# MA.912.DP.2.1 Export Print
For two or more sets of numerical univariate data, calculate and compare the appropriate measures of center and measures of variability, accounting for possible effects of outliers. Interpret any notable features of the shape of the data distribution.

### 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.

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

• 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. o 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= Q3Q1
• 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.

• 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?

• 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. *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.
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))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.DP.2.AP.1: For two sets of numerical univariate data, calculate and compare the mean, median and range, then select the shape of the data from given graphs.

## Related Resources

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

## Formative Assessments

How Many Jeans?:

Students are asked to select a measure of center to compare data displayed in dot plots and to justify their choice.

Type: Formative Assessment

Texting During Lunch Histograms:

Students are asked to select measures of center and spread to compare data displayed in histograms and to justify their choices.

Type: Formative Assessment

Texting During Lunch:

Students are asked to select a measure of center to compare data displayed in frequency tables and to justify their choice.

Type: Formative Assessment

Total Points Scored:

Students are given a set of data and are asked to determine how the mean is affected when an outlier is removed.

Type: Formative Assessment

Using Spread to Compare Tree Heights:

Students are asked to compare the spread of two data distributions displayed using box plots.

Type: Formative Assessment

Using Centers to Compare Tree Heights:

Students are asked to compare the centers of two data distributions displayed using box plots.

Type: Formative Assessment

Comparing Distributions:

Students are given two histograms and are asked to describe the differences in shape, center, and spread.

Type: Formative Assessment

## Original Student Tutorials

Movies Part 2: What’s the Spread?:

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

Movies Part 1: What's the Spread?:

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: Experts

Tree Rings Research to Inform Land Management Practices:

In this video, fire ecologist Monica Rother describes tree ring research and applications for land management.

Type: Perspectives Video: Expert

Birdsong Series: Statistical Analysis of Birdsong:

Wei Wu discusses his statistical contributions to the Birdsong project which help to quantify the differences in the changes of the zebra finch's song.

Type: Perspectives Video: Expert

Birdsong Series: STEM Team Collaboration :

Researchers Frank Johnson, Richard Bertram, Wei Wu, and Rick Hyson explore the necessity of scientific and mathematical collaboration in modern neuroscience, as it relates to their NSF research on birdsong.

Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

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?

Type: Perspectives Video: Expert

Statistics and Scientific Data:

Hear this oceanography student float some ideas about how statistics are used in research.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Nestle Waters & Statistical Analysis:

Hydrogeologist from Nestle Waters discusses the importance of statistical tests in monitoring sustainability and in maintaining consistent water quality in bottled water.

Type: Perspectives Video: Professional/Enthusiast

Statistical Art: Four Words:

Graphic designer and artist, Drexston Redway infuses statistics into his artwork to show population distribution and overlap of poverty and ethnicity in Tallahassee, FL.

Type: Perspectives Video: Professional/Enthusiast

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

Sampling Amphibian Populations to Study Human Impact on Wetlands:

Ecologist Rebecca Means discusses the use of statistical sampling and comparative studies in field biology.

Type: Perspectives Video: Professional/Enthusiast

Winning the Race with Data Logging and Statistics:

Data logging has transformed competitive racing! These SCCA drivers discuss how they use computers to compare multiple sets of data after test runs.

Type: Perspectives Video: Professional/Enthusiast

Revolutionize Wing Design with Equations and Statistics:

Brandon Reese, a PhD candidate in the FAMU-FSU College of Engineering, discusses the significance of both Bernoulli's equation and statistical analysis for the design of a "smart wing."

Type: Perspectives Video: Professional/Enthusiast

## Teaching Idea

An Ecological Field Study with Statistical Analysis of Two Populations:

Students will design an investigation that compares a characteristic of two populations of the same species. Students will collect data in the field and analyze the data using descriptive statistics.

Type: Teaching Idea

## MFAS Formative Assessments

Comparing Distributions:

Students are given two histograms and are asked to describe the differences in shape, center, and spread.

How Many Jeans?:

Students are asked to select a measure of center to compare data displayed in dot plots and to justify their choice.

Texting During Lunch:

Students are asked to select a measure of center to compare data displayed in frequency tables and to justify their choice.

Texting During Lunch Histograms:

Students are asked to select measures of center and spread to compare data displayed in histograms and to justify their choices.

Total Points Scored:

Students are given a set of data and are asked to determine how the mean is affected when an outlier is removed.

Using Centers to Compare Tree Heights:

Students are asked to compare the centers of two data distributions displayed using box plots.

Using Spread to Compare Tree Heights:

Students are asked to compare the spread of two data distributions displayed using box plots.

## Original Student Tutorials Mathematics - Grades 9-12

Movies Part 1: What's the Spread?:

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.

Movies Part 2: What’s the Spread?:

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

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

## Original Student Tutorials

Movies Part 2: What’s the Spread?:

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

Movies Part 1: What's the Spread?:

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

Mathematically Exploring the Wakulla Caves:

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?