Fractile vs. equal

Fractile vs. Equal
Lesson Description
Students work with data that represent the ages of 24 people to learn the difference between categorizing data in fractile intervals and equal intervals. Students discuss divid-ing bonus points among class members to understand what per capita means. Then students look at per capita personal income by state using the GeoFRED™ mapping tool. They compare per capita personal income displayed with data in equal intervals and with data in fractile intervals.
Note: Data used in this lesson are from 2002. More recent data were not used because these data are updated periodically. Therefore, answer keys for more recent data would change. Grade Level
Concepts
Equal intervalsFractile intervalsIncomePer capita personal income Objectives
Explain the differences between equal intervals and fractile intervals.
Define personal income and per capita personal income.
Compare tables and maps of the same data displayed with equal intervals and with fractile intervals.
Content Standards
National Standards in Economics
Standard 13: Income for most people is determined by the market value of the
productive resources they sell. What workers earn depends, primarily, on the mar-
ket value of what they produce and how productive they are.
• Benchmark 2, Grade 8: To earn income, people sell productive resources. These include their labor, capital, natural resources and entrepreneurial talents.
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
National Geography Standards
Element One: The World in Spatial Terms
Standard 1: Globes and maps have been among the most ubiquitous tools for
learning geography. They have been joined by aerial photographs, remotely sensed
images, and geographic information systems. As technology makes them easier to
make, maps and other geographic representations appear practically everywhere.
Time Required
Materials
A copy of Handout 1 for each pair of students A copy of Handouts 2 and 3 for each student Procedures
Explain that a marketing research firm interviewed several people at a large suburban mall regarding their shopping habits. During the interviews, they asked each partici-pant to provide his or her age. Display a visual of Handout 1: Age Cards and explain that this visual lists the ages of one group of participants that was interviewed. Explain that students are going to use the age information to learn about ways to organize data to make it easy to understand and analyze. Point out that there are at least two ways to organize this list of ages or set of data. One way is to establish categories or age-range values and to then place each age into the appropriate category—for example, 20-25 years, 25-30 years and so on. Each category has the same age range, i.e. six years.
Another possibility is to place an equal number of data points/ages in each cat-egory and to then build the ranges accordingly. For example, suppose there are five people in the 16-19 age range, five people in the 20-30 range, five people in the 31-39 range, etc. In this case, the range for the categories varies, but the quantity of ages in each category is the same for all categories.
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Divide the students into pairs. Distribute a copy of Handout 1: Age Cards and a pair of scissors to each pair of students. Tell half of the pairs in the room to cut out their cards and organize the data into categories, with each category having the same number of age cards/data points. Tel these student pairs to write the value of the beginning and ending range for each category. For example, if each range is supposed to have three data points/ages in it, then one category might be 15-19 years. With this type of categorizing, the range in number of years per category might vary from category to category, but there wil be the same number of data points, or cards, in each category. Point out that the group-ings must have consecutive ages—that is 15-19, 20-27, etc., vs. 15, 20, 27, etc. Point out that there are different possible answers. Tell pairs in the other half of the class to cut the cards apart and divide the cards into categories of equal range. Tell them to write the range for each category, such as 20-25 years and so on. Remind them that each age-range/category should be equal—that is, it should include the same number of years. For example, a category titled “20-27 years” would include eight ages (20, 21, 22, 23, 24, 25, 26, 27) and therefore other ranges should include eight ages, too. Point out that there are dif-ferent possible answers.
Allow time for students to work. Ask several pairs from each half of the class to share their work.
Display Handout 2: Equal versus Fractile and refer to Table A and review the catego-ries/intervals in column 1. Ask the students if the age range (number of ages) for each category is the same. (Yes, five ages.) Distribute a copy of Handout 2 to each student and have students look at Table A. Review the intervals in column 1 and organize their cards into those intervals. Discuss the fol owing: • The first interval is 15-19 years.
• How many age cards fall in this interval? (three—16, 17 and 19) • Write these numbers in the second column of Table A across from the 15-19 • Continue with similar discussions for each interval in Table A. Refer to Handout 2: Equal vs. Fractile—Answer Key to check student work and record information on the chart. Refer to Table B on the visual of Handout 2 and review the intervals in column 3. Instruct student pairs to organize their cards into these intervals. Ask the students to report the number of age cards in each category. Write that information in col-umn 4 across from each interval. Refer to Handout 2: Equal vs. Fractile—Answer Key. Discuss the following: Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
• How do the age intervals differ in Table A and Table B? (In Table A, column 1, each of the age intervals are the same size. In Table B, column 3, the age inter-vals are of different sizes.) • How do the number of data points in each age interval differ in Table A and in Table B? (In Table A, there is a different number of data points in the age inter-vals. In Table B, there is the same number of data points in each age interval.) Explain that when data are divided into equal-size ranges or intervals, the method
of categorizing the data is called equal interval. In the first row of Table A, next
to the words “Table A,” write the title “Equal Interval.” Explain that when data are
divided into intervals so that there is the same number of data points in each inter-
val, the method of categorizing the data is called fractile interval. In the first row
of Table B, next to the words “Table B,” write the title “Fractile Interval.”
10. Point out that in both cases, the set of cards or data points were the same, but the way in which the data were organized was different. Point out that neither way of organizing the data—equal interval or fractile interval—is wrong or right. But the organization may give those reviewing the data different ideas about the data and allow people to tell slightly different stories about the data. 11. Tell the students that they are going to look at some data in GeoFRED. Explain that GeoFRED is an internet tool that displays data on maps. In this case, students will be looking at per capita personal income data. Define “per capita” as per person. To help students understand the idea of per capita, use the following example: (Note: For this example, determine the total number of bonus points available by multiplying the number of students in your class by 10.) • Over the semester so far, students have earned a total of 240 bonus points. • To determine how the bonus points relate to the number of students in the class, we need to determine per capita bonus points. • The same number of points will be available for each student in the class. How many points will be available per student? (number of points ÷ number of students = 10 points) 12. Open an internet link and enter the address: http://geofred.stlouisfed.org/. Tell students that this is the GeoFRED site of the Federal Reserve Bank of St. Louis. GeoFRED is designed to map data; that is, to take data about states and show the data on a map. Explain that the current map shows unemployment data by state for a given year for the continental United States (Hawaii and Alaska are not shown when the map first appears, but can be seen by zooming in so that these states show on the map.) Point out that the map has a legend that tells readers what the colors on the map mean.
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
13. Tell students that one set of data available on GeoFRED is per capita personal income by state. Remind the students that per capita means per person. Explain
that income is the payment people earn for the work that they do. Per capita
personal income is the total income earned by individuals in a state, region or
country during a year, divided by the population of the state, region or country. For
this map, we will use per capita personal income by state.
14. Click on the “Edit Data/Layers” tab at the top of the GeoFRED screen. From the drop-down boxes, select “State,” “Per Capita Personal Income,” “Annual” and “2002.” Click on “Update Map.” Refer students to the legend/key for the map. Point out that the data are reported in dollars. Distribute a calculator to each student. Discuss the following: • Into how many intervals are the data divided? (five) • Are the intervals equal? (Yes, each interval is $4,429.) • Which color represents the highest category of per capita personal income? • Which color represents the lowest category of per capita personal income? • How many states are in the lowest category in 2002? (14—Alabama, Arizona, Arkansas, Idaho, Kentucky, Louisiana, Mississippi, Montana, New Mexico, North Dakota, Oklahoma, South Carolina, Utah and West Virginia) • Click on the Northeast to zoom in and ask how many states are in the highest cat- egory in 2002? (one state—Connecticut) • Knowing that the intervals are equal in size and that the number of data points in each interval vary, is the data categorized into equal intervals or fractile inter-vals? (equal intervals) • Would you describe this map as being primarily light-colored (cream, gold or light yellow) or primarily dark-colored (light orange, dark orange, rust/brown)? (primarily light-colored) • If the map is primarily light-colored, what might you summarize about state- level per capita incomes in the United States in 2002? (Answers may vary. Some students might say that per capita incomes in the United States are rela-tively low and homogeneous or similar.) 15. Explain that GeoFRED also al ows users to organize data in fractile categories. Ask students what that means. (Sizes of the interval may vary, but the number of data points—in this case states—in each interval will be equal.) 16. Click on the “Edit Data/Layers” tab at the top of the GeoFRED screen. In the drop-down box, across the top, click on “Classes” tab. Point out that the current selection is “Equal,” “5” and “Yellow/Orange/Brown.” Select “Fractile,” “5” and “Yel ow/Orange/Brown.” Click on “Update Map.” Discuss the following: Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
• Which color or colors represent higher per capita incomes? (the darkest color) • Which colors represent lower per capita incomes? (the lightest color) • What is the interval for each category? (category 1: $3,084; category 2: $2,921; category 3: $1,610; category 4: $2,970; and category 5; $11,559) • How many states are in the lowest category of personal income? (10-Idaho, Montana, Utah, New Mexico, Arkansas, Mississippi, Louisiana, Kentucky, West Virginia and South Carolina) • How many states are in the highest category of personal income? (10—Colo- rado, Connecticut, Delaware, Maryland, Massachusetts, Minnesota, New Hampshire, New Jersey, New York and Virginia) • What is different about the number of data points (states) in each interval for this map? (the number of states in each interval are the same) • What is different about the intervals for this map? (no longer equal) • Would you describe this map as being primarily light-colored or primarily dark- • What might you summarize about state-level per capita income in the United States in 2002 from looking at this map? (Answers wil vary but might include that there are more differences in income or less homogeneity than when the data are categorized in equal intervals.) 17. Point out that if people do not carefully review the keys for these two maps (equal interval and fractile interval), the information might be misinterpreted. It is impor-tant that people ask questions about data that is presented to them, such as how the data are organized and why the data are organized in that way. 18. Review important content by asking the following questions: • How does categorizing data in equal intervals differ from categorizing data in fractile intervals? (With equal categorization, intervals or ranges of the same size are established and data are categorized in those ranges. There will be varying number of data points in each range. With fractile categorization, inter-vals or ranges are established so that there are an equal number of data points in each range or interval.) • What does per capita mean? (per person) • What is personal income? (the payment people earn for the work that they do) • What is per capita personal income? (the total income earned by people in a state, region or country, divided by the population of the state, region or country) Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Assessment
19. Distribute a copy of Handout 3: Assessment to each student. Open an internet browser and enter the address: http://geofred.stlouisfed.org/. Click on the “Edit Data/Layers” tab at the top of the GeoFRED screen. In the drop-down box across the top, click on the “Data” tab. Select “Total Gross Domestic Product by State.” For frequency, select “Annual” and for year, select “2002.” Click on “Update Map.” Refer students to the map key and ask the questions on Part I of Handout 3 as follows. After each question is asked, allow time for students to write their responses on the handout. Use Handout 3: Assessment—Answer Key to review student answers. • How many categories of data are there? • What is the interval for each category? • Which color represents the highest amount of Gross Domestic Product by state? • Which color represents the lowest amount of Gross Domestic Product by state? • Based on this map, does there appear to be much difference from state to state? • Why is this an example of equal interval categorization? 20. Click on the “Edit Data/Layers” tab at the top of the GeoFRED screen. In the drop- down box across the top, select the “Classes” tab. Select “fractile intervals” and update the map. Refer students to Part II of Handout 3. Read each question and allow time for students to record their answers on the handout. Refer to Handout 3—Answer Key to check student responses.
• How many categories of data are there? • What is the interval for each category? • Which color represents the highest amount of Gross Domestic Product by state? • Which color represents the lowest amount of Gross Domestic Product by state? • Based on this map, does there appear to be much difference from state to • Why is this an example of fractile interval categorization? Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 1: Age Cards
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 2: Equal vs. Fractile
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 2: Equal vs. Fractile—Answer Key
Number of Cards/
Number of Cards/
Category/Interval
Category/Interval
Data Points
Data Points
Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 3: Assessment
Part I:
Directions: Refer to the GeoFRED map being displayed in class to answer each question.
2. How many categories of data are there? 3. What is the interval for each category? 4. Which color represents the highest amount of Gross Domestic Product by state? 5. Which color represents the lowest amount of Gross Domestic Product? 6. Based on this map, does there appear to be much difference from state to state? 7. Why is this an example of equal interval categorization? Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 3: Assessment—cont.
Part II:
Directions: Refer to the GeoFRED map being displayed in class to answer each
question below.
9. How many categories of data are there? 10. What is the interval for each category? 11. Which color represents the highest amount of Gross Domestic Product by state? 12. Which color represents the lowest amount of Gross Domestic Product by state? 13. Based on this map, does there appear to be much difference from state to state? 14. Why is this an example of fractile interval categorization? Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 3: Assessment—Answer Key
Part I:
Directions: Refer to the GeoFRED map being displayed in class to answer each question.
1. How are the data reported? in dollars 2. How many categories of data are there? five 3. What is the interval for each category? $264,179 4. Which color represents the highest amount of Gross Domestic Product by state? 5. Which color represents the lowest amount of Gross Domestic Product? the 6. Based on this map, does there appear to be much difference from state to state? No, most states fall into the lowest category. 7. Why is this an example of equal interval categorization? The range for each category is equal, and the number of data points (states) in each category is different. Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.
Fractile vs. Equal
Handout 3: Assessment—Answer Key—cont.
Part II:
Directions: Refer to the GeoFRED map being displayed in class to answer each
question below.
8. How are the data reported? in dollars 9. How many categories of data are there? five 10. What is the interval for each category? category 1: $25,479; category 2: $49,541; category 3: $92,581; category 4: $114,281; and category 5: $1,044,011 11. Which color represents the highest amount of Gross Domestic Product by state? 12. Which color represents the lowest amount of Gross Domestic Product by state? 13. Based on this map, does there appear to be much difference from state to state? Yes, there is much more variance in the colors of the map. There are more states displayed in the darker colors and fewer states displayed in the lightest color. 14. Why is this an example of fractile interval categorization? The range for each cat- egory is different, and the number of data points [states] in each category is equal. Permission is granted to reprint or photocopy this lesson in its entirety for educational purposes, provided the user credits the Federal Reserve Bank of St. Louis, www.stlouisfed.org/education.

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