### 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

*For each of the following data sets, create a stemplot and identify any outliers.*

The miles-per-gallon ratings for 30 cars are shown below (lowest to highest):

19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43.

The height in feet of 25 trees is shown below (lowest to highest):

25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54.

The data are the prices of different laptops at an electronics store. Round each value to the nearest 10.

249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610

The following data are daily high temperatures in a town for one month:

61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95.

*For the next three exercises, use the data to construct a line graph.*

In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in Table 2.40.

Number of Times in Store | Frequency |
---|---|

1 | 4 |

2 | 10 |

3 | 16 |

4 | 6 |

5 | 4 |

In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table 2.41.

Years Since Last Purchase | Frequency |
---|---|

0 | 2 |

1 | 8 |

2 | 13 |

3 | 22 |

4 | 16 |

5 | 9 |

Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.42.

Number of TV Shows | Frequency |
---|---|

0 | 12 |

1 | 18 |

2 | 36 |

3 | 7 |

4 | 2 |

The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.43 shows the four seasons, the number of students who have birthdays in each season, and the percentage of students in each group. Construct a bar graph showing the number of students.

Seasons | Number of Students | Proportion of Population |
---|---|---|

Spring | 8 | 24% |

Summer | 9 | 26% |

Autumn | 11 | 32% |

Winter | 6 | 18% |

Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.8, construct a bar graph showing the percentages.

David County has six high schools. Each school sent students to participate in a county-wide science competition. Table 2.44 shows the percentage breakdown of competitors from each school and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school.

High School | Science Competition Population | Overall Student Population |
---|---|---|

Alabaster | 28.9% | 8.6% |

Concordia | 7.6% | 23.2% |

Genoa | 12.1% | 15.0% |

Mocksville | 18.5% | 14.3% |

Tynneson | 24.2% | 10.1% |

West End | 8.7% | 28.8% |

Use the data from the David County science competition supplied in Exercise 2.10. Construct a bar graph that shows the county-wide population percentage of students at each school.

### 2.2 Histograms, Frequency Polygons, and Time Series Graphs

65 randomly selected car salespersons were asked the number of cars they generally sell in one week. 14 people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Complete the table.

Data Value (Number of Cars) | Frequency | Relative Frequency | Cumulative Relative Frequency |
---|---|---|---|

What is the difference between relative frequency and frequency for each data value in Table 2.45?

What is the difference between cumulative relative frequency and relative frequency for each data value?

To construct the histogram for the data in Table 2.45, determine appropriate minimum and maximum *x*- and *y*-values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

Construct a frequency polygon for the following.

Pulse Rates for Women Frequency 60–69 12 70–79 14 80–89 11 90–99 1 100–109 1 110–119 0 120–129 1 Actual Speed in a 30-MPH Zone Frequency 42–45 25 46–49 14 50–53 7 54–57 3 58–61 1 Tar (mg) in Nonfiltered Cigarettes Frequency 10–13 1 14–17 0 18–21 15 22–25 7 26–29 2

Construct a frequency polygon from the frequency distribution for the 50 highest-ranked countries for depth of hunger.

Depth of Hunger | Frequency |
---|---|

230–259 | 21 |

260–289 | 13 |

290–319 | 5 |

320–349 | 7 |

350–379 | 1 |

380–409 | 1 |

410–439 | 1 |

Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlaid frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men?

Life Expectancy at Birth – Women | Frequency |
---|---|

49–55 | 3 |

56–62 | 3 |

63–69 | 1 |

70–76 | 3 |

77–83 | 8 |

84–90 | 2 |

Life Expectancy at Birth – Men | Frequency |
---|---|

49–55 | 3 |

56–62 | 3 |

63–69 | 1 |

70–76 | 1 |

77–83 | 7 |

84–90 | 5 |

Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births.

Sex/Year | 1855 | 1856 | 1857 | 1858 | 1859 | 1860 | 1861 |

Female | 45,545 | 49,582 | 50,257 | 50,324 | 51,915 | 51,220 | 52,403 |

Male | 47,804 | 52,239 | 53,158 | 53,694 | 54,628 | 54,409 | 54,606 |

Total | 93,349 | 101,821 | 103,415 | 104,018 | 106,543 | 105,629 | 107,009 |

Sex/Year | 1862 | 1863 | 1864 | 1865 | 1866 | 1867 | 1868 | 1869 |

Female | 51,812 | 53,115 | 54,959 | 54,850 | 55,307 | 55,527 | 56,292 | 55,033 |

Male | 55,257 | 56,226 | 57,374 | 58,220 | 58,360 | 58,517 | 59,222 | 58,321 |

Total | 107,069 | 109,341 | 112,333 | 113,070 | 113,667 | 114,044 | 115,514 | 113,354 |

Sex/Year | 1871 | 1870 | 1872 | 1871 | 1872 | 1827 | 1874 | 1875 |

Female | 56,099 | 56,431 | 57,472 | 56,099 | 57,472 | 58,233 | 60,109 | 60,146 |

Male | 60,029 | 58,959 | 61,293 | 60,029 | 61,293 | 61,467 | 63,602 | 63,432 |

Total | 116,128 | 115,390 | 118,765 | 116,128 | 118,765 | 119,700 | 123,711 | 123,578 |

The following data sets list full-time police per 100,000 citizens along with incidents of a certain crime per 100,000 citizens for the city of Detroit, Michigan, during the period from 1961 to 1973.

Year | 1961 | 1962 | 1963 | 1964 | 1965 | 1966 | 1967 |

Police | 260.35 | 269.8 | 272.04 | 272.96 | 272.51 | 261.34 | 268.89 |

Incidents | 8.6 | 8.9 | 8.52 | 8.89 | 13.07 | 14.57 | 21.36 |

Year | 1968 | 1969 | 1970 | 1971 | 1972 | 1973 |

Police | 295.99 | 319.87 | 341.43 | 356.59 | 376.69 | 390.19 |

Incidents | 28.03 | 31.49 | 37.39 | 46.26 | 47.24 | 52.33 |

- Construct a double time series graph using a common
*x*-axis for both sets of data. - Which variable increased the fastest? Explain.
- Did Detroit’s increase in police officers have an impact on the incident rate? Explain.

### 2.3 Measures of the Location of the Data

Listed are 29 ages for Academy Award-winning best actors *in order from smallest to largest:*

18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

- Find the 40
^{th}percentile. - Find the 78
^{th}percentile.

Listed are 32 ages for Academy Award-winning best actors *in order from smallest to largest:*

18, 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

- Find the percentile of 37.
- Find the percentile of 72.

Jesse was ranked 37^{th} in his graduating class of 180 students. At what percentile is Jesse’s ranking?

- For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
- The 20
^{th}percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20^{th}percentile in the context of the situation. - A bicyclist in the 90
^{th}percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90^{th}percentile in the context of the situation.

- For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
- The 40
^{th}percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40^{th}percentile in the context of the situation.

On an exam, would it be more desirable to earn a grade with a high or a low percentile? Explain.

Mina is waiting in line at the Department of Motor Vehicles. Her wait time of 32 minutes is the 85^{th} percentile of wait times. Is that good or bad? Write a sentence interpreting the 85^{th} percentile in the context of this situation.

In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78^{th} percentile. Should Li be pleased or upset by this result? Explain.

In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90^{th} percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90^{th} percentile in the context of this problem.

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:

- Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an
*admissions index*score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12 percent of high school students in the state. In this context, what percentile does the top 12 percent represent? - Students whose GPAs are at or above the 96
^{th}percentile of all students at their high school are eligible, called eligible in the local context, even if they are not in the top 12 percent of all students in the state. What percentage of students from each high school are*eligible in the local context*?

Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34^{th} percentile. The 34^{th} percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34 percent of the houses or 66 percent of the houses?

Use Exercise 2.25 to calculate the following values.

First quartile = ________

Third quartile = ________

10^{th} percentile = ________

### 2.4 Box Plots

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars.

Construct a box plot below. Use a ruler to measure and scale accurately.

Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas but not in others? How can you tell?

### 2.5 Measures of the Center of the Data

Find the mean for the following frequency tables:

Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 Daily Low Temperature Frequency 49.5–59.5 53 59.5–69.5 32 69.5–79.5 15 79.5–89.5 1 89.5–99.5 0 Points per Game Frequency 49.5–59.5 14 59.5–69.5 32 69.5–79.5 15 79.5–89.5 23 89.5–99.5 2

*Use the following information to answer the next three exercises:* The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest:
16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40

Identify the median.

*Use the following information to answer the next three exercises:* Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Calculate the following.

sample mean = $\overline{x}$ = ________

mode = ________

### 2.6 Skewness and the Mean, Median, and Mode

*Use the following information to answer the next three exercises.* State whether the data are symmetrical, skewed to the left, or skewed to the right.

16, 17, 19, 22, 22, 22, 22, 22, 23

When the data are skewed left, what is the typical relationship between the mean and median?

What word describes a distribution that has two modes?

Describe the relationship between the mode and the median of this distribution.

Describe the shape of this distribution.

Are the mean and the median the exact same in this distribution? Why or why not?

Describe the relationship between the mode and the median of this distribution.

The mean and median for the data are the same.

3, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7

Is the data perfectly symmetrical? Why or why not?

Which is the greatest, the mean, the mode, or the median of the data set?

11, 11, 12, 12, 12, 12, 13, 15, 17, 22, 22, 22

Which is the least, the mean, the mode, and the median of the data set?

56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67

Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?

In a perfectly symmetrical distribution, when would the mode be different from the mean and median?

### 2.7 Measures of the Spread of the Data

For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability.

Scientists randomly select five groups of 10 women from a population of 1,000 women to record their body fat percentage. The scientists compute the mean body fat percentage from each group. The differences in outcomes may be attributed to which type of variability?

A pharmaceutical company randomly assigns participants to one of two groups: one is a control group receiving a placebo, and another is a treatment group receiving a new drug to lower blood pressure. The differences in outcomes may be attributed to which type of variability?

Jaiqua and Harold are trying to determine how ramp steepness affects the speed of a ball rolling down the ramp. They measure the time it takes for the ball to roll down ramps of differing slopes. When Jaiqua rolls the ball and Harold works the stopwatch, they get different results than when Harold rolls the ball and Jaiqua works the stopwatch. The differences in outcomes may be attributed to which type of variability?

Twenty people begin the same workout program on the same day and continue for three months. During that time, all participants worked out for the same amount of time and did the same number of exercises and repetitions. Each person was weighed at both the beginning and the end of the program. The differences in outcomes regarding the amount of weight lost may be attributed to which type of variability?

*Use the following information to answer the next two exercises.* The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.

29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150

Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

Find the value that is one standard deviation below the mean.

Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team?

Baseball Player | Batting Average | Team Batting Average | Team Standard Deviation |
---|---|---|---|

Fredo | .158 | .166 | .012 |

Karl | .177 | .189 | .015 |

Use Table 2.60 to find the value that is three standard deviations

- above the mean, and
- below the mean

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 Daily Low Temperature Frequency 49.5–59.5 53 59.5–69.5 32 69.5–79.5 15 79.5–89.5 1 89.5–99.5 0 Points per Game Frequency 49.5–59.5 14 59.5–69.5 32 69.5–79.5 15 79.5–89.5 23 89.5–99.5 2