# MA.912.DP.3.3

Given a two-way relative frequency table or segmented bar graph summarizing categorical bivariate data, interpret joint, marginal and conditional relative frequencies in terms of a real-world context.

### Examples

Algebra 1 Example: Given the relative frequency table below, the ratio of true positives to false positives can be determined as 7.2 to 4.55, which is about 3 to 2, meaning that a randomly selected person who tests positive for diabetes is about 50% more likely to have diabetes than not have it.

 Positive Negative Total Has diabetes 7.2% 1.8% 9% Doesn't have diabetes 4.55% 86.45% 91%

### Clarifications

Clarification 1: Instruction includes problems involving false positive and false negatives.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Categorical Data

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grades 7 and 8, students explored the relationship between experimental and theoretical probabilities. In Algebra I, students study associations in bivariate categorical data using conditional relative frequencies and they apply these to real-world contexts. In later courses, students will study experimental and theoretical conditional probabilities in real-world contexts.
• In this benchmark, students will calculate and interpret joint, marginal and conditional relative frequencies in terms of a real-world context.
• When medical studies are involved the following terminology is used.
• True Positive (TP) means that a person who has a disease is correctly identified as having that disease by the test.
• False Positive (FP) means that a person who does not have a disease is incorrectly identified as having that disease by the test.
• True Negative (TN) means that a person who does not have a disease is correctly identified as not having that disease by the test.
• False Negative (FN) means that a person who has a disease is incorrectly identified as not having that disease by the test.
• In order to interpret the joint, marginal and conditional relative frequencies, students must know the difference between them.
• Marginal relative frequencies Guide students to understand that the total column and total row are in the “margins” of the table, thus they are referred to as the marginal relative frequencies.
• Joint relative frequencies Guide students to connect that the word joint refers to the coming together of more than one, therefore the term joint relative frequency refers to combination of two categories or conditions happening together.
• Conditional relative frequencies Guide students to connect that the word conditional refers imposing a constraint on one of the two variables, therefore the term conditional relative frequency refers to the percentage in a category for one variable when one has put a constraint on the other variable.
• For example, the relative frequency table below describes whether a student completed a test review sheet or not and whether they passed the test. To determine the conditional relative frequency of those who passed the test given that they completed the review, one could take the ratio between the joint relative frequency for passing the test in the first row and the marginal relative frequency in the first row, which is $\frac{\text{54%}}{\text{60%}}$ which is equivalent to 90%. One could say that 90% of those who completed the review passed the test.

### Common Misconceptions or Errors

• Students may not understand how the two-way relative frequency table connects with true positives/false positives and false negatives/true negatives.
• Students may not understand how to create the proper ratio for a given purpose.
• Students may not understand how to connect the ratio to the context.

### Strategies to Support Tiered Instruction

• Teacher co-creates a graphic organizer to distinguish between the positives and negatives.

• Teacher models distinguishing between relative frequencies given basic equations.

• Teacher provides opportunity to highlight the middle of the two-way table to distinguish the conditional and joint relative frequencies from marginal relative frequencies. The teacher models a joint, conditional or marginal relative frequency given the instructional task/problem.
• For example, when determining the conditional relative frequency that given that a student is an 8th grader, that student studies on the weekend, teacher can highlight the numbers in the two-way table below. Then, students can determine the relative frequency to be 51.5% since $\frac{\text{170}}{\text{330}}$ = 0.5151.

• Teacher co-creates a graphic organizer for each of the types of relative frequencies (joint, marginal and conditional) and highlights key similarities and differences.

• The relative frequency table below describes the members of the local sports club, showing whether they are male or female and whether they play disc golf or not.

• Part A. Compare the conditional relative frequency of being male given that they play disc golf and being female given that they play disc golf.
• Part B. Is there an association between being female and playing disc golf?
• Part C. Determine a possible reason for the association found in Part B. How do you think this may compare with any association that might exist in the general population between being female and playing disc golf?

### Instructional Items

Instructional Item 1
• In 2016, a clinical study in a small town of New Hampshire found that about 13% of its population had the flu and 87% did not. The study also showed that 19% of the people who had the flu tested negative while 8% of the people who did not have the flu tested positive. The relative frequency table below summarizes the data.

Determine the ratio of true positives to false positives and interpret that in terms of the context.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1210300: Probability and Statistics Honors (Specifically in versions: 2014 - 2015, 2015 - 2019, 2019 - 2022, 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.DP.3.AP.3: Given a segmented bar graph summarizing categorical bivariate data, select the interpretation in terms of a real-world context.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

School Start Time:

Students are asked to describe an association between two variables given a table of relative frequencies by column.

Type: Formative Assessment

Conditional Relative Frequency:

Students are asked to use a two-way frequency table to interpret two different conditional relative frequencies.

Type: Formative Assessment

Siblings and Pets:

Students are asked to describe an association between two variables given a table of relative frequencies by row.

Type: Formative Assessment

## Lesson Plans

Investigating Relationships With Two-Way Frequency Tables:

In this lesson, students are introduced to two-way frequency tables. They will calculate joint, marginal, and relative frequencies and draw conclusions about the relationship between two categorical variables.

Type: Lesson Plan

Can You Walk in My Shoes?:

Students use real-life data to create dot-plots and two-way tables. Students will collect data at the beginning of the lesson and use that data to create double dot plots and frequency tables, finding and interpreting relative frequencies.

The assignment allows students to work collaboratively and cooperatively in groups. They will communicate within groups to compare shoes sizes and ages to acquire their data. From the collection of data they should be able to predict, analyze and organize the data into categories (two-way tables) or place on a number line (dot-plot).

As the class assignment concludes, a discussion of the final class display should take place about the purchasing of shoes versus ages and the relationship that either exists or doesn't exist.

Type: Lesson Plan

How Random is "Shuffle Mode"?:

Today's teenager is a savvy consumer of digital music and the constantly-evolving technology that plays it. Ask a typical student what they know about iTunes versus Pandora versus Spotify—most of them will have an opinion on the "best" service for listening to songs. This lesson links students' existing interest in music with the mathematical topics of frequency and relative frequency.

The activity assumes that students know what Shuffle Mode does when they listen to digital music. Shuffle Mode is a function on digital music players that "shuffles" or randomly rearranges the order of a list of songs. Each time a person presses Shuffle Mode, the playlist is rearranged. If we assume a music player's Shuffle Mode is truly random, the chances of any particular song being played would equal 1 divided by the total number of songs (1/total #). This is analogous to rolling a fair die; each number on the die has an equal probability of being rolled (1/6 or 16.7%).

Type: Lesson Plan

Dropping Out or Staying In: Two-Way Table Analysis:

This lesson will require students to calculate relative frequencies and determine if an association exists within a two-way table. The students will analyze the frequencies and write a response justifying the associations and trends found within the table.

Type: Lesson Plan

In this lesson, students will collect data and construct two-way frequency tables. They will analyze the two-way frequency table by calculating relative conditional frequencies.

Type: Lesson Plan

What's Your Story?: Exploring Marginal and Conditional Distributions Through Social Networks:

In this interactive lesson, students explore marginal and conditional distributions. Students will calculate the relative frequency of data collected about cell phone use and social media access. These categories can be adjusted as necessary.

Type: Lesson Plan

High School Dropouts:

Students will examine dropout rates in the United States in 2012 by gender and race using data provided by the National Center for Education Statistics. Students will create conditional relative frequency tables to interpret the data and identify associations between genders, races, and dropout rates.

Type: Lesson Plan

In groups, students will analyze associations between categorical data by constructing two-way frequency tables and two-way relative frequency tables. Students will analyze and interpret the results and present their findings to their classmates.

Type: Lesson Plan

Breakfast for Champions?:

Students will create and interpret two-way frequency tables using joint, marginal, and conditional frequencies in context. They will investigate whether breakfast is for champions.

Type: Lesson Plan

Using Two-Way Frequency Tables to Analyze Data:

The television program, 60 Minutes reports that parents are intentionally holding their children back in kindergarten to give them a competitive advantage in sports later on in life. The students will use data collected to decide if this is truly a trend in the United States.

Type: Lesson Plan

Comedy vs. Action Movies Frequency Interpretation:

Using a completed survey of male and female student interest in comedy vs. action movies, the students will create a two-way frequency table using actual data results, fraction results, and percent results. The students will then act as the movie producer and interpret the data to determine if it is in their best interest to make a comedy or action movie. As the Summative Assessment, the student will take on the job/role of an actor/actress and interpret the data to support their decision.

Type: Lesson Plan

Show Me the Money:

Students will create a statistical question and collect and analyze data using relative frequency tables. They will present their argument in hopes of earning a cash prize for their philanthropy. An iterative process of critique and refinement will take place. A student packet is included that guides all parts of the lesson.

Type: Lesson Plan

Are you a CrimiNole or Gatorbait? Two rivalries in one table!:

This is an introduction to two-way frequency tables. The lesson will be delivered using a PowerPoint presentation. The teacher will introduce and define joint and marginal frequency, demonstrate how two-way frequency tables are constructed from a given set of data, calculate relative frequencies, and draw conclusions based on the information in the table. Students will practice these skills through guided practice with the teacher, independent practice, and complete a summative assessment to measure student learning. All resources, including the PowerPoint, have been provided.

Type: Lesson Plan

Can You Make Heads or Tails of It?:

Students learn how to make two-way tables, frequency, and relevant frequency tables. Students make predictions, collect data, and display it in two-way tables for interpretation.

Type: Lesson Plan

Two-Way Frequency Table and Relative Frequency:

In this lesson, the student will learn how to set up a two-way frequency table from two categorical variables and use the two-way frequency table to calculate frequency counts and relative frequency. The vocabulary terms learned in this lesson are two-way frequency table, relative frequency, joint frequency, marginal frequency, and conditional frequency.

Type: Lesson Plan

Is It a Guess or Statistics?:

This lesson teaches random sampling which leads to making inferences about a larger group or population. Students will determine the best measure of center to use for a data set. Students will collect data, select a data display and then analyze the data.

Type: Lesson Plan

How hot are hot dogs?:

In this lesson, students will learn how to convert simple and two-way frequency tables into relative frequency tables using data collected in the classroom.

Type: Lesson Plan

Legos, Lunch, and Lollipops: an introduction to two-way frequency tables:

Students will learn how to read, complete, and interpret two-way frequency tables.

Type: Lesson Plan

## Original Student Tutorial

Data and Frequencies:

Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.

Type: Original Student Tutorial

Musical Preferences:

This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.

Music and Sports:

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

## MFAS Formative Assessments

Conditional Relative Frequency:

Students are asked to use a two-way frequency table to interpret two different conditional relative frequencies.

School Start Time:

Students are asked to describe an association between two variables given a table of relative frequencies by column.

Siblings and Pets:

Students are asked to describe an association between two variables given a table of relative frequencies by row.

## Original Student Tutorials Mathematics - Grades 9-12

Data and Frequencies:

Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Data and Frequencies:

Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.

Type: Original Student Tutorial

Musical Preferences:

This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.

Music and Sports:

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Musical Preferences:

This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.