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Clarifications
Clarification 1: The measure of center is limited to mean and median. The measure of variation is limited to range, interquartile range, and standard deviation.Clarification 2: Shape features include symmetry or skewness and clustering.
Clarification 3: Within the Probability and Statistics course, instruction includes the use of spreadsheets and technology.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Interquartile range
- Mean
- Measures of center
- Measures of variability
- Median
- Outlier
- Range
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 7, students determined an appropriate measure of center or measure of variation to summarize numerical data. In Math for College Liberal Arts, students calculate and compare the measures of center or measures of variation of two or more sets of numerical univariate data. In other classes, students will use technology to calculate and compare two or more sets of numerical data.- The expectation of this course is to use range and interquartile range as the measures of variation.
- Instruction includes discussion of a measure of center which is a numerical value used to describe the overall clustering of data in a set, or the overall central value of a set of data. Measures of center include the mean and the median.
- Instruction includes finding the mean, or the arithmetic average. The mean is found by adding the data values and dividing by the number of values. The mean is affected by outliers.
- For example, for the data set {3,5,9,8,17,23,25,7}, the mean is calculated by dividing the sum of the values, 97, by the number of values, 8: 97/ 8=12.125.
- Instruction includes finding the median of a set of values. The median is found by finding the middle value of the data when sorted from smallest to largest. The median is not as affected by outliers.
- For example, for the sorted data set {21,23,25,28,27,35,45}, the median is the middle value: 28.
- If there is an even number of values, find the mean of the middle two values.
- For example, for the data set {3,5,9,8,17,23,25,7}, to find the median the data must first be sorted from smallest to largest: {3,5,7,8,9,17,23,25}. The middle two values are 8 and 9 so the median is 17/2=8.5.
- Instruction includes measures of variation including the range and the Interquartile Range (IQR). A greater measure of variation shows a greater spread of the data.
- The range is found by finding the difference between the highest and lowest value.
- For example, given the following 5-number summary: 25,40,60,84,100, the range is 100−25=75.
- The Interquartile Range (IQR) is the range of the middle 50% of the data and is found by subtracting IQR= Q3−Q1.
- For example, given the following 5-number summary: 25,40,60,84,100, IQR=84−40=44.
Common Misconceptions or Errors
- Students may not put the data values in ascending order when finding the median.
Instructional Tasks
Instructional Task 1 (MTR.5.1, MTR.7.1)- Route 44 and Route 65 are two popular roads in All City, USA. The highway patrol gathered the following data points to show the speeds of cars, in miles per hour, when they were involved in a car accident.
- Part A. Find the 5-number-summary for each road.
- Part B. What do you notice about the differences in speeds on the two roads?
- Part C. What conclusions might you make about the roads?
Instructional Task 2 (MTR.4.1, MTR.6.1)
- Two companies are posting job openings for entry level workers. Both companies post their average salary as $35,000. Company A has a mean of $35,000 and a median of $17,000. Company B has a mean of $35,000 and a median of $35,000. Which company would be a better company to work for and why?
Instructional Items
Instructional Item 1- Calculate and compare the medians and interquartile ranges for the number of hours per week for the two age groups.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Experts
Perspectives Video: Professional/Enthusiasts
Problem-Solving Tasks
Teaching Idea
STEM Lessons - Model Eliciting Activity
In this Model Eliciting Activity, MEA, students will analyze and use factors of various counties to recommend the top 3 to buy for a home given a client’s preferences. Students will use weighted averages, key statistics like median and mean, and correlation to conduct a thorough analysis of the data to justify their recommendations.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. 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. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will use data to decide the ideal candidate for a college scholarship by computing the mean and the standard deviation. The student will present the data using the normal distribution and make recommendations based on the findings. Students will recognize that not all data can be presented in this format.
Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.
In this Model Eliciting Activity, MEA, students will apply basic arithmetic and averages to assess and rank the home field advantages of various baseball parks. They will analyze hitting statistics and use weighted averages and composite scores to determine rankings.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. 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. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will analyze sets of data and draw conclusions to justify the top candidates for positions of President, Treasurer, and Secretary of the school's Student Government Association.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. 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. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
MFAS Formative Assessments
Students are given two histograms and are asked to describe the differences in shape, center, and spread.
Students are asked to select a measure of center to compare data displayed in dot plots and to justify their choice.
Students are asked to select a measure of center to compare data displayed in frequency tables and to justify their choice.
Students are asked to select measures of center and spread to compare data displayed in histograms and to justify their choices.
Students are given a set of data and are asked to determine how the mean is affected when an outlier is removed.
Students are asked to compare the centers of two data distributions displayed using box plots.
Students are asked to compare the spread of two data distributions displayed using box plots.
Original Student Tutorials Mathematics - Grades 9-12
Follow Jake as he displays real-world data by creating box plots showing the 5 number summary and compares the spread of the data from surveys of the ages of moviegoers in part 1 of this interactive tutorial.
This is part 1 of 2-part series, click HERE to view part 2.
Follow Jake along as he relates box plots with other plots and identifies possible outliers in real-world data from surveys of moviegoers' ages in part 2 in this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1.
Student Resources
Original Student Tutorials
Follow Jake along as he relates box plots with other plots and identifies possible outliers in real-world data from surveys of moviegoers' ages in part 2 in this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1.
Type: Original Student Tutorial
Follow Jake as he displays real-world data by creating box plots showing the 5 number summary and compares the spread of the data from surveys of the ages of moviegoers in part 1 of this interactive tutorial.
This is part 1 of 2-part series, click HERE to view part 2.
Type: Original Student Tutorial
Perspectives Video: Expert
The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
