2.1 Introduction to Descriptive Statistics and Frequency Tables

Descriptive statistics are ways of organizing summarizing and presenting data.  There are two main types: visual and numerical.  Usually we want to first examine a dataset visually then describe it numerically.  Appropriate methods often depend on the type of data you are working with, however frequency tables are a quick easy way to organize any type of data.

2.2 Displaying and Describing Categorical Data

Two basic visual methods we have for displaying categorical statistics are:

• Pie charts
• Bar charts

When describing a categorical distribution we want to note:

• Mode
• Level of variability (diversity)

2.3 Displaying Quantitative Data

The following are common methods of displaying quantitative data

• Stem-and-leaf plots
• Dot plots
• Line graphs
• Histograms
• Frequency polygons
• Time series plots

Some work better to show certain aspects, or for different sample sizes than others.

2.4 Describing Quantitative Distributions

When describing a quantitative distribution we want to at least note 4 things: the shape of the distribution, the presence of outliers, the center, and the spread.  A helpful acronym to remember this is SOCS:

• Shape –  Can be identified visually, want to note symmetry or lack thereof (skewness) and modality
• Outliers – Extreme outliers can be seen visually
• Center – Central tendency can be estimated visually
• Spread – Dispersion can be estimated visually and roughly quantified with the range

2.5 Measures of Location and Outliers

2.6 Measures of Center

The mean and the median can be calculated to help you find the “center” of a data set. The mean may often be the best representation of the center of a dataset, but the median is often more appropriate when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your data set.

The mean of a dataset can can be approximated from a frequency table by:

[latex]\(\mu =\frac{\sum fm}{\sum f}\)[/latex]

Where:

• f = interval frequencies
• m = interval midpoints.

2.7 Measures of Spread

The variance and standard deviation are numerical measures of the spread or dispersion of a dataset. There are different equations to use if you are calculating the standard deviation of a sample or of a population. You find the sample and population standard deviations, respectively:

• s = [latex]\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\overline{x}\right)}^{2}}{n-1}}[/latex]
• σ = [latex]\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\mu \right)}^{2}}{N}}[/latex]

To find the standard deviation of a frequency table:

[latex]{s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}[/latex] where [latex]\begin{array}{l}{s}_{x}=\text{ sample standard deviation}\\ \overline{x}\text{ = sample mean}\end{array}[/latex]

Z-scores are a measure of location that puts an observation in units of standard deviations relative to the mean.  We can use these to compare things from different distributions.

Try to define the terms below on your own. Scroll over any term to check your response!

2.1 Introduction 

• Descriptive statistics
• Graphical descriptive methods
• Numerical descriptive methods
• Distribution
• Frequency
• Relative frequency
• Cumulative relative frequency
• Lower class limit
• Upper class limit
• Class width
• Class midpoint

2.2 Displaying and Describing Categorical Data

• Categorical data
• Mode
• Variability

2.3 Displaying Quantitative Data

• Quantitative data
• Discrete data
• Ordinal categorical data

2.4 Describing Quantitative Distributions

• Shape
• Outliers
• Center
• Spread
• Modality

2.5 Measures of Location and Outliers

2.6 Measures of Center

• Mean (average)
• Median
• Sample mean
• Population mean
• Robust
• Mode
• Modality

2.7 Measures of Spread

• Variation (variability, spread)
• Standard deviation
• Sample
• Population
• Variance
• Population
• Sample
• Z-score

2.1 Introduction 

1. The two types of descriptive statistical methods are:

Answer:

• Graphical
• Numerical

2.2 Displaying and Describing Categorical Data

1. The two basic options for graphing categorical data are

Answer:

• Graphical
• Numerical

2. When describing categorical data we want to note:

Answer:

• Mode
• Level of variability

2. When describing the level of variability in categorical data we want to think about it as:

Answer:

• Diversity

2.3 Displaying Quantitative Data

1. Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data, since books are counted.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
4, 4, 4, 4, 4, 4
5, 5, 5, 5, 5
6, 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

The following histogram displays the number of books on the x-axis and the frequency on the y-axis.

2. We will construct an overlay frequency polygon comparing the scores from the figure below with the students’ final numeric grade.

6. 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?^[3]

7. The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO₂ emissions for the United States.^[4]

8. 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.^[5]

Solution:

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2.5 Measures of Location and Outliers

1. Test scores for a college statistics class held during the day are: 99, 56, 78, 55.5, 32, 90, 80, 81, 56, 59, 45, 77, 84.5, 84, 70, 72, 68, 32, 79, 90. Test scores for a college statistics class held during the evening are: 98, 78, 68, 83, 81, 89, 88, 76, 65, 45, 98, 90, 80, 84.5, 85, 79, 78, 98, 90, 79, 81, 25.5. ^[7]

2. The following data set shows the heights in inches for the boys in a class of 40 students: 66, 66, 67, 67, 68, 68, 68, 68, 68, 69, 69, 69, 70, 71, 72, 72, 72, 73, 73, 74. The following data set shows the heights in inches for the girls in a class of 40 students: 61, 61, 62, 62, 63, 63, 63, 65, 65, 65, 66, 66, 66, 67, 68, 68, 68, 69, 69, 69. Construct a box plot using a graphing calculator for each data set, and state which box plot has the wider spread for the middle 50% of the data.

3. Graph a box-and-whisker plot for the data values shown.

10, 10, 10, 15, 35, 75, 90, 95, 100, 175, 420, 490, 515, 515, 790

The five numbers used to create a box-and-whisker plot are:

• Min: 10
• Q₁: 15
• Med: 95
• Q₃: 490
• Max: 790

Solution: The following graph shows the box-and-whisker plot.

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

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

6. In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results.

1. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected.
2. Have more Americans or more Germans surveyed been to over eight foreign countries?
3. Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other?

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7. Given the following box plot, answer the questions.

1. Think of an example (in words) where the data might fit into the above box plot. In 2–5 sentences, write down the example.
2. What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart?

1. Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
2. Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.

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8. Given the following box plots, answer the questions.

1. In complete sentences, explain why each statement is false.
   1. Data 1 has more data values above two than Data 2 has above two.
   2. The data sets cannot have the same mode.
   3. For Data 1, there are more data values below four than there are above four.
2. For which group, Data 1 or Data 2, is the value of “7” more likely to be an outlier? Explain why in complete sentences.

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9. A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.

1. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series.
2. Which group is most likely to have an outlier? Explain how you determined that.
3. Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other?
4. Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread?
5. Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread?
6. Look at the BMW 5 series. Estimate the interquartile range (IQR).
7. Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this?
8. Look at the BMW 5 series. Which interval has the fewest data in it? How do you know this?
   1. 31–35
   2. 38–41
   3. 41–64

1. Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50% of buyers are more variable than the ages of the lower 50%.
2. The BMW 3 series is most likely to have an outlier. It has the longest whisker.
3. Comparing the median ages, younger people tend to buy the BMW 3 series, while older people tend to buy the BMW 7 series. However, this is not a rule, because there is so much variability in each data set.
4. The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
5. The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
6. IQR ~ 17 years
7. There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is concentrated.
8. The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25% fall between 41 and 64. Since 25% of values fall between 31 and 38, we know that fewer than 25% fall between 31 and 35.

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11. Santa Clara County, CA, has approximately 27,873 Japanese-Americans. Their ages are as follows:

Figure 2.77

Age Group	Percent of Community
0–17	18.9
18–24	8.0
25–34	22.8
35–44	15.0
45–54	13.1
55–64	11.9
65+	10.3

1. Construct a histogram of the Japanese-American community in Santa Clara County, CA. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph?
2. What percentage of the community is under age 35?
3. Which box plot most resembles the information above?

12. For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars.

Data: 389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000.

14. For the two data sets in Example 1 (test scores), find the following:

15. Find the interquartile range for the following two data sets and compare them.

16. Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:

a. Find the 28^th percentile.

b. Find the median.

c. Find the third quartile.

Solution:

a. Notice the 0.28 in the “cumulative relative frequency” column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28^th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28^th percentile is between the last six and the first seven. The 28^th percentile is 6.5.

b. Look again at the “cumulative relative frequency” column and find 0.52. The median is the 50^th percentile or the second quartile. 50% of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and eleven of the 7s. The median or 50^th percentile is between the 25^th, or seven, and 26^th, or seven, values. The median is seven.

c. The third quartile is the same as the 75^th percentile. You can “eyeball” this answer. If you look at the “cumulative relative frequency” column, you find 0.52 and 0.80. When you have all the fours, fives, sixes and sevens, you have 52% of the data. When you include all the 8s, you have 80% of the data. The 75^th percentile, then, must be an eight. Another way to look at the problem is to find 75% of 50, which is 37.5, and round up to 38. The third quartile, Q₃, is the 38^th value, which is an eight. You can check this answer by counting the values. (There are 37 values below the third quartile and 12 values above.

17. Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65^th percentile.

18. Using the table below:

24. For runners in a race, a higher speed means a faster run.

a. Is it more desirable to have a speed with a high or a low percentile when running a race?

32. The median age for U.S. Black citizens currently is 30.9 years; for U.S. White citizens it is 42.3 years.

a. Based upon this information, give two reasons why the Black median age could be lower than the White median age.

b. Does the lower median age for Blacks necessarily mean that Blacks die younger than Whites? Why or why not?

c. How might it be possible for Blacks and Whites to die at approximately the same age, but for the median age for Whites to be higher?

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33. Six hundred adult Americans were asked by telephone poll, “What do you think constitutes a middle-class income?” The results are in the figure below. Also, include left endpoint, but not the right endpoint.

Figure 2.84

Salary ($)	Relative Frequency
< 20,000	0.02
20,000–25,000	0.09
25,000–30,000	0.19
30,000–40,000	0.26
40,000–50,000	0.18
50,000–75,000	0.17
75,000–99,999	0.02
100,000+	0.01

1. What percentage of the survey answered “not sure”?
2. What percentage think that middle-class is from $25,000 to $50,000?
3. Construct a histogram of the data.
   1. Should all bars have the same width, based on the data? Why or why not?
   2. How should the <20,000 and the 100,000+ intervals be handled? Why?
4. Find the 40^th and 80^th percentiles
5. Construct a bar graph of the data

Solutions:

1. 1 – (0.02+0.09+0.19+0.26+0.18+0.17+0.02+0.01) = 0.06
2. 0.19+0.26+0.18 = 0.63
3. Check student’s solution.

4. 40^th percentile will fall between 30,000 and 40,000

   80^th percentile will fall between 50,000 and 75,000

5. Check student’s solution.

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34. Given the following box plot:

1. which quarter has the smallest spread of data? What is that spread?
2. which quarter has the largest spread of data? What is that spread?
3. find the interquartile range (IQR).
4. are there more data in the interval 5–10 or in the interval 10–13? How do you know this?
5. which interval has the fewest data in it? How do you know this?
   1. 0–2
   2. 2–4
   3. 10–12
   4. 12–13
   5. need more information

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35. The following box plot shows the U.S. population for 1990, the latest available year.

1. Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know?
2. 12.6% are age 65 and over. Approximately what percentage of the population are working age adults (above age 17 to age 65)?

Solutions:

1. more children; the left whisker shows that 25% of the population are children 17 and younger. The right whisker shows that 25% of the population are adults 50 and older, so adults 65 and over represent less than 25%.
2. 62.4%

36. On a 20 question math test, the 70th percentile for number of correct answers was 16. Interpret the 70th percentile in the context of this situation.

47. The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers: 0.5, 0.7, 1.1, 1.2, 1.2, 1.3, 1.3, 1.5, 1.5, 1.7, 1.7, 1.8, 1.9, 2.0, 2.2, 2.5, 2.6, 2.8, 2.8, 2.8, 3.5, 3.8, 4.4, 4.8, 4.9, 5.2, 5.5, 5.7, 5.8, 8.0

48. For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest): 32, 32, 33, 34, 38, 40, 42, 42, 43, 44, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 52, 52, 52, 53, 54, 56, 57, 57, 60, 61. Construct a stem plot for the data.

49. The table below shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.

51. Using this data set, construct a histogram.

53. Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each.

54. Following are the 2010 obesity rates by U.S. states and Washington, DC.

^[9]

Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the x-axis with the states. Answers will vary.

55. Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99.

56. The table below contains the 2010 obesity rates in U.S. states and Washington, DC.

^[10]

57. For each of the following data sets, create a stem plot and identify any outliers.The miles per gallon rating 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

Figure 2.102

Stem	Leaf
1	9 9 9
2	0 1 1 5 5 5 6 6 8 9
3	1 1 2 2 3 4 5 6 7 7 8 8 8 8
4	1 3 3

a. 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

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

Figure 2.103

Stem	Leaf
2	5 5 6 7 7 8
3	0 0 1 2 3 3 5 5 5 7 7 9
4	1 6 9
5	6 7 7
6	1

c. The 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

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58. The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. The figure below 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.

Figure 2.104

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, construct a bar graph showing the percentages.

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59. David County has six high schools. Each school sent students to participate in a county-wide science competition. The figure below 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.

Figure 2.105

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 above. Construct a bar graph that shows the county-wide population percentage of students at each school.

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2.6 Measures of Center

1. The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.

3, 4, 5, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 17, 17, 18, 19, 19, 19, 21, 21, 22, 22, 23, 24, 24, 24, 24

2. In a sample of 60 households, one house is worth $2,500,000. Half of the rest are worth $280,000, and all the others are worth $315,000. Which is the better measure of the “center”: the mean or the median?

3. The number of books checked out from the library from 25 students are as follows: 0, 0, 0, 1, 2, 3, 3, 4, 4, 5, 5, 7, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12. Find the mode.

4. Find the mean for the following frequency tables.

1. Figure 2.107

   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

2. Figure 2.108

   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

3. Figure 2.109

   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

5. 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

6. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, nineteen generally sell four cars, twelve generally sell five cars, nine generally sell six cars, and eleven generally sell seven cars. Calculate the following:

7. The most obese countries in the world have obesity rates that range from 11.4% to 74.6%. This data is summarized in the following table. ^[11]

1. What is the best estimate of the average obesity percentage for these countries?
2. The United States has an average obesity rate of 33.9%. Is this rate above average or below?
3. How does the United States compare to other countries?

8. The following figure gives the percent of children under five considered to be underweight. What is the best estimate for the mean percentage of underweight children? ^[12]

9. Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?

a.

b.

^[13]

c.

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

12. When the data are symmetrical, what is the typical relationship between the mean and median?

Solution: When the data are symmetrical, the mean and median are close or the same.

13. What word describes a distribution that has two modes?

14. Use the following graph to answer a-c.

a. Describe the shape of this distribution.

• Solution: The distribution is skewed right because it looks pulled out to the right.

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

c. Describe the relationship between the mean and the median of this distribution.

• Solution: The mean is 4.1 and is slightly greater than the median, which is four.

15. Data: 11, 11, 12, 12, 12, 12, 13, 15, 17, 22, 22, 22

a. Is the data perfectly symmetrical? Why or why not?

b. Which is the largest, the mean, the mode, or the median of the data set?

16. Data: 56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67

a. Is the data perfectly symmetrical? Why or why not?

b. Which is the largest, the mean, the mode, or the median of the data set?

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

19. The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years.

1. What does it mean for the median age to rise?
2. Give two reasons why the median age could rise.
3. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?

20. Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information.

1. How can you determine which survey was correct ?
2. Explain what the difference in the results of the surveys implies about the data.
3. If the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia’s sample? How do you know?
   

   

4. If the two box plots depict the distribution of values for each supervisor, which one depicts Ercilia’s sample? How do you know?
   

   

21. We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section.

22. How much time does it take to travel to work? The figure below shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

23. Find the midpoint for each class. These will be graphed on the x-axis. The frequency values will be graphed on the y-axis values.

2.7 Measures of Spread

1. Use the following data (first exam scores) from Susan Dean’s spring pre-calculus class: 33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100.

a. Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places.

b. Calculate the following to one decimal place:

1. 1. The sample mean
   2. The sample standard deviation
   3. The median
   4. The first quartile
   5. The third quartile
   6. IQR

c. Construct a box plot and a histogram on the same set of axes. Make comments about the box plot, the histogram, and the chart.

Solutions:

a.

Figure 2.122

Data	Frequency	Relative Frequency	Cumulative Relative Frequency
33	1	0.032	0.032
42	1	0.032	0.064
49	2	0.065	0.129
53	1	0.032	0.161
55	2	0.065	0.226
61	1	0.032	0.258
63	1	0.032	0.29
67	1	0.032	0.322
68	2	0.065	0.387
69	2	0.065	0.452
72	1	0.032	0.484
73	1	0.032	0.516
74	1	0.032	0.548
78	1	0.032	0.580
80	1	0.032	0.612
83	1	0.032	0.644
88	3	0.097	0.741
90	1	0.032	0.773
92	1	0.032	0.805
94	4	0.129	0.934
96	1	0.032	0.966
100	1	0.032	0.998 (Why isn’t this value 1?)

b.

1. 1. The sample mean = 73.5
   2. The sample standard deviation = 17.9
   3. The median = 73
   4. The first quartile = 61
   5. The third quartile = 90
   6. IQR = 90 – 61 = 29

c. The x-axis goes from 32.5 to 100.5; y-axis goes from –2.4 to 15 for the histogram. The number of intervals is five, so the width of an interval is (100.5 – 32.5) divided by five, is equal to 13.6. Endpoints of the intervals are as follows: the starting point is 32.5, 32.5 + 13.6 = 46.1, 46.1 + 13.6 = 59.7, 59.7 + 13.6 = 73.3, 73.3 + 13.6 = 86.9, 86.9 + 13.6 = 100.5 = the ending value; No data values fall on an interval boundary.

The long left whisker in the box plot is reflected in the left side of the histogram. The spread of the exam scores in the lower 50% is greater (73 – 33 = 40) than the spread in the upper 50% (100 – 73 = 27). The histogram, box plot, and chart all reflect this. There are a substantial number of A and B grades (80s, 90s, and 100). The histogram clearly shows this. The box plot shows us that the middle 50% of the exam scores (IQR = 29) are Ds, Cs, and Bs. The box plot also shows us that the lower 25% of the exam scores are Ds and Fs.

2. The following data show the different types of pet food stores in the area carry: 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12. Calculate the sample mean and the sample standard deviation to one decimal place.

3. 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.

4. 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?

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

6. The population parameters below describe the full-time equivalent number of students (FTES) each year at ABC University from 1976–1977 through 2004–2005.

• μ = 1000 FTES
• median = 1,014 FTES
• σ = 474 FTES
• first quartile = 528.5 FTES
• third quartile = 1,447.5 FTES
• n = 29 years

7. Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best GPA when compared to other students at his school? Explain how you determined your answer.

12. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:

16. Refer to the figures below and determine which of the following (a-d) are true and which are false. Explain your solution to each part in complete sentences.

1. The medians for all three graphs are the same.
2. We cannot determine if any of the means for the three graphs is different.
3. The standard deviation for graph b is larger than the standard deviation for graph a.
4. We cannot determine if any of the third quartiles for the three graphs is different.

Solutions:

1. True
2. True
3. True
4. False

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19. X = the number of days per week that 100 clients use a particular exercise facility.

21. This figure contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.

Figure 2.139

Year	Total Number of Deaths
2000	231
2001	21,357
2002	11,685
2003	33,819
2004	228,802
2005	88,003
2006	6,605
2007	712
2008	88,011
2009	1,790
2010	320,120
2011	21,953
2012	768
Total	823,856

Answer each of the following questions and check your answers below.

a. What is the frequency of deaths measured from 2006 through 2009?
b. What percentage of deaths occurred after 2009?
c. What is the relative frequency of deaths that occurred in 2003 or earlier?
d. What is the percentage of deaths that occurred in 2004?
e. What kind of data are the numbers of deaths?
f. The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?

Solution:

---

22. The following figure contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.

Figure 2.140

Year	Total Number of Crashes	Year	Total Number of Crashes
1994	36,254	2004	38,444
1995	37,241	2005	39,252
1996	37,494	2006	38,648
1997	37,324	2007	37,435
1998	37,107	2008	34,172
1999	37,140	2009	30,862
2000	37,526	2010	30,296
2001	37,862	2011	29,757
2002	38,491	Total	653,782
2003	38,477

Answer the following questions.

1. What is the frequency of deaths measured from 2000 through 2004?
2. What percentage of deaths occurred after 2006?
3. What is the relative frequency of deaths that occurred in 2000 or before?
4. What is the percentage of deaths that occurred in 2011?
5. What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.

---

23. Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

Figure 2.141: Part-time Student Course Loads

# of Courses	Frequency	Relative Frequency	Cumulative Relative Frequency
1	30	0.6
2	15
3

Fill in the blanks in the figure above.

1. What percent of students take exactly two courses?
2. What percent of students take one or two courses?

---

24. Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. The figure below shows the ages of the chief executive officers for the first 60 ranked firms.

Figure 2.142

Age	Frequency	Relative Frequency	Cumulative Relative Frequency
40–44	3
45–49	11
50–54	13
55–59	16
60–64	10
65–69	6
70–74	1

1. What is the frequency for CEO ages between 54 and 65?
2. What percentage of CEOs are 65 years or older?
3. What is the relative frequency of ages under 50?
4. What is the cumulative relative frequency for CEOs younger than 55?
5. Which graph shows the relative frequency and which shows the cumulative relative frequency?

25. The figure below contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

26. The following data are the shoe sizes of 50 male students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.
9, 9, 9.5, 9.5, 10, 10, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5
11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5
12, 12, 12, 12, 12, 12, 12, 12.5, 12.5, 12.5, 12.5, 14

27. The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
3, 3, 3, 3, 3, 3, 3, 3
20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.

28. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, nineteen generally sell four cars, twelve generally sell five cars, nine generally sell six cars, and eleven generally sell seven cars. Complete the table.

29. Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:

32. Use the data to construct a line graph.

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

Figure 2.156

Number of times in store	Frequency
1	4
2	10
3	16
4	6
5	4

Solution:

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

Figure 2.159

Number of TV Shows	Frequency
0	12
1	18
2	36
3	7
4	2

Solution:

Image References

Figure 2.55: Figure 2.6 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-2-histograms-frequency-polygons-and-time-series-graphs

Figure 2.58: Figure 2.9 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-2-histograms-frequency-polygons-and-time-series-graphs

Figure 2.68: Figure 2.8 from OpenIntro Introductory Statistics (2019) (CC BY-SA 3.0). Retrieved from https://cnx.org/contents/pJuo4h-U@4.478:UMM7d-Hy/Display-Data

Figure 2.70: Figure 2.14 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-4-box-plots

Figure 2.71: Figure 2.17 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-4-box-plots

Figure 2.72: Figure 2.45 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-homework

Figure 2.73: Figure 2.46 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-homework

Figure 2.74: Figure 2.47 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-homework

Figure 2.75: Figure 2.46 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-homework

Figure 2.78: Figure 2.47 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-bringing-it-together-homework

Figure 2.85: Figure 2.43 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-homework

Figure 2.86: Figure 2.44 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-homework

Figure 2.96: Figure from OpenStax Introductory Business Statistics (2012) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-business-statistics/pages/2-homework

Figure 2.99: Figure 2.58 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions

Figure 2.100: Figure 2.59 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions

Figure 2.101: Figure 2.60 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions

Figure 2.106: Figure 2.54 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions

Figure 2.112: Figure 2.24 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-6-skewness-and-the-mean-median-and-mode

Figure 2.114: Figure 2.25 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-6-skewness-and-the-mean-median-and-mode

Figure 2.115: Figure 2.7.9 from LibreTexts Introductory Statistics (2020) (CC BY 4.0). Retrieved from https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Introductory_Statistics_(OpenStax)/02%3A_Descriptive_Statistics/2.07%3A_Skewness_and_the_Mean_Median_and_Mode

Figure 2.117: Figure 2.9.1 from LibreTexts Introductory Business Statistics (2020) (CC BY 4.0). Retrieved from https://biz.libretexts.org/Courses/Gettysburg_College/MGT_235%3A_Introductory_Business_Statistics/02%3A_Descriptive_Statistics/2.09%3A_Homework

Figure 2.118: Figure 2.51 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-bringing-it-together-homework

Figure 2.121: Figure 2.58 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-solutions

Figure 2.123: Figure 2.8.2 from LibreTexts Introductory Statistics (2020) (CC BY 4.0). Retrieved from https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Introductory_Statistics_(OpenStax)/02%3A_Descriptive_Statistics/2.08%3A_Measures_of_the_Spread_of_the_Data

Figure 2.134: Figure from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-solutions#element-324s-solution

Figure 2.135: Figure 2.52 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/2-bringing-it-together-homework

Figure 2.143: Figure 1.11 from OpenStax Introductory Business Statistics (2012) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-business-statistics/pages/1-homework

Figure 2.153: Figure 2.36 from OpenStax Introductory Business Statistics (2012) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-business-statistics/pages/2-solutions#eip-457-solution

Figure 2.154: Figure 2.37 from OpenStax Introductory Business Statistics (2012) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-business-statistics/pages/2-solutions#eip-457-solution

Figure 2.157: Figure 2.51 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions#fs-idp113295424-solution

Figure 2.160: Figure 2.52 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/2-solutions#fs-idp113295424-solution

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