# MA.6.DP.1.5

Create box plots and histograms to represent sets of numerical data within real-world contexts.

### Examples

The numerical data set {15,0,32,24,0,17,42,0,29,120,0,20}, collected based on minutes spent on homework, can be represented graphically using a box plot.

### Clarifications

Clarification 1: Instruction includes collecting data and discussing ways to collect truthful data to construct graphical representations.

Clarification 2: Within this benchmark, it is the expectation to use appropriate titles, labels, scales and units when constructing graphical representations.

Clarification 3: Numerical data is limited to positive rational numbers.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Status: State Board Approved

## Benchmark Instructional Guide

• Box Plot
• Data
• Histogram

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 5, students collected and represented numerical data using tables, line graphs or line plots and interpreted this data by determining the mean, median, mode and range. In grade 6, students include box plots and histograms to the types of data displays they are creating. In grade 7, students are asked to choose and create an appropriate graphical representation of both numerical and categorical data, which includes bar graphs and circle graphs (MTR.1.1, MTR.2.1, MTR.7.1).
• Instruction includes developing statistical questions that generate numerical data.
• Instruction includes opportunities for students to collect their own data to create a graphical display. This increases student interest in analyzing the data (MTR.1.1).
• Scales can be represented using brackets, ranges, inclusive or exclusive endpoints.
• For example, the numerical data set from the example is {15, 0, 32, 24, 0, 17, 42, 0, 29, 120, 0, 20}. A student has determined to use intervals of 0-20, 20-40, 40-60, 60-80, 80-100, 100-120.
• Students could use brackets to represent inclusive endpoints and parentheses to represent exclusive endpoints. Students can write the intervals as [0-20); [20-40); [40-60); [60-80); [80-100); [100-120].
• Students can write the intervals as 0-20, 21-40, 41-60, 61-80, 91-100, 101-120 since the data only contains whole numbers.
• It is important to note that students may create intervals (or bins) that do not necessitate the use of these as it is not an issue with the given data set.
• Students should discuss how there could be differences in distributions when the bins of the histograms may vary.
• Given a set of data, allow students to choose to create either a histogram or box plot, ensuring both are being created. Then discuss pros and cons for why one may be preferred over another. Practice this using data sets containing whole numbers as well as with data sets containing positive rational numbers. Students should be provided with different ways to organize the data in order to best create the graphical representation.
• Instruction allows for analysis of the truthfulness or reasonableness of the data set. Students should understand whether the data can be used to showcase real situations.
• Instruction includes the use of online tools to quickly show the difference in distributions when changing the size of the bins in a histogram.

### Common Misconceptions or Errors

• Students may choose bin sizes that do not effectively show the distribution of the data.
• For example, students gathered gas mileage (in miles per gallon) data for 50 cars as shown in the histograms below. Histogram A easily shows the gas mileage for the majority of cars, and a much smaller count being above 40 miles per gallon. Histogram B uses very small bins which leads the consumer to believe that there is more variability in the data than truly exists.

• Students may incorrectly include the same number in 2 bins. For example, if bins are 010, 10-20, 20-30, etc., they must decide whether 10 is included in the first bin or if numbers greater than 0, but less than 10 are included in the first bin.
• Students may incorrectly calculate the median given a data set with an even number of values.
• Students may incorrectly believe more data will create a larger box in a box plot.
• Students may neglect to order values from least to greatest when creating a box plot.
• Students may neglect to include titles and labels in the graphical representations.

### Strategies to Support Tiered Instruction

• Teacher reviews the difference between histograms and bar graphs, creating an anchor chart with properties of a histogram for students to refer to.
• Teacher reinforces how scales are represented with specific endpoints. The endpoints they chose to use, or as defined in a problem, tell them if the point is included in the bin or not. Include notation of endpoints on anchor chart to display in the classroom.
• If there are an even number of total data points, teacher models how the median is found by finding the mean of the two middle data points. Teachers provide opportunities for students to practice this skill by gradually releasing them until they are proficient and gain understanding.
• Teacher co-constructs vocabulary guide/anchor chart with students who need additional support understanding the vocabulary for measures of center and variation.
• Examples of guides and charts are shown below.

Instructional Task 1 (MTR.3.1, MTR.7.1)
Data from the International Shark Attack File on the number of shark attacks in Florida is given in the table below.

• Part A. Construct a box plot to summarize this data.
• Part B. Identify and explain what each of the key numbers you used to make the box plot means in the context of the data.
• Part C. Describe the distribution of the number of shark attacks in Florida between 2001 and 2013. Be sure to describe the spread and distribution of the data.

Instructional Task 2 (MTR.4.1, MTR.7.1)

Below are the 25 birth weights, in ounces, of all the Labrador Retriever puppies born at Kingston Kennels in the last six months.

• Part A. Use a box plot or histogram to summarize these birth weights and explain how you chose which type of graph to use.
• Part B. Describe the distribution of birth weights for puppies born at Kingston Kennels in the last six months. Be sure to describe the spread and distribution of the data.
• Part C. What is a typical birth weight for puppies born at Kingston Kennels in the last six months? Explain why you chose this value.

### Instructional Items

Instructional Item 1
Every year a local basketball team plays 82 games. During the past two decades, the number of wins each year was:
42, 54, 51, 72, 67, 61, 43, 38, 53, 57, 41, 68, 54, 52, 47, 60, 46, 53, 73, 65
Make a histogram to summarize the data.

*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.
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.DP.1.AP.5: Create histograms to represent sets of numerical data with 10 or fewer elements.

## Related Resources

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

## Formative Assessments

Shark Attack Data:

Students are asked to construct a box plot corresponding to a given set of data.

Type: Formative Assessment

Students are asked to construct a histogram corresponding to a given set of data.

Type: Formative Assessment

## Lesson Plans

Using Box Plots and the Mean Absolute Deviation to Interpret Data:

This lesson explores the use of box plots and the mean absolute deviation to compare two data sets and draw inferences.

Type: Lesson Plan

Measurement Data Error:

In this interdisciplinary lesson, students will practice the skill of data collection with a variety of tools and by statistically analyzing the class data sets will begin to understand that error is inherent in all data.

Type: Lesson Plan

Analyzing Box Plots:

This lesson is designed for students to demonstrate their knowledge of box plots.

• Students will need to create four box plots from given data.
• Students will need to analyze the data displayed on the box plots by comparing similarities and differences.
• Students will work with a partner to complete the displays and the follow-up questions.

Type: Lesson Plan

Exercise Your Brain, Analyze Your Heart Rate:

Students will compile the data gathered from measuring their resting heart rates and heart rates after exercising into box plots. Using these displays, they will analyze the data's center, shape, and spread.

Type: Lesson Plan

Bowling for Box Plots:

Students will learn about the effects of an outlier and interpret differences in shape, center, and spread using a bowling activity to gather data. The students will learn to score their games, report their scores, and collectively measure trends and spread by collaborating to create a box plot. They will analyze and compare box plots, and determine how much of an effect an extreme score (outlier) can have on the overall box plot of the data.

Type: Lesson Plan

"What's My Grade" is a lesson that will focus on a sample student's grades to demonstrate how a final grade is calculated as well as explore possible future grades. Students will create the distributions of each grade category using histograms. They will also analyze grades using mean and standard deviation. Students will use statistics to determine data distribution while comparing the center and spread of two or more different data sets.

Type: Lesson Plan

How tall is an 8th grader?:

Ever wonder about the differences in heights between students in grade 8? In this lesson, students will use data they collect to create and analyze multiple box plots using 5-number summaries. Students will make inferences about how height and another category may or may not be related.

Type: Lesson Plan

Plane Statistics:

This lesson starts with an activity to gather data using paper airplanes then progresses to using appropriate statistics to compare the center and spread of the data. Box plots are used in this application lesson of concepts and skills previously acquired.

Type: Lesson Plan

Which One: Box plot, Dot Plot, or Histogram?:

Students will be asked to obtain data and create a human box plot, which will be analyzed and explained using statistical terms. Students will then understand the differences and advantages to using the box plot, histogram, and dot plot. Students will also practice selecting the most appropriate graphical representation for a set of data.

Type: Lesson Plan

This resource can be used to teach students how to create and compare box plots. After completing this lesson, students should be able to answer questions in both familiar and unfamiliar situations.

Type: Lesson Plan

The Distance a Coin Will Travel:

This lesson is a hands-on activity that will allow students to collect and display data about how far different coins will travel. The data collected is then used to construct double dot plots and double box plots. This activity helps to facilitate the statistical implications of data collection and the application of central tendency and variability in data collection.

Type: Lesson Plan

Which is Better? Using Data to Make Choices:

Students use technology to analyze measures of center and variability in data. Data displays such as box plots, line plots, and histograms are used. The effects of outliers are taken into consideration when drawing conclusions. Students will cite evidence from the data to support their conclusions.

Type: Lesson Plan

How long did you study?:

Students will create and analyze histograms based on student study time when preparing for the Algebra EOC. Students will be given a set of data and guided notes

Type: Lesson Plan

How many licks does it take to get to the center?:

Students will create different displays, line plots, histograms, and box plots from data collected about types of lollipops. The data will be analyzed and compared. Students will determine "Which lollipop takes the fewest number of licks to get to the center: a Tootsie Pop, a Blow Pop, or a Dum Dum?"

Type: Lesson Plan

Birthday Party Decisions:

Students will create and compare four different boxplots to determine the best location for a birthday party.

Type: Lesson Plan

Outliers in the Outfield – Dealing With Extreme Data Points:

Students will explore the effects outliers have on the mean and median values using the Major League Baseball (MLB) salary statistics. They will create and compare box plots and analyze measures of center and variability. They will also be given a set of three box plots and asked to identify and compare their measures of center and variablity.

Type: Lesson Plan

This lesson allows students to have a hands-on experience collecting real-world data, creating graphical representations, and analyzing their data. Students will make predictions as to the outcome of the data and compare their predictions to the actual outcome. Students will create and analyze line plots, histograms, and box plots.

Type: Lesson Plan

Digging the Plots:

Students construct box plots and use the measure(s) of center and variability to make comparisons, interpret results, and draw conclusions about two populations.

Type: Lesson Plan

A Walk Down the Lane:

Students will collect data, and create box plots. Students will make predictions about which measurement best describes the spread and center of the data. Students will use this information to make predictions.

Type: Lesson Plan

How Old are the Players?:

For this lesson, students will research the ages of players on two basketball teams. They will find the five-number summary, the mean, and determine if there are outliers in the data set. Two box plots will be created and the measures of center and variation analyzed.

Type: Lesson Plan

Centers, Spreads, and Outliers:

The students will compare the effects of outliers on measures of center and spread within dot plots and box plots.

Type: Lesson Plan

The Penny Lab:

Students will design an investigation to collect and analyze data, determine results, write a justification and make a presentation using U.S. pennies.

Paired student teams will determine the mass of 50 U.S. pennies. Students will also collect other data from each penny such as minted year and observable appearance. Students will be expected to organize/represent their data into tables, histograms and other informational structures appropriate for reporting all data for each penny. Students will be expected to consider the data, determine trends, and research information in order to make a claim that explains trends in data from minted U.S. pennies.

Hopefully, student data reports will support the knowledge that the metallic composition of the penny has changed over the years. Different compositions can have significantly different masses. A sufficiently random selection of hundreds of pennies across the class should allow the students to discover trends in the data to suggest the years in which the composition changed.

Type: Lesson Plan

Baking Soda and Vinegar: A statistical approach to a chemical reaction.:

Students experiment with baking soda and vinegar and use statistics to determine which ratio of ingredients creates the most carbon dioxide. This hands-on activity applies the concepts of plot, center, and spread.

Type: Lesson Plan

Homework or Play?:

Students will be given data and then plot the data using a graphical method of choice (dot plot, bar graph, box plot, etc.) The students will work in groups and then analyze and summarize the data.

Type: Lesson Plan

Interpreting Box Plots:

Students will analyze various real world scenario data sets and create, analyze, and interpret the components of the box plots. Students will use data from morning routines, track times, ages, etc. Lesson includes a PowerPoint, homework, and assessments.

Type: Lesson Plan

Exploring Box plots:

This lesson involves real-world data situations. Students will use the data to create, explore, and compare the key components of a box plot.

Type: Lesson Plan

The Debate: Who is a Better Baller?:

In this activity the students will use NBA statistics on Lebron James and Tim Duncan who were key players in the 2014 NBA Finals, to calculate, compare, and discuss mean, median, interquartile range, variance, and standard deviation. They will also construct and discuss box plots.

Type: Lesson Plan

Who's Better?--Using Data to Determine:

This lesson is intended for use after students are able to construct data plots (histograms, line plots, box plots). Students are tasked with not only constructing data plots, but also matching data plots to data sets. In the summative assessment, students are given two data sets and asked to select which of three data plots (histogram, line plot, or box plot) would best be used to compare the data. After choosing and constructing their plot, students are then tasked with forming a conclusion based on the plots they have constructed.

Type: Lesson Plan

Burgers to Smoothies.:

Students will create double box plots to compare nutritional data about popular food choices.

Type: Lesson Plan

Advantages and Disadvantages of Dot Plots, Histograms, and Box Plots:

Students will compare the advantages and disadvantages of dot plots, histograms, and box plots. During this lesson, students will review the statistical process and learn the characteristics of a statistical question; whether it be numerical or categorical. Students will apply the information learned in a project that involves real-world issues and make an analysis based on the data collected.

Type: Lesson Plan

Inferences:

This lesson shows students how to conduct a survey and display their results. The lesson takes the students through:

1. What is a statistical question?
2. General population versus sample population.
3. What is a hypothesis?
4. What is a survey?
5. How to make inferences.

Type: Lesson Plan

Box Plots:

An introduction lesson on creating and interpreting box plots.

Type: Lesson Plan

Using Box Plots to Interpret Data:

This lesson explores the creation of box plots to compare two data sets and draw inferences.

Type: Lesson Plan

## Original Student Tutorials

Castles, Catapults and Data: Histograms Part 1:

Learn how to create a histogram to display continuous data from projectiles launched by a catapult in this interactive tutorial.

This is part 1 in a 2-part series. Click HERE to open part 2.

Type: Original Student Tutorial

It's Raining....Cats and Dogs:

Learn how to make and interpret boxplots in this pet-themed, interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Histograms Show Trends in Fisheries Data Over Time:

NOAA Fishery management relies on histograms to show patterns and trends over time of fishery data.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Normal? Non-Normal Distributions & Oceanography:

What does it mean to be normally distributed?  What do oceanographers do when the collected data is not normally distributed?

Type: Perspectives Video: Professional/Enthusiast

Graphs Help Identify Cost-Effective Sea Turtle Conservation Strategies:

This marine biologist discusses her use of graphical representations to help determine the most cost-effective management strategies for sea turtle conservation.

Type: Perspectives Video: Professional/Enthusiast

Haircut Costs:

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Puppy Weights:

Using the information provided, create an appropriate graphical display and answer the questions regarding shape, center and variability.

## Teaching Ideas

Pump Up the Volume:

This activity is a statistical analysis of recorded measurements of a single value - in this case, a partially filled graduated cylinder.

Type: Teaching Idea

Communicating about Numbers-SeaWorld Classroom Activity:

Students communicate mathematical ideas and visually represent ideas by constructing charts, graphs, and scale drawings based on information cards about various marine animals.

Type: Teaching Idea

## Tutorials

Constructing a Box Plot:

This video demonstrates how to construct a box plot, formerly known as a box and whisker plot.

Type: Tutorial

Histograms:

Learn how to create histograms, which summarize data by sorting it into groups.

Type: Tutorial

## MFAS Formative Assessments

Students are asked to construct a histogram corresponding to a given set of data.

Shark Attack Data:

Students are asked to construct a box plot corresponding to a given set of data.

## Original Student Tutorials Mathematics - Grades 6-8

Castles, Catapults and Data: Histograms Part 1:

Learn how to create a histogram to display continuous data from projectiles launched by a catapult in this interactive tutorial.

This is part 1 in a 2-part series. Click HERE to open part 2.

It's Raining....Cats and Dogs:

Learn how to make and interpret boxplots in this pet-themed, interactive tutorial.

## Student Resources

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

## Original Student Tutorials

Castles, Catapults and Data: Histograms Part 1:

Learn how to create a histogram to display continuous data from projectiles launched by a catapult in this interactive tutorial.

This is part 1 in a 2-part series. Click HERE to open part 2.

Type: Original Student Tutorial

It's Raining....Cats and Dogs:

Learn how to make and interpret boxplots in this pet-themed, interactive tutorial.

Type: Original Student Tutorial

Haircut Costs:

This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.

Puppy Weights:

Using the information provided, create an appropriate graphical display and answer the questions regarding shape, center and variability.

## Tutorials

Constructing a Box Plot:

This video demonstrates how to construct a box plot, formerly known as a box and whisker plot.

Type: Tutorial

Histograms:

Learn how to create histograms, which summarize data by sorting it into groups.

Type: Tutorial

## Parent Resources

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