MA.912.DP.2.7

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Compute the correlation coefficient of a linear model using technology. Interpret the strength and direction of the correlation coefficient.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Data Analysis and Probability
Standard: Solve problems involving univariate and bivariate numerical data.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Association 
  • Bivariate data 
  • Line of fit 
  • Scatter plot
 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 8, students constructed scatter plots and informally fit a straight line into scatter plots with linear associations. Association in a scatter plot was described as positive/negative, linear/nonlinear and strong/weak. In Mathematics for College Statistics, students continue to describe the correlation, or association, among the numerical bivariate data using the correlation coefficient to assist in determining direction and the size of the residuals to aid in describing the strength. 
  • In MA.912.DP.2.6, students use the scatter plot, residuals and slope to describe the strength and trend of the linear model. This benchmark expands on those ideas using the correlation coefficient. 
  • Instruction relies on using technology to calculate the correlation coefficient. Using a formula can be quite tedious, and may rely upon calculations that students are not familiar with at this point in their learning. 
  • This benchmark is only meant to address linear models. 
  • Instruction emphasizes the correlation coefficient, or r, is always between −1 and 1. A perfect positive linear association has a correlation coefficient of 1; a perfect negative linear association has correlation coefficient of −1. Weaker relationships have correlations that are closer to 0. 
  • Instruction emphasizes that the sign of the correlation coefficient corresponds to the direction/trend of the scatter plot and the sign of the slope. It should also be noted that the sign of the correlation coefficient does not affect the strength of the correlation. 
  • While there are not an agreed upon set of boundaries when classifying the strength of a correlation, it may be helpful to set up guidelines for students. 
    • For example, a statistics course may generally teach that if |r| ≥ .80 then the relationship is strong, if .50 ≤ |r| ≤ .79 then the relationship is moderately strong, and if |r| ≤ .49 then the relationship is weak. 
  • Instruction includes describing the direction of the correlation as positive or negative and the strength of the correlation as strong, possibly moderate or weak. It should be mentioned that the formal name of this computation is Pearson’s correlation coefficient, and it is used to measure the strength and direction of linear models. However, there are other calculations that exist for measuring the strength of nonlinear models.
 

Common Misconceptions or Errors

  • Students may incorrectly assume that the sign of the correlation has an effect on the strength. 
  • Students may have issues comparing the decimal values of correlation coefficients. 
  • Students may have trouble rounding decimals or rounding to the correct decimal place.
 

Instructional Tasks

Instructional Task 1 (MTR.6.1, MTR.7.1) 
  • The strongest hurricanes from 2010 to 2019 were studied, and the central pressure (mb) and maximum wind speed (kt) were recorded and are given in the table below. 
    • Part A. Use technology to create a scatter plot, and use the graph to estimate the correlation coefficient. 
    • Part B. Use technology to calculate the correlation coefficient. 
    • Part C. Describe the direction of the correlation coefficient. 
    • Part D. Describe the strength of the correlation coefficient. 
    • Part E. How does your estimate in Part A compare to the calculation in Part B? 
    • Part F. Use technology to find the line of best fit. Do you feel the linear model is appropriate based on the graph and the correlation coefficient? Justify your reasoning.
 

Instructional Items

Instructional Item 1 
  • Match the correlation coefficient with the appropriate scatter plot.  
    • r = -1
    • r = .94
    • r = -.75 
    • r = .11 
                                                                                                                                                   

                                                                                      Scatter plot A                         Scatter plot B 

                                                                                                                                          

                                                                                                                                 

                                                                                     Scatter plot C                         Scatter plot D 

                                                                                                                                
Instructional Item 2 
  • Use technology to calculate the correlation coefficient of the data set. Interpret the direction and strength.


 

*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.
1210300: Probability and Statistics Honors (Specifically in versions: 2014 - 2015, 2015 - 2019, 2019 - 2022, 2022 - 2024, 2024 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 - 2024, 2024 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

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

Formative Assessments

July December Correlation:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

Type: Formative Assessment

How Big Are Feet?:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

Type: Formative Assessment

Correlation Order:

Students are asked to estimate a correlation coefficient for each of four data sets and then order the coefficients from least to greatest in terms of the strength of relationship.

Type: Formative Assessment

Correlation for Life Expectancy:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

Type: Formative Assessment

Lesson Plans

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Spreading the Vote Part 3:

Students will explore voter turnout data for three gubernatorial elections before and after the passage of the 19th amendment. They will fit linear functions to the data and compute predicted values for raw and percentage of voter turnout. Students will draw some conclusions concerning the relationship between eligible voters and voter turnout, including possible causes behind the fluctuation in voter participation in this integrated lesson

Type: Lesson Plan

Compacting Cardboard:

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

A Day at the Park:

In this activity, students investigate a set of bivariate data to determine if there is a relationship between concession sales in the park and temperature. Students will construct a scatter plot, model the relationship with a linear function, write the equation of the function, and use it to make predictions about values of variables.

Type: Lesson Plan

Basketball - it's a tall man's sport - or is it?:

The students will use NBA player data to determine if there is a correlation between the height of a basketball player and his free throw percentage. The students will use technology to create scatter plots, find the regression line and calculate the correlation coefficient.

Basketball is a tall man's sport in most regards. Shooting, rebounding, blocking shots - the taller player seems to have the advantage. But is that still true when shooting free throws?

Type: Lesson Plan

Heart Rate and Exercise: Is there a correlation?:

Students will use supplied heart rate data to determine if heart rate and the amount of time spent exercising each week are correlated. Students will use GeoGebra to create scatter plots and lines of fit for the data and examine the correlation. Students will gather evidence to support or refute statistical statements made about correlation. The lesson provides easy to follow steps for using GeoGebra, a free online application, to generate a correlation coefficient for two given variables.

Type: Lesson Plan

Span the Distance Glider - Correlation Coefficient:

This lesson will provide students with an opportunity to collect and analyze bivariate data and use technology to create scatter plots, lines of best fit, and determine the correlation strength of the data being compared. Students will have a hands on inquire based lesson that allows them to create gliders to analyze data. This lesson is an application of skills acquired in a bivariate unit of study.

Type: Lesson Plan

Study of Crowd Ratings at Disney:

In this lesson, students develop a strong use of the vocabulary of correlation by investigating crowd ratings at Disney. Students will determine weekly crowd rating regression lines and correlations and discuss what this means for a Disney visit.

Type: Lesson Plan

Hand Me Your Data:

Students will gather and use data to calculate a line of fit and the correlation coefficient with their classmates' height and hand size. They will use their line of fit to make approximations.

Type: Lesson Plan

Calculating Residuals and Constructing a Residual Plot with Soccer Seats:

Students will learn all about residuals. The definition, how to calculate them, how to plot and analyze residuals, and how to use them to assess the fit of a linear function. They will do this within the context of comparing the location of a seat in a soccer stadium with its price.

Type: Lesson Plan

Why do I have to have a bedtime?:

This predict, observe, explain lesson that allows students to make predictions based on prior knowledge, observations, discussions, and calculations. Students will receive the opportunity to express themselves and their ideas while explaining what they learned. Students will make a prediction, collect data, and construct a scatter plot. Next, students will calculate the correlation coefficient and use it to describe the strength and magnitude of a relationship.

Type: Lesson Plan

What's So Funny About Correlation?:

Students investigate correlation and causation through the medium of cartoons. Students construct arguments in favor of and against causal relationships between two strongly correlated events and decide which one is more reasonable. Students create cartoons representing the idea that correlation does not imply causation.

Type: Lesson Plan

Scatter Plots and Correlations:

Students create scatter plots, and lines of fit, and then calculate the correlation coefficient. Students analyze the results and make predictions. This lesson includes step-by-step directions for calculating the correlation coefficient using Excel, GeoGebra, and a TI-84 Plus graphing calculator. Students will make predictions for the number of views of a video for any given number of weeks on the charts.

Type: Lesson Plan

Scrambled Coefficient:

Students will learn how the correlation coefficient is used to determine the strength of relationships among real data. Students use card sorting to order situations from negative to positive correlations. Students will create a scatter plot and use technology to calculate the line of fit and the correlation coefficient. Students will make a prediction and then use the line of fit and the correlation coefficient to confirm or deny their prediction.

Students will learn how to use the Linear Regression feature of a graphing calculator to determine the line of fit and the correlation coefficient.

The lesson includes the guided card sorting task, a formative assessment, and a summative assessment.

Type: Lesson Plan

How technology can make my life easier when graphing:

Students will use GeoGebra software to explore the concept of correlation coefficient in graphical images of scatter plots. They will also learn about numerical and qualitative aspects of the correlation coefficient, and then do a matching activity to connect all these representations of the correlation coefficient. They will use an interactive program file in GeoGebra to manipulate the points to create a certain correlation coefficient. Step-by-step instructions are included to create the graph in GeoGebra and calculate the r correlation coefficient.

Type: Lesson Plan

Smarter than a Statistician: Correlations and Causation in the Real World!:

Students will learn to distinguish between correlation and causation. They will build their skills by playing two interactive digital games that are included in the lesson. The lesson culminates with a research project that requires students to find and explain the correlation between two real world events.

Type: Lesson Plan

House Hunting!:

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

Type: Lesson Plan

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Perspectives Video: Professional/Enthusiast

Determining Strengths of Shark Models based on Scatterplots and Regression:

Chip Cotton, fishery biologist, discusses his use of mathematical regression modeling and how well the data fits his models based on his deep sea shark research.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Task

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Type: Problem-Solving Task

STEM Lessons - Model Eliciting Activity

House Hunting!:

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

MFAS Formative Assessments

Correlation for Life Expectancy:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

Correlation Order:

Students are asked to estimate a correlation coefficient for each of four data sets and then order the coefficients from least to greatest in terms of the strength of relationship.

How Big Are Feet?:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

July December Correlation:

Students are asked to compute and interpret the correlation coefficient for a given set of data.

Student Resources

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

Problem-Solving Task

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Type: Problem-Solving Task

Parent Resources

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

Problem-Solving Task

Coffee and Crime:

This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.

Type: Problem-Solving Task