VII

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Course Learning Outcomes for Unit VII Upon completion of this unit, students should be able to:

1. Apply mathematical applications used in real-world situations. 1.1 Apply measures of central tendency to compare data.

6. Calculate basic statistical measures and analyze distribution graphs.

6.1 Identify the difference between a sample and a population. 6.2 Represent data visually using a variety of methods including frequency

tables and stem and leaf displays to compare data. 6.3 Compute the range of a data set. 6.4 Identify how the standard deviation measures the spread of a

distribution. 6.5 Apply the coefficient of a variation to compare the standard deviations

of different distributions.

Unit Lesson Suppose you owned a coffee shop in a nearby neighborhood. In order to boost sales, it would be important to know the time of day most people drink coffee or what their most popular coffee flavor is. To find these answers, we must use the principles of descriptive statistics. Descriptive statistics use data to calculate different facts about certain populations. Not all statistics are meaningful when taken out of context, but some statistics can give us an idea of surroundings, people, events, or other areas of interest. 14.1 Organizing and Visualizing Data: In this section, we will introduce the study of statistics and some key terms that are used when describing statistics. Populations and Samples Statistics is the study of gathering, organizing, analyzing, and making predictions from numerical information called data. For example, assume that we wanted to find out how many men in Florida drink coffee. First, we would need to gather the information. One way to gather information would be to conduct a survey to all men in Florida asking them if they drink coffee. Next, we would need to organize the data by identifying how many men answered yes to drinking coffee. Then, we would analyze the data by finding the percentage of men that drink coffee. Finally, we could make a prediction. For instance, if we found that 65% of men drink coffee in Florida, we could make a prediction that says that the majority of all men in the United States drink coffee.

Reading Assignment See information below.

Key Terms 1. Bar graph 2. Biased 3. Box-and-whisker plot 4. Coefficient of variation 5. Data 6. Five-number summary 7. Frequency distribution 8. Frequency table 9. Histogram

10. Mean, median, mode, range

11. Population 12. Relative frequency

distribution 13. Sample 14. Standard deviation 15. Stem-and-leaf display

UNIT VII STUDY GUIDE

Descriptive Statistics

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We will use the study below to introduce some key terms. These key terms are important to understand when performing any statistical study.

Study: Suppose that several colleges in the southeast are considering building dog parks on campus. The college presidents want to know if a dog park would be utilized. First, we are tasked to find the percentage of college students that own dogs.

Key Term Definition Example

Survey A tool that is administered to a sample population and used to find data about the study.

A tool used to ask college students if they own a dog.

Population The entire group of people mentioned in the study.

All college students who attend college in the southeast

Sample A part or subset of population that will be involved in the study. *Samples are used because it is not practical to survey every person in the population. *Samples should be chosen carefully because the sample represents the entire population.

College students who attend the Florida State University and the University of Central Florida.

Frequency Tables After a study is conducted, we need to gather the data. Usually, the data consists of large sets of numerical information. In this section, we will learn how to use frequency tables to gather and organize the data in a meaningful way. There are two types of frequency tables: frequency table and relative frequency table. A frequency table shows how many times a certain value occurs. A relative frequency table shows the percent of the time that each item occurs. The figure below represents an example of a frequency table. It shows how many students in a class received a particular grade letter. Notice that the column on the right identifies the different types of grade letters that received and the column on the left identifies how many students received that letter.

Grade Letter Frequency

A 18 B 24 C 15 D 12 F 6 Total 75

The next figure represents an example of a relative frequency table. Relative frequency is found dividing the frequency of a value by the total of all frequencies. We will use the same example represented by the figure above to find the relative frequencies of each grade letter.

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Grade Letter Frequency Relative Frequency

A 18 18

75 = 0.24

B 24 24

75 = 0.32

C 15 15

75 = 0.2

D 12 12

75 = 0.16

F 6 6

75 = 0.08

Total 75 1.00

Example: Construct a frequency table and a relative frequency table using the data given below: 7,8,6,5,7,10,2,7,9,5,8,8,10,9,6,5,10,7,9,8. Solution: First, we will need to construct a frequency table. To do this, draw a table and list all the different values you see in the right column. We will label the column ‘x’ and list the given values in order from smallest to largest.

x Frequency

2 3 4 5 6 7 8 9 10

Note: The ‘x’ values should also be listed in consecutive order to not

confuse the reader. Start with 2 in the table because it is the smallest in

the set. Stop at 10 because it is the largest value in the set. Include 3

and 4 because the values are in increments of one.

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Next, list the number of times each value occurs in the frequency column to finish the frequency table:

x Frequency

2 1 3 0 4 0 5 3 6 2 7 4 8 4 9 3 10 3

The problem also asks to construct a relative frequency table. To do this, use the frequency we just found. We will add an additional column and row:

x Frequency Relative Frequency

2 1 1

20 = 0.05

3 0 0

20 = 0

4 0 0

20 = 0

5 3 3

20 = 0.15

6 2 2

20 = 0.10

7 4 4

20 = 0.20

8 4 4

20 = 0.20

9 3 3

20 = 0.15

10 3 3

20 = 0.15

Total 20 1.00

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Our answer includes only two columns: ‘x’ and relative frequency. Therefore, our formal answer is written as follows:

x Relative Frequency

2 1

20 = 0.05

3 0

20 = 0

4 0

20 = 0

5 3

20 = 0.15

6 2

20 = 0.10

7 4

20 = 0.20

8 4

20 = 0.20

9 3

20 = 0.15

10 3

20 = 0.15

Sometimes, we will gather data that contains many different values. If this happens, it may be beneficial to construct a frequency table based on grouping. Example: The individual race times (in minutes) are provided for 20 people. Construct a frequency table to represent the data. 12, 48, 34, 33, 20, 18, 15, 42, 23, 20, 19, 30, 32, 33, 48, 15, 20, 23, 39, 24 Solution: The smallest value in our list is 12. The largest value in our list is 48. If we listed the values in consecutive order by a unit one (as in the previous example), we would have 48 – 12 = 36 rows in our frequency table. Therefore, we will group the values and list the grouping in the right column of frequency table. You may group the data by any increment; however, the larger the increment, the less rows we will have to place in our frequency table. Our smallest value is 12, so we can round down to 10 because 10 is divisible by 5. Therefore, we can group by increments of 5. We will first start with our smallest value of 12 and round down to 10. Ten will be our starting value to our first group. We want to include five numbers in each group. Start at 10 and count until we have five numbers: 10, 11, 12, 13, and 14. This tells us that our ending value in group one is 14. Therefore, group one is 10-14. Place this group in your frequency table and count the number of times that 10, 11, 12, 13, or 14 appear in the data set.

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Race Times (minutes)

Frequency

10-14 1

Next, find the second group. To do this, list the number that comes after 14. The number that comes after 14 is 15. Therefore, the starting number in group two is 15. Now, count until five numbers are listed: 15, 16, 17, 18, and 19. Our ending number is 19. Therefore, group two is listed as 15-19. Place this in the frequency table and count the number of times 15, 16, 17, 18, or 19 appears.

Race Times (minutes)

Frequency

10-14 1 15-19 4

Continue this pattern until all values are included in the table. The final solution is listed below.

Race Times (minutes)

Frequency

10-14 1 15-19 4 20-24 6 25-29 0 30-34 5 35-39 1 40-44 1 45-49 2

Representing Data Visually Bar graphs and histograms are charts that allow the viewer to get a snapshot of the data that was collected from a particular study. Graphs also allow the viewer to make some observations about the data in a quick and easy way. Example: Construct a bar graph for the data given in the frequency table below that contains the weight loss by 30 participants in the Biggest Loser show.

Weight Loss (lbs.) 0 1 2 3 4 5 6 7 8 9 10

Frequency 2 3 3 0 1 6 1 4 3 2 5

Solution: To construct a bar graph, list the frequencies on the vertical axis and the pounds lost on the horizontal axis. Graphs always start at zero and count up. Label the vertical axis (frequency) with numbers 0 through 6 since 6 is the largest frequency. Label the horizontal axis (pounds lost) with numbers 0 through 10. Next, represent the frequency of each pound lost by drawing a bar. The final solution is shown below:

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As stated before, visualizing data in the form of a graph makes it easy to make observations. For example, we can see that the most amount of people (6 people) lost 5 pounds. We can also easily see that no one lost 3 pounds. Example: Because of budget cutbacks, the campus rec center surveyed the number of students using the center hourly between 9 PM and midnight for a semester to decide if it should reduce its hours. Use the results in the given graph to answer the following questions.

a. What was the smallest number of students in the rec center and how many times did it occur?

0 occurred seven times

b. What was the smallest student count between 5 and 8 inclusive? How many

times did that occur?

The smallest student count between 5 and 8 is 7. This only occurred once between 5 and 8.

c. For how many hours was the survey taken?

We will find the total of all frequencies to solve:

7+8+5+3+8+10+7+9+5+3+2=67

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d. For what fractional part of the total number of hours were there less than five students present?

𝟕 + 𝟖 + 𝟓 + 𝟑

𝟔𝟕 =

𝟐𝟑

𝟔𝟕

Stem and Leaf Plots Stems and leaf plots are another way to organize data. These plots separate the number into two parts. We call these parts the stem and the leaf. For example, the number 89 has two parts: 8 and 9. We call the 8 a stem and the 9 a leaf. As shown, the parts are separated by the place value of each digit that make up the number. Example: Construct a stem leaf plot for the following values: 89, 87, 82, 83. Solution: First, identify the stems. The stem of a number is the first digit on the left. Therefore, every number in the set has a stem of 8. Next, identify the leaves. The leaves are the second digit of the number. Therefore, the leaves of the set are 9, 7, 2 and 3. We will write the stem on the left side of the diagram and the leaves on the right side from smallest to largest. Our answer is:

Example: Represent the two sets of data on a single stem-and-leaf display.

A: 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33, 42, 18, 22, 39, 21, 26,18, 43

B: 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21, 29, 22, 24, 33, 47, 23, 22, 18, 33 Solution: The stem of each number listed is the first digit. The leaves are the last digit. Since we are constructing a plot for two sets of data, we need to form the plot so that all the stems are in the middle of the plot. The leaves for set A will be on the right side of the stem, and the leaves of set B will be on the left side of the stem. First, list all the stem values. The stem values are the first digit of every number. Do not repeat the values in the chart. These values are highlighted in red.

A: 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33, 42, 18, 22, 39, 21, 26, 18, 43

B: 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21, 29, 22, 24, 33, 47, 23, 22,18, 33 The unique stem values are 1, 2, 3, and 4.

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Second, list all the leaves in set A and B. Make sure to pair them with their corresponding stem, and list the leaves with the smallest digit closest to the stem. Our answer is:

14.2 Measures of Central Tendency: Now that we know how to organize data, we can start calculating different statistics. The calculations we will discuss in this section are mean, median, and mode. The mean, median, and mode are called measures of central tendency are used by statisticians to describe different sets of data. Mean The mean of a set of data is the average. The mean of set of numbers is found by dividing the sum of all numbers by the number of values in the set. Example: Find the mean of the following data set: 3, 13, 2, 6, 9, 10, 16, 8 Solution: First, find the sum of all the numbers.

3 + 13 + 2 + 6 + 9 + 10 + 16 + 8 = 67 Next, find the number of values in the set.

There are 8 numbers in the set. Last, divide 67 by 8 to get the mean.

67 ÷ 8 = 𝟖. 𝟑𝟕𝟓 Example: Exam scores: Assume that in your history of film class you have earned test scores of 78, 82, 56, and 72, and only one test remains. If you need a mean score of 70 to earn a C, then what must you obtain on the final test? Solution:

Let 𝑥 be the grade on the final test. We then construct an equation using the formula for finding an average. 78+82+56+72+𝑥

5 = 70 Plug in the 4 known test values, 𝑥 for the missing test, 5

because there are five test scores to be averaged, and 70 for the average.

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288+𝑥

5 = 70 Add the numbers in the top of the fraction.

288 + 𝑥 = 70 ∙ 5 Multiply each side of the equation by 5 to clear the fraction. 288 + 𝑥 = 350 Simplify the right side by multiplying. 𝑥 = 350 − 288 Subtract 288 from both sides to isolate 𝑥. 𝑥 = 62 Simplify by subtracting on the right side to find the value of 𝑥. You must make a 62 or higher on the final test to earn a C. Notation Mathematicians developed a formula to express the mean. This formula uses a Greek symbol ( ∑) to show that we need to find the sum of a set of numbers. This symbol, ∑ , is capital sigma and is called the summation symbol. Example: Let x = 3, 7, 8, 9. Find ∑𝑥 . Solution: To find ∑𝑥, we will add all values together in set x. Therefore,

The formula for the mean is:

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Median The median is the middle number of a data set. It is found by listing the values of a set in order from smallest to largest and then finding the middle value. There are two cases to consider:

1. If there is an odd number of values, then the median is the value in the middle position.

2. If there is an even number of values, then the median is in the average of the two middle values.

Example: Let x = 3, 7, 8, 2, 1. Find the median. Solution: List all values from smallest to largest.

1, 2, 3, 7, 8 The middle value is the median, since there is an odd number of values in the data set.

Example: Let x = 3, 7, 8, 9. Find the median. Solution: List all values from smallest to largest.

3, 7, 8, 9 The median is the average of the two middle values because there is an even number of values in the data set.

The middle values are 7 and 8. Find the average by adding the values together and dividing by two.

𝑀𝑒𝑑𝑖𝑎𝑛 = 7 + 8

2 =

15

2 = 𝟕. 𝟓

The median of the data set is 7.5.

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Mode The mode is the number that occurs most often in a data set. If two numbers occur the most often, then the data set has multiple modes. If the more than two values occur most often, then there is no mode. Example: Let x = 3, 2, 5, 8, 12, 3, 5, 12, 12, 2, 2 Solution: There are two modes, namely 2 and 12. They both occur 3 times. Example: Find the mean, median, and mode for the following:

12, 4, 4, 8, 4, 7, 9, 8, 7, 7 Solution:

First place the data in order: 4, 4, 4, 7, 7, 7, 8, 8, 9, 12 Calculate the mean:

�̅� = ∑ 𝑥

𝑛 =

4 + 4 + 4 + 7 + 7 + 7 + 8 + 8 + 9 + 12

10 =

70

10 = 7

Calculate the median: Since the data set contains an even number of elements, we average the two numbers in the middle to find the median.

7 + 7

2 =

14

2 = 7

Calculate the mode: The highest occurring numbers (4 and 7) each occur twice. The modes are 4 and 7. Using Frequency Tables to Compute Mean, Median, and Mode It can be time consuming to calculate the mean and median of large sets of data. If the data can be organized in a frequency table, it is much faster to find these values. Steps for computing the mean of a frequency distribution – The mathematical formula for finding the mean of a frequency distribution is

�̅� = ∑(𝑥 ∙ 𝑓)

∑ 𝑓

The steps will demonstrate how to find the mean of the following frequency table.

x Frequency

2 18

3 32

4 4

5 12

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1. Create a new column in the frequency table and label it “Product.” Multiply the number in the “x” column by the number in the frequency column. Do this for all rows in the frequency table.

x Frequency

(𝑓) Product

(𝑥 ∙ 𝑓) 2 18 2 ∙ 18 = 36 3 32 3 ∙ 32 = 96 4 4 4 ∙ 4 = 16 5 12 5 ∙ 12 = 60

2. Create a new row in the frequency table and label it “Total.” Add all of the

products together and place this number in the table. This is notated by ∑(𝑥 ∙ 𝑓).

x Frequency

(𝑓) Product

(𝑥 ∙ 𝑓)

2 18 2 ∙ 18 = 36

3 32 3 ∙ 32 = 96

4 4 4 ∙ 4 = 16

5 12 5 ∙ 12 = 60

Total ∑(𝑥 ∙ 𝑓) = 36 + 96 + 16 + 60 = 𝟐𝟎𝟖

3. Find the sum of all the frequencies in the table.

x Frequency

(𝑓) Product

(𝑥 ∙ 𝑓) 2 18 2 ∙ 18 = 36 3 32 3 ∙ 32 = 96 4 4 4 ∙ 4 = 16 5 12 5 ∙ 12 = 60

Total ∑ 𝑓 = 66 36 + 96 + 16 + 60 = 208

4. Divide the number you found in step two (208) by the number you found in step three (66).

�̅� = ∑(𝑥 ∙ 𝑓)

∑ 𝑓 = 208 ÷ 66 = 𝟑. 𝟏𝟓

Steps for computing the median using a frequency table – The steps will demonstrate how to find the median of the following frequency table.

x Frequency

2 18

3 32

4 4

5 12

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1. Create a row and label it “Total.” Add all frequencies together and place the sum in the table.

x Frequency

2 18

3 32

4 4

5 12 Total 66

2. Divide the total by 2.

66 ÷ 2 = 𝟑𝟑

3. Step 2 tells us that are 33 values before and after the median. Since the value

in step two is even (66), we need to find the number in the data set that is in position 33 and 34.

x Frequency

2 18

3 32

There are eighteen 2s and thirty-two 3s. Therefore, position 33 is a 3 and position 34 is a 3.

𝑀𝑒𝑑𝑖𝑎𝑛 = 3 + 3

2 =

6

2 = 𝟑

Computing the mode using a frequency table – The mode is the ‘x’ value that has the highest frequency.

x Frequency

2 18 3 32

4 4

5 12

The mode is 3. Example: Find the mean, median, and mode for the following frequency distributions.

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Solution: Calculate the mean: The mean is found using the following formula:

�̅� = ∑(𝑥 ∙ 𝑓)

∑ 𝑓 =

𝑠𝑢𝑚 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠

Take the 𝑥 value from the left column and multiply by the corresponding 𝑓 value in the right column.

∑(𝑥 ∙ 𝑓) = 2 ∙ 3 + 5 ∙ 2 + 7 ∙ 3 + 8 ∙ 2 + 9 ∙ 1 + 10 ∙ 5

+ 11 ∙ 2 = 6 + 10 + 21 + 16 + 9 + 50 + 22

= 134 To find 𝑛, add up all the values from the right column (column 𝑓)

𝑛 = ∑ 𝑓 = 3 + 2 + 3 + 2 + 1 + 5 + 2 = 18

�̅� = ∑(𝑥 ∙ 𝑓)

∑ 𝑓 =

134

18 ≈ 𝟕. 𝟒

Calculate the median: We need to find the middle two numbers from the table. Since the table has 18 numbers, then the median is the average of the 9th and 10th numbers. By adding up the frequency values from the right column of the table we find that the 9th and 10th numbers are both 8. Therefore, the median is the average of 8 and 8.

8 + 8

2 =

16

2 = 𝟖

Calculate the mode: The number we the highest frequency is 10. The Five Number Summary The Five Number Summary consists of five numbers that describe a data set. More specifically, it identifies the minimum value, first quartile (written Q1), median, third quartile (written Q3), and maximum value of a data set.

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The five-number summary is represented by a graph called the box-and-whisker plot. An example is shown below.

Example: Construct a box and whisker plot using the following data:

6, 20, 9, 18, 17, 25, 20, 4, 22, 25, 13, 5, 6, 9, 13, 23, 20, 13, 18, 14 Solution: First, arrange the data in ascending order.

4, 5, 6, 6, 9, 9, 13, 13, 13, 14, 17, 18, 18, 20, 20, 20, 22, 23, 25, 25

Graph depicting ages (Pirnot, 2014, p. 721)

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Next, take the data set and find the middle number or numbers and then find the middle of the lower half (Q1) and the middle of the upper half (Q3).

Because our data set has an even number of elements, the median will be the average of the two numbers in the middle:

𝑀𝑒𝑑𝑖𝑎𝑛 = 14 + 17

2 = 15.5

We now have the information for the five-number summary and can use this information to construct a box plot. Minimum value: 4 Q1: 9 Median: 15.5 Q3: 20 Maximum value: 25

14.3 Measures of Dispersion: In this section, we will review how to find the range, standard deviation, and coefficient of variation of a dataset. These are statistics used when describing the dispersion or spread of data. The Range of Data Set The range of data set is found by subtracting the smallest value from the largest value of the data set. Example: Find the range for the following data set. {5, 7, 9, 4, 6, 8, 7, 10} Solution: First, rearrange the data and place it in order from smallest to largest. {4, 5, 6, 7, 7, 8, 9,10}

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Next, calculate the range by subtracting the smallest value from the largest value as follows:

𝑙𝑎𝑟𝑔𝑒𝑠𝑡 – 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 = 10 – 4 = 𝟔 The range is 6.

Standard Deviation The standard deviation also measures how much distance is between each data value and the mean of the data set. The formula for the standard deviation is as follows:

Example: Find the standard deviation for the following data set. {5, 7, 9, 4, 6, 8, 7, 10} Solution: We will use the following formula for the finding the standard deviation:

𝑠 = √ ∑(𝑥 − 𝑥)2

𝑛 − 1

Steps:

1. Calculate the mean ( 𝑥 ).

First, find the sum of all values in the set.

∑ 𝑥 = 4 + 5 + 6 + 7 + 7 + 8 + 9 + 10 = 56

Next, divide the sum by the number of values in the set (n = 8).

Therefore, 𝒙 = 𝟕

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2. Create a table. List the data values in one column and label the next two columns as follows:

3. Complete column two (𝑥 − 𝑥 ). To do this, subtract the mean (𝑥 = 7) from every data value in column one. It is ok if you get negative values.

4. Complete column three: (𝑥 − 𝑥)2 . To do this, square the value you calculated in column two. *Note: To square a value, multiply that number to itself.

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5. Plug in the values you calculated in the standard deviation formula and solve.

The standard deviation is 2. We can also use what we have learned about frequency tables to calculate the standard deviation. Example: The following frequency table summarizes the number of job offers made to graduates of a Microsoft network administrator certification program. Complete the table entries to find the mean and standard deviation for this distribution.

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Solution:

1. Calculate the mean:

2. Complete the table as follows:

To find the standard deviation we will use the following formula. Remember that the number of data values that are represented by 𝑛 is ∑ 𝑓.

The standard deviation is 1.64. Example: The following table gives the annual incomes for eight families, in thousands of dollars. Find the number of standard deviations family H’s income is from the mean.

Solution: We will let x represent the annual income. First calculate the mean of the data set. Recall that the mean is found by adding up all values in the set and dividing by the number of values.

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Next, create and complete a table as shown below:

Last, calculate the standard deviation by using the following formula and substitute our values:

To find the number of standard deviations family H’s income is from the mean, we will subtract the mean from family H’s income, then divide that result by the standard deviation as follows:

51 − 49

1.60 = 1.25

Therefore, family H’s income is 1.25 standard deviations above the mean. The Coefficient of Variation The coefficient of variation is used when comparing the standard deviation of two data sets. The data values of a set are more varied, if the coefficient of variation is large. The formula for the coefficient of variation is

𝐶𝑉 = 𝑠

𝑥 ∙ 100%.

Therefore, to calculate the coefficient of variation (CV), divide the standard deviation by the mean and multiply by 100. The CV will always be a percentage. Example: A particular brand of laptop was sampled with regard to the time it can be used before it requires recharging. The mean time was calculated to be 7.8 hours with a standard deviation of 1.78 hours. Calculate the coefficient of variation for this example.

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Solution: Use the formula for the coefficient of variation:

𝐶𝑉 = 𝑠

𝑥 ∙ 100%

Now, substitute the given values into the equation to find the answer as follows:

𝐶𝑉 = 𝑠

𝑥 ∙ 100% =

1.78

7.8 ∙ 100% = 22.82%

Reading Assignment Chapter 14: Descriptive Statistics: What a Data Set Tells Us

 Section 14.1 Organizing and Visualizing Data*, pp. 702-713

 Section 14.2 Measures of Central Tendency, pp. 714-726

 Section 14.3 Measures of Dispersion, pp. 727-737