# MA.7.DP.1.1

Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, taking into consideration the context and any outliers.

### Clarifications

Clarification 1: Instruction includes recognizing whether a measure of center or measure of variation is appropriate and can be justified based on the given context or the statistical purpose.

Clarification 2: Graphical representations are limited to histograms, line plots, box plots and stem-and-leaf plots.

Clarification 3: The measure of center is limited to mean and median. The measure of variation is limited to range and interquartile range.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Box Plot
• Data
• Histogram
• Interquartile Range (IQR)
• Line Plot
• Mean
• Measures of Center
• Measures of Variability
• Median
• Outlier
• Quartiles
• Range (of data set)
• Stem-and-Leaf Plot

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students found and interpreted mean, median, mode and range, as well as determined and described how changes in data values impacted measures of center and variation. In grade 7, students determine an appropriate measure of center or measure of variation to summarize numerical data, taking into consideration the context and any outliers. Instruction builds on student knowledge from MA.6.DP.1.6. In grade 8, students will be introduced to numerical bivariate data and will depict it with line graphs and lines of fit. In high school, students will select an appropriate method to represent both univariate and bivariate numerical data, and interpret the different components in the display.
• The difference between range and interquartile range is just as important as, and very similar to, the difference between mean and median. In both cases, the difference has to do with whether or not one thinks outliers should be ignored (MTR.1.1, MTR.6.1).
• Outliers should be mostly ignored if a researcher is more interested in only the “typical” members of a population, as might be the case in politics or advertising. In these cases, the median is often the best choice as a measure of center, and the IQR is often the best choice as a measure of variation. These measures are little affected by outliers.
• Outliers should not be ignored if a researcher is concerned about the risks associated with “extreme” cases or concerned with the effects that outliers have on the average, as is the case in the insurance industry or in medical trials. In these cases, the mean is often a better choice than the median as a measure of center, and the range is better than the IQR as a measure of variation.
• All four of these measures are widely used in data analysis.
• Instruction focuses on identifying outliers qualitatively rather than quantitatively. To determine quantitatively if a data point is an outlier, a teacher may use the following definition. A data value is considered to be an outlier if it lies 1.5 times the IQR below
Q1, (Q1 − (1.5 · IQR)), or above Q3, (Q3 + (1.5 · IQR)).
• For example, the table below showcases a five-number summary of a data set.

Within the data set, the IQR is 12. To calculate if a value is an outlier, start with finding 1.5 · 12, which is 18. From there, a data value is considered an outlier if it is less than 37 − 18, or 19. It is also considered an outlier if it is greater than 49 + 18, or 67.
The maximum value in this data set is an outlier within the data set, though there could be additional values when you see the entire data set.
• Instruction includes cases where students are able to gather their own data for analysis (MTR.7.1).
• Instruction includes activities that require students to match graphs and explanations, or measures of center/variation and explanations prior to interpreting graphs based upon the appropriate measures of center or spread calculated (MTR.2.1, MTR.4.1).

### Common Misconceptions or Errors

• Some students may incorrectly calculate the measures of center and variation.
• Students may incorrectly believe all graphical displays are symmetrical. To address this misconception, students should use graphs of various shapes, including those with outliers, which will show this to be false. Start with small data sets related to familiar contexts to discuss how the data should be represented and to show how extreme values can alter the measures.
• Students may incorrectly identify data points as outliers.

### Strategies to Support Tiered Instruction

• Instruction includes opportunities for students to calculate the measures of center or the measures of variation for the initial and the changed data sets before comparing the impact of an outlier or an additional data point.
• Teacher provides examples of several visual displays or graphs to discuss the shapes of each one. Opportunities should be provided for students to see the various shapes with and without outliers so they can see that not all graphical displays are symmetrical.
• Teacher provides instruction on the definition of an outlier and interpretation on when to consider outliers (refer to the Instructional Strategies). Teacher provides examples of how outliers can be displayed within different types of data displays.
• Instruction includes co-creating a graphic organizer for measures of center and measures of variation and including the generalized impact of outliers on each.
• For example:

Instructional Task 1 (MTR.7.1)
Teacher Background Information
Unlike many elections for public office where a person is elected strictly based on the results of a popular vote (i.e., the candidate who earns the most votes in the election wins), in the United States, the election for President of the United States is determined by a process called the Electoral College. According to the National Archives, the process was established in the United States Constitution “as a compromise between election of the President by a vote in Congress and election of the President by a popular vote of qualified citizens.” (Archives -electoral college accessed July 1, 2021).
Each state receives an allocation of electoral votes in the process, and this allocation is determined by the number of members in the state’s delegation to the U.S. Congress. This number is the sum of the number of U.S. Senators that represent the state (always 2, per the Constitution) and the number of Representatives that represent the state in the U.S. House of Representatives (a number that is directly related to the state’s population of qualified citizens as determined by the US Census). Therefore the larger a state’s population of qualified citizens, the more electoral votes it has. Note: the District of Columbia (which is not a state) is granted 3 electoral votes in the process through the 23rd Amendment to the Constitution.

The following table shows the allocation of electoral votes for each state and the District of Columbia for the 2012, 2016 and 2020 presidential elections. (Archives -electoral college accessed July 1, 2021).

• Part A. Which state has the most electoral votes? How many votes does it have?
• Part B. Based on the given information, which state has the second highest population of qualified citizens?
• Part C. Here is a line plot of the distribution.

• a. What is the shape of this distribution: symmetric or skewed?
• b. Imagine that someone you are speaking with is unfamiliar with these shape terms. Describe clearly and in the context of this data set what the shape description you have chosen means in terms of the distribution.
• Part D. Does the line plot lead you to think that any states are outliers in terms of their number of electoral votes? Explain your reasoning, and if you do believe that there are outlier values, identify the corresponding states.
• Part E. What measure of center (mean or median) would you recommend for describing this data set? Why did you choose this measure?
• Part F. Determine the value of the median for this data set (electoral votes).

### Instructional Items

Instructional Item 1
The household incomes of 15 families in a local neighborhood was recorded to the nearest \$1,000 in the table below.

Determine the most appropriate measure of center to describe this data set. What is the value of that measure?

Instructional Item 2
The household incomes of 15 families in a local neighborhood was recorded to the nearest \$1,000 in the table below.

At the end of the month, a new family is moving in whose household income is \$475,000. Determine the most appropriate measure of center to describe this data set. What is the value of that measure? Justify your choice.

*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.
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 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))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.DP.1.AP.1: Use context to determine the appropriate measure of center (mean or median) or range to summarize a numerical data set with 10 or fewer elements, represented numerically or graphically.

## Related Resources

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

## Formative Assessment

Height Research:

Students are asked to describe a method for collecting data in order to estimate the average height of 12 year old boys in the U.S.

Type: Formative Assessment

## Lesson Plans

A MEANingful Discussion about Central Tendency:

Using relatable scenarios, this lesson explores the mean and median of a data set and how an outlier affects each measure differently.

Type: Lesson Plan

Students will utilize historical trade flow data (import and export) to interpret, create, and draw conclusions about foreign policy, specifically the World Trade Organization. Students will write a claim using the data to make suggestions regarding foreign trade import and export in this integrated lesson plan.

Type: Lesson Plan

Build a New School:

Students will calculate, interpret, and use measures of center and spread of different populations to determine in which city in Manatee County new schools should be built. Students will also use percentages to estimate the future population of school-aged children which will be used to determine where new schools should be built.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here: https://www.cpalms.org/cpalms/mea.aspx to learn more about MEAs and how they can transform your classroom.They resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator.

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

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

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

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

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

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

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

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

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

Water Troubles:

This Model Eliciting Activity (MEA) presents students with the real-world problem of contaminated drinking water.  Students are asked to provide recommendations for a non-profit organization working to help a small Romanian village acquire clean drinking water.  They will work to develop the best temporary strategies for water treatment, including engineering the best filtering solution using local materials.  Students will utilize measures of center and variation to compare data, assess proportional relationships to make decisions, and perform unit conversions across different measurement systems.

Type: Lesson Plan

A MEANingful Discussion about Central Tendency:

Using relatable scenarios, this lesson explores the mean and median of a data set and how an outlier affects each measure differently.

Type: Lesson Plan

## Perspectives Video: Teaching Ideas

Using Visual Models to Determine Mode, Median and Range:

Unlock an effective teaching strategy for teaching median, mode, and range in this Teacher Perspectives Video for educators.

Type: Perspectives Video: Teaching Idea

Rubber Band Races for Testing Measurement Accuracy:

<p>This activity will send your measurement lab to new distances.</p>

Type: Perspectives Video: Teaching Idea

Speed Trap:

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

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.

## 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

A Certain Uncertainty:

Students will measure the mass of one nickel 10 times on a digital scale precise to milligrams. The results will be statistically analyzed to find the error and uncertainty of the scale.

Type: Teaching Idea

All Numbers Are Not Created Equal:

Although a sheet of paper is much thinner than the divisions of a ruler, we can make indirect measurements of the paper's thickness.

Type: Teaching Idea

## STEM Lessons - Model Eliciting Activity

Water Troubles:

This Model Eliciting Activity (MEA) presents students with the real-world problem of contaminated drinking water.  Students are asked to provide recommendations for a non-profit organization working to help a small Romanian village acquire clean drinking water.  They will work to develop the best temporary strategies for water treatment, including engineering the best filtering solution using local materials.  Students will utilize measures of center and variation to compare data, assess proportional relationships to make decisions, and perform unit conversions across different measurement systems.

## MFAS Formative Assessments

Height Research:

Students are asked to describe a method for collecting data in order to estimate the average height of 12 year old boys in the U.S.

## Student Resources

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

Speed Trap:

The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.

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.

## Parent Resources

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