the returns for the professionals exhibit a greater variability.

Why did the pros have more variability than the darts? After all, in finance, high variability is not necessarily advantageous because it is associated with greater risk. The professionals evidently chose higher-risk stocks with greater potential for high returns—but also greater potential for losing money.

CASE STUDY

FIGURE 20 Comparative dotplots of the net change in prices.

141

3.2 Measures of Variability

142

Chapter 3 Describing Data Numerically

143

3.2 Measures of Variability

Saturated Fat and Calories. The table contains the calories and saturated fat in a sample of 10 food items. Use this information for Exercises 47 and 48.

47.

Find the following measures of spread for calories:

a. Range

b. Sample variance

c. Sample standard deviation

48.

Calculate the following measures of spread for saturated fat:

a. Range

b. Sample variance

c. Sample standard deviation 

Figure 1.79: satfatcorr

Food item

Calories

Grams of saturated fat

Chocolate bar (1.45 ounces)

216

7.0

Meat & veggie pizza (large slice)

364

5.6

New England clam chowder (1 cup)

149

1.9

Baked chicken drumstick (no skin, medium size)

 75

0.6

Curly fries, deep-fried (4 ounces)

276

3.2

Wheat bagel (large)

375

0.3

Chicken curry (1 cup)

146

1.6

Cake doughnut hole (one)

 59

0.5

Rye bread (1 slice)

 67

0.2

Raisin bran cereal (1 cup)

195

0.3

Source: Food-a-Pedia.

Video Game Sales. Refer to the video game sales data in Exercises 41 and 42 for Exercises 49–52.

49.

The sample variance of sales was expressed in “game units squared.” Do you find this concept easy to understand? Which measure do you find to be more easily understood and interpreted for these data, the variance or the standard deviation?

50.

Consider the histogram of total units sold for all the top 30 video games.

a. Is the distribution bell-shaped?

b. Can we apply the Empirical Rule?

c. Can we apply Chebyshev’s Rule?

51.

Use the sample of size five and Chebyshev’s Rule to find the minimum percentage of total sales that are between 0.0048 million and 1.8352 million.

52.

Refer to Table 3 of Chapter 1 on page 8. Calculate the actual proportion of total sales that are between 0.0048 million and 1.8352 million. Does this fit the answer you got using Chebyshev’s Rule?

Figure 1.81: CASE STUDY

Darts and the DJIA. Refer to the darts and DJIA data in Exercises 43 and 44 for Exercises 53–56.

53.

Based on your measures of spread in Exercises 43 and 44, which stock market return reflects greater variability, the darts or the DJIA?

54.

The histogram shows the population distribution of the stock market changes for the darts. Can we live with the assumption that the distribution is bell-shaped?

55.

Based on the sample of size 8, use the Empirical Rule to approximate the percentage of darts stock returns that lie between −13.41 and 32.31.

56.

Can the Empirical Rule tell us what approximate percentage of the darts stock returns lie between −1.98 and 20.88? Explain.

Age and Height. Refer to the age and height data in Exercises 45 and 46 for Exercises 57–60.

57.

The histogram shows the population distribution of the women’s ages.

a. Is the distribution bell-shaped?

b. Can we apply the Empirical Rule?

c. Can we apply Chebyshev’s Rule?

58.

Based on the sample of size 8, use Chebyshev’s Rule to find the minimum percentage of the women’s ages that lie between 16.78 and 37.48.

59.

The histogram shows the population distribution of the women’s heights.

a. Though it’s not perfect, can we live with the assumption that the distribution is bell-shaped?

b. Can we apply the Empirical Rule?

c. Can we apply Chebyshev’s Rule?

60.

Based on the sample of size 8, use the Empirical Rule to approximate the percentage of the women’s heights that lie between 59.449 inches and 69.677 inches.

Saturated Fat and Calories. Refer to the food data in Exercises 47 and 48 for Exercises 61 and 62.

61.

The histogram contains the grams of saturated fat for the 10 foods in the sample.

a. Is the distribution bell-shaped?

b. Can we apply the Empirical Rule?

c. Can we apply Chebyshev’s Rule?

62.

Use Chebyshev’s Rule to find the minimum percentage of food items with saturated fat between −1.51 and 5.75. (Note that, because grams of saturated fat cannot be negative, this is the same as between 0 and 5.75.)

Fuel Economy. Refer to Table 7 on page 123 to answer Exercises 63–65. The data represent a sample.

63.

Find the following measures of spread for the number of cylinders:

a. Range

b. Variance

c. Standard deviation

64.

Find the following measures of spread for the engine size:

a. Range

b. Variance

c. Standard deviation

65.

Find the following measures of spread for the fuel economy:

a. Range

b. Variance

c. Standard deviation

Ant Size. Use the following information for Exercises 66 and 67. A study compared the size of ants from different colonies. The masses (in milligrams) of samples of ants from two different colonies are shown in the accompanying table.4 

Colony A

Colony B

109

134

148

115

120

 94

110

101

 94

113

110

158

 61

111

 97

 67

 72

106

136

114

66.

Calculate the range for each ant colony.

a. Which has the greater range?

b. Which colony has the greater variability according to the range?

67.

Calculate the standard deviation for each colony.

a. Which has the greater standard deviation?

b. Which colony has the greater variability according to the standard deviation? Does this concur with your answer from the previous exercise?

c. Without calculating the variances, say which colony has the greater variance. How do you know this?

68.

Computational Formula for the Population Variance and Standard Deviation: Wins in Baseball. The following table provides the number of wins for all the teams in the American League East Division for the 2013 season, which we can consider to be a population.

Team

Wins

Boston Red Sox

97

Tampa Bay Rays

92

Baltimore Orioles

85

New York Yankees

85

Toronto Blue Jays

74

Source: MLB.mlb.com.

An alternative computational formula for the population variance is as follows:

a. Use the computational formula to find the population variance for the number of wins.

b. Use your result from (a) to find the population standard deviation for the number of wins.

Note: x2 means that you square each data value and then add up the squared data values, and (x)2 means that you add up all the data values and then square the sum.

69.

Computational Formula for the Sample Variance and Standard Deviation. Refer to the previous exercise. Suppose a random sample of size n 5 3 from these teams yields the New York Yankees, the Tampa Bay Rays, and the Baltimore Orioles.

An alternative computational formula for the sample variance is as follows:

a. Use the computational formula to find the sample variance for the number of wins.

b. Use your result from (a) to find the sample standard deviation for the number of wins.

c. Interpret your result from (b).

70.

Challenge Exercise. Refer to the table in Exercise 68. Suppose we are taking a sample of size n 5 2.

a. Which sample of two teams will yield the largest sample standard deviation? Explain your reasoning.

b. Which sample of two teams will yield the smallest sample standard deviation? Explain your reasoning.

71.

Empirical Rule: October in Santa Monica. The National Climate Data Center reports that the mean October temperature in Santa Monica, California, is 63 degrees Fahrenheit, with a standard deviation of 3 degrees. Suppose the data distribution is bell-shaped. If possible, estimate the percentage of October days with temperatures within the following ranges. If not possible, explain why.

a. Between 60 and 66 degrees

b. Between 57 and 69 degrees

c. Between 55 and 71 degrees

72.

Empirical Rule: Energy Consumption. The U.S. Department of Energy reports that the mean annual energy consumption per person in the United States is 1400 watts. Assume that the standard deviation is 200 watts and the data distribution is bell-shaped. Estimate the percentage of Americans with energy consumption within the following ranges.

a. Between 1200 and 1600 watts

b. Between 1000 and 1800 watts

c. Above 1000 watts

73.

Chebyshev’s Rule. Refer to Exercise 71. Suppose that we did not know that the October temperature in Santa Monica is bell-shaped. If possible, find minimums for (a)–(c) in Exercise 71.

74.

Chebyshev’s Rule. Refer to Exercise 72. Suppose that we did not know that the annual energy consumption is bell-shaped. If possible, find minimums for (a)–(c) in Exercise 72.

Energy Consumption. Refer to Table 16, which shows the per capita energy consumption (watts per person) for samples of countries on three continents for Exercises 75−78. 

Figure 1.82: energyconsumption

TABLE 16 Per capita energy consumption for three samples of countries

Asia

Europe

North America

China 447

Germany 861

USA 1402

Japan 774

France 804

Canada 1871

South Korea 1038

United Kingdom 622

Mexico 131

Source: The World Factbook.

75.

Construct dotplots of the energy consumption for each continent. Which continent would you say has the greatest spread (variability)? Why?

76.

Find the range and variance of the per capita energy consumption for each of the continents. Do your findings agree with your judgment from the previous exercise?

77.

Without performing any calculations, use your results from the previous exercise to state which continent has (a) the largest standard deviation, and (b) the smallest standard deviation.

78.

Now suppose we omit Mexico from the data.

a. Without recalculating them, describe how this would affect the values of the measures of spread you found for the North American countries.

b. Now recalculate the three measures of spread for the North American countries. Was your judgment in (a) supported?

Women’s Volleyball Team Heights. Refer to Table 10 on page 126 for Exercises 79−81.

79.

Suppose a new player joins the NCU team. She is 7 feet tall (84 inches) and replaces the 72-inch-tall player.

a. Would you expect the standard deviation to go up or down, and why?

b. Now find the standard deviation for the team including the new player. Was your intuition correct?

80.

Linear Transformations. Add 4 inches to the height of each player on the WMU team.

a. Recalculate the range and standard deviation.

b. Formulate a rule for the behavior of these measures of variability when a constant (such as 4) is added to each member of the data set.

81.

Linear Transformations. Starting with the original data, double the height of each player on the NCU team.

a. Recalculate the range and standard deviation.

b. Formulate a rule for the range and standard deviation when the data values are doubled.

Coefficient of Variation. The coefficient of variation enables analysts to compare the variability of two data sets that are measured on different scales. The coefficient of variation (CV) itself does not have a unit of measure. Larger values of CV indicate greater variability or spread. The coefficient of variation is given as

Use this measure of variability for Exercises 82 and 83.

82.

Coefficient of Variation for Fuel Economy Data. Refer to Table 7 on page 123.

a. Calculate the coefficient of variation for the following variables: cylinders, engine size, and city mpg.

b. According to the coefficient of variation, which variable has the greatest spread? The least variability?

83.

Coefficient of Variation for Energy Consumption. Refer to Table 16 on page 146.

a. Calculate the coefficient of variation for the per capita energy consumption for each continent.

b. According to the coefficient of variation, which continent has the greatest spread? Does this agree with your measures of spread from Exercise 76?

Mean Absolute Deviation. Recall that the variance and standard deviation use squared deviations because the mean deviation for any data set is zero. Another way to avoid negative deviations offsetting positive ones is to use the absolute value of the deviations. The mean absolute deviation (MAD) is a measure of spread that looks at the average of the absolute values of the deviations:

Use this measure of variability for Exercises 84 and 85.

84.

Mean Absolute Deviation for the Fuel Economy Data. Refer to Table 7 on page 123.

a. Find the mean absolute deviation for cylinders, engine size, and city mpg.

b. According to the mean absolute deviation, which variable has the greatest variability? The least variability?

85.

Mean Absolute Deviation for Energy Consumption. Refer to Table 16 on page 146.

a. Calculate the mean absolute deviation for each continent.

b. According to the mean absolute deviation, which continent has the greatest spread? Does this agree with your measures of spread from Exercise 76?

Coefficient of Skewness. The coefficient of skewness quantifies the skewness of a distribution. It is defined as

Most skewness values lie between 23 and 3. Negative values of skewness are associated with left-skewed distributions, whereas positive values are associated with right-skewed distributions. Values close to zero indicate distributions that are nearly symmetric. Use this information for Exercises 86−88.

86.

Coefficient of Skewness. For the following distributions, compute the coefficient of skewness and comment on the skewness of the distribution.

a. Mean 5 0, Median 5 0, Standard deviation 5 1

b. Mean 5 1, Median 5 0, Standard deviation 5 1

c. Mean 5 0, Median 5 1, Standard deviation 5 1

d. Mean 5 75, Median 5 80, Standard deviation 5 10

e. Mean 5 100, Median 5 100, Standard deviation 5 15

f. Mean 5 3.2, Median 53.0, Standard deviation 5 1.0

87.

What is the coefficient of skewness for any distribution where the mean equals the median, regardless of the nonzero value of the standard deviation?

88. Coefficient of Skewness for the Case Study Data. The median price change for the professional analysts is 9.60, the median for the dart throwers is 3.25, and the median for the DJIA is 7.00. Use this information, along with the information in Figure 21 on page 140 to answer the following.

a. Calculate the coefficient of skewness for each of the Pros, the Darts, and the DJIA.

b. Comment on the skewness of each distribution.

BRINGING IT ALL TOGETHER

In Exercises 89 and 90, we bring together all the measures of spread we have learned in the chapter and the new ones we learned in the exercises.

89. Fuel Economy Data. You calculated the range, variance, and standard deviation for this data in Exercises 63−65. You calculated the coefficient of variation in Exercise 82 and the mean absolute deviation in Exercise 84. Use this information to do the following.

a. Construct a table of the five measures of dispersion (range, sample variance, sample standard deviation, coefficient of variation, and mean absolute deviation) for the number of cylinders, the engine size, and the city mpg.

b. Which measures of dispersion suggest that the city mpg is the most dispersed variable? Engine size? Number of cylinders?

 90. Energy Consumption Data. You calculated the range and variance for this data in Exercise 76. You calculated the coefficient of variation in Exercise 83 and the mean absolute deviation in Exercise 85. Use this information to do the following:

a. Using the variance, calculate the standard deviation energy consumption for each continent.

b. Construct a table of the five measures of spread (range, sample variance, sample standard deviation, coefficient of variation, and mean absolute deviation) for each continent.

c. Do the measures of spread agree on which distribution has the greatest variability?

d. Bringing together all your statistics about measures of spread, what is your conclusion about the variability in Europe, compared with the other two continents?

CONSTRUCT YOUR OWN DATA SETS

 91. Construct two data sets, A and B, that you make up on your own, so that the range of A is greater than the range of B. Verify this.

 92. Construct two data sets, A and B, that you make up on your own, so that the standard deviation of A is greater than the range of B. Verify this.

 93. Construct two data sets, A and B, that you make up on your own, so that the mean of A is greater than the mean of B, but the standard deviation of B is greater than that of A. Verify this.

 94. Construct two data sets, A and B, that you make up on your own, so that the mean of A is greater than the mean of B, and the standard deviation of A is greater than that of B. Verify this.

 95. Construct two data sets, A and B, that you make up on your own, so that the range of A is greater than the range of B, but the standard deviation of B is greater than that of A. Verify this. (Hint: Remember the sensitivity of the standard deviation to extreme values.)

WORKING WITH LARGE DATA SETS

The Professionals versus the Darts. We will assess how well the Empirical Rule performs, using the Chapter 3 Case Study data set. Open the Darts data set. Use technology to do the following.

Figure 1.83: darts

 96. Find the mean and standard deviation for each of the Pros, the Darts, and the DJIA.

 97. Construct histograms of each of the Pros, the Darts, and the DJIA. Conclude that we can live with the assumption of a bell-shaped distribution for all three groups.

 98. For the Pros, do the following:

a. Calculate the following quantities: μ − 1σ, μ + 1σ, μ − 2σ, μ + 2σ, μ − 3σ, and μ + 3σ.

b. State what approximate percentages lie within those intervals, according to the Empirical Rule.

c. Count how many stock returns actually lie within each of those intervals. Divide these counts by the population size 100 to obtain the actual percentages.

d. Compare the approximate percentages estimated by the Empirical Rule with the actual percentages from the population data.

 99. Repeat the same comparison (a)–(d) from Exercise 98, but this time for the Darts.

100.

Repeat the same comparison (a)–(d) from Exercise 98, but this time for the DJIA.

3.2 Measures of Variability

antcolony

WHAT IF

?

3.2 Measures of Variability

CASE STUDY

148

Chapter 3 Describing Data Numerically

CASE STUDY

3.3

Working with Grouped Data

OBJECTIVES By the end of this section, I will be able to . . .

1

Calculate the weighted mean.

2

Estimate the mean for grouped data.

3

Estimate the variance and standard deviation for grouped data.

1

The Weighted Mean

Sometimes, not all the data values in a data set are of equal importance. Certain data values may be assigned greater importance or weight than others when calculating the mean. For example, have you ever figured out what your final grade for a course was based on the percentages listed in the syllabus? What you actually found was the weighted mean of your grades.

EXAMPLE 17  Weighted mean of course grades

The syllabus for the Introduction to Management course at a local college specifies that the midterm exam is worth 30%, the term paper is worth 20%, and the final exam is worth 50% of your course grade. Now, say you did not get serious about the course until after Halloween, so that you got a 40 on the midterm. You then began working harder, and got a 70 on the term paper. Finally, you remembered that you had to pay for the course again if you did not pass and had to retake it, so you worked really hard for the last month of the course and got a 90 on the final exam. Calculate your course average, that is, the weighted mean of your grades.

Solution

The data values are 40, 70, and 90. The weights are 0.30, 0.20, and 0.50. Your course weighted mean is then calculated as follows:

Because the final exam had the most weight, you were able to raise your course weighted mean to 71, and you passed the course.

The author’s syllabus for his Business Statistics I course during Summer 2014 stated that the quiz average was worth 50% of the course grade, with the midterm worth 20% and the final exam worth 30%. One of the students had a 90 quiz average, a 70 midterm grade, and an 85 final exam grade. Calculate the student’s course grade.

(The solution is shown in Appendix A.)

2

Estimating the Mean for Grouped Data

Thus far in Chapter 3, we have computed measures of center and spread from a raw data set. However, data are often reported using grouped frequency distributions. Without the original data, we cannot calculate the exact values of the measures of center and spread. The remainder of this section examines methods for approximating the mean, variance, and standard deviation of grouped data—that is, population data summarized using frequency distributions.

For each class in the frequency distribution, we estimate the class mean using the class midpoint. The class midpoint, denoted x, is defined as the mean of two adjoining lower class limits.

The product of the class frequency, f, and class midpoint, x, is used as an estimate of the sum of the data values within that class. Summing these products across all classes and dividing by the size of the data set thus provides us with an estimated mean for data grouped into a frequency distribution.

EXAMPLE 18  Calculating the estimated mean for grouped data

The first two columns of Table 17 contain the frequency distribution of the number of Americans younger than 85 years old who were living in the United States in 2013, by age group, as reported by the U.S. Census Bureau.

a. Find the class midpoints.

b. Calculate the product of each class frequency with its midpoint.

c. Find the sum of the frequencies, f, and the sum of the products, (f ? x).

d. Divide (f ? x) by f to find the estimated mean age of all Americans under the age of 85.

Solution

a. The midpoint for the first class (ages 0–20) is the mean of the lower class limits for this class (0) and the adjoining class (20). That is, the midpoint is (0 + 20)∙2 = 10. Similarly, the midpoint for