# Common Core High School Statistics & Probability Teaching and Learning Resources

## Domain: ID: Interpreting Categorical and Quantitative Data

• ID-A: Summarize, represent, and interpret data on a single count or measurement variable

• ID-B: Summarize, represent, and interpret data on two categorical and quantitative variables

• ID-C: Interpret linear models

### ID-A: Summarize, represent, and interpret data on a single count or measurement variable

Standards:

• ID-A.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
• ID-A.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
• ID-A.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
• ID-A.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary.  Note: Terms followed by * are defined at MathsIsFun.com.  Also see the Stattrek.com Statistics Dictionary for terms followed by **.  Key vocabulary for this domain.  Use with ID-A, ID-B, ID-C:

 bivariate and univariate data* histogram* linear fit regression equation box plot mean normal distribution* scatterplot box-and-whisker plot median outlier standard deviation* correlation coefficient mode quartiles slope dot plot* interquartile range first quartile stemplot frequency table** 2-way table** third quartile joint frequency** marginal frequency** conditional frequency** relative frequency**

Academo.org: Standard Deviation Calculator.  After entering the numbers, this tool shows the number of numbers, the mean, variance, and standard deviation.  A brief explanation of the standard deviation formula is included.  Aligns with ID-A.2 and ID-A.4.

Alcula.com:

BBC GCSE Bitesize Maths, Statistics and Probability: Averages: Mean, Mode, and Median --There is an explanation (called Revise), activity, and test.

Calculator.net: The following also includes explanations of theory for each calculator

CalculatorSoup: Descriptive Statistics Calculator.  A complete calculator showing min, max, range, mean, median, mode, standard deviation, variance, quartiles, interquartiles, outliers, and more.  A frequency table is also shown.  Aligns with ID-A.2.

College Preparatory Math: Student Tutorials: TI-84 Graphing Calculator: Univariate Data: Videos and screen shots illustrating use of the calculator to set up histograms, box plots, comparing two box plots, and summarizing data numerically.  Use with ID-A.1.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  Flaws of Averages: "This learning video presents an introduction to the Flaws of Averages using three exciting examples: the “crossing of the river” example, the “cookie” example, and the “dance class” example. Averages are often worthwhile representations of a set of data by a single descriptive number. The objective of this module, however, is to simply point out a few pitfalls that could arise if one is not attentive to details when calculating and interpreting averages."  Aligns with ID-A.2.

National Library of Virtual Manipulatives: Box Plots and Histograms (Java required). Use with ID-A.1.

National Center for Education Statistics: Create a graph

Ohio Resource Center on YouTube: Tutorials for High School Mathematics.  Select the following:

OnlineStatBook: Graphing Distributions is a chapter that addresses stem and leaf displays, histograms, frequency polygons, box plots, dot plots, bar charts, line graphs, and more.  Exercises are included.  The project to develop the book was led by David Lane at Rice University.

PhET Interactive Simulations: Plinko Probability. Per its description: "Drop balls through a triangular grid of pegs and see them accumulate in containers. Switch to a histogram view and compare the distribution of balls to an ideal binomial distribution. Adjust the binary probability and develop your knowledge of statistics!"  Aligns with ID-A.1, ID-A.2, and IC-A.2.

Purple Math:

Shodor Interactivate: Statistics and Probability: Interpreting Categorical and Quantitative Data: Summarize, represent, and interpret data on a single count or measurement variable.  A series of nine lessons and 14 activities with virtual manipulatives to investigate concepts such as box plots, histograms, stem-and-leaf plots, measures of center and spread, the Bell curve, univariate and bivariate data, normal distributions, skewed distributions.  Aligns with ID-A.

Stat Trek: Tutorials:

Statistics Canada: Statistics, Power from Data:

Thinking Mathematics: Standard Deviation Formula Explained: A YouTube video.  Aligns with ID-A.2 and ID-A.4.

University of Illinois at Urbana-Champaign, Jay Hill: Introduction to Descriptive Statistics--mean, median, mode, central tendency, variation, range, variance, and standard deviation, simply explained.

U.S. Census Bureau:

• Commuting to Work: Box Plots, Central Tendency, and Outliers.  Per its description: "Students will calculate various measures of central tendency using data on the number of people who bike to work in select states. Students will then create a box plot to represent the data set and answer conceptual questions about the impact of the data set’s outlier."  Aligns with ID-A.1, ID-A.3, and math practice standards MP2 (reason abstractly and quantitavely), MP4 (model with math) and MP6 (attend to precision).
• Describing and Comparing Data Distributions.  Per its description: "Students will use data on the organization, spending, and populations of governments at different levels (city or town, county, and state) to compare and contrast the distributions of these variables in graphs, analyzing the shape, center, and spread of each."  Aligns with ID-A.2, ID-A.3, and math practice standards MP2 (reason abstractly and quantitavely) and MP6 (attend to precision).
• Over the Hill: Aging on a Normal Curve.  Per its description: "Students will use census data from a sample of 136 U.S. counties and other sample data to make estimates about the U.S. population that is 65 or older in all other counties and about other variables, using normal distribution models."  Aligns with ID-A.4 and math practice standard MP2 (reason abstractly and quantitavely).

Teaching Channel Video: Statistical Analysis to Rank Baseball Players:  This video's lesson objective: Rank the greatest NY Yankee homerun hitters using statistical analysis.  Questions for learners to consider are included.  Use to address ID-A.1, ID-A.2, and ID-A.3.

Online Statistics Calculators:  There are five calculators available for measures of central tendency and dispersion, box and whisker plots, linear regression, correlation coefficients, and scatter-plots.  Aligns with ID-A.1 and ID-A.2.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 14 manipulatives addressing ID-A.1, 5 manipulatives for ID-A.2, 9 manipulatives for ID-A.3, 12 manipulatives for ID-A.4.  Among those:

YouTube video: GCSE Maths Median and IQR: This short video shows how to find a median, lower quartile, upper quartile, then inter-quartile range.

Multiple Choice:

MathBitsNotebook: Algebra 1: Statistics: Data Distributions includes lessons, then practice problems on categorizing data, box plots, outliers, measures of center and shapes of distributions, standard deviation, and interpreting graphs.  Use with ID-A.1, ID-A.2, and ID-A.3.

MathsIsFun.com:

• Quartiles: How to calculate quartiles and interquartile range and their association to a box and whisker plot.  Following the explanation are ten multiple choice exercises.  Use with ID-A.1 and ID-A.2.
• Outliers: Contains a concise explanation of the role outliers play in data analysis involving mean, median, and mode.  Following the explanation are eight multiple choice exercises.  Use with ID-A.3.
• Standard deviation and variance:  After explanation and examples, there are 10 multiple choice exercises.  Use with ID-A.4.
• Normal Distribution: Contains a concise explanation of the normal distributions, standard deviation (how to compute it, and accompanied by a standard deviation calculator), and z-scores with visuals and worked examples.  Following the explanation are ten multiple choice exercises.  Use with ID-A.4.
• Standard Normal Distribution Table: includes a manipulative  to work with z-scores on a normal distribution and a table of values associated with percent of population with worked example.  Following the explanation are ten multiple choice exercises.  Use with ID-A.4.

Constructed-response:

AlgebraLab.org: Mean, Median, Mode.  Lesson, interactive online practice problems.  Show the Related AlgebraLab documents for activities, additional practice problems and word problems.  Aligns with ID-A.2.

Khan Academy: HSS-S-ID includes HSS-ID.A.1 through HSS-ID.A.4 multiple choice and constructed-response Common Core aligned problems.

Also see the Analyzing Categorical Data unit with practice questions with videos.

Statistics: Power from Data:

• Ch. 9: Graph Types: Explanations of graph types.  A tool is presented for learners to create graphs.  Practice exercises are included.
• Ch. 11: Measures of Central Tendency: Explanations on mean, median, mode with exercise problems and answers.
• Ch. 12: Measures of Spread: Explanations on range and quartiles, variance and standard deviation, box and whisker plots with exercise problems and answers.

Zweigmedia.com:  Tutorials and examples using fill-in or multiple choice to test understanding.  Learners can also choose to do game versions of some topics.  Use with ID-A:

Illustrative Mathematics: Statistics and Probability:

Mathematics Assessment Project: Standards: High School: Statistics & Probability:

Mathematics Vision Project, Secondary 1 Student Edition:

• Module 8: Modeling Data: This module contains 8 classroom tasks.  Module 7 addresses standards ID-A.1, ID-A.2, ID-A.3 and ID-B.5, ID-B.6 and ID-C.7, ID-C.8.  Task 1: Texting by the Numbers; and Task 2: Data Distributions align to standards ID-A.1, ID-A.2, and ID-A.3.

Mathematics Vision Project, Secondary 3 Student Edition:

• Module 8: Statistics: This module contains 8 classroom tasks.  Tasks 1-4 align with standard ID-A.4.

NCTM's Reasoning and Sense Making Task Library: Eruptions: Old Faithful includes the task overview, teacher notes for its use, and student activity sheet.  Aligns with IC-A.1, ID-A.1, and mathematical practice standards 1, 3, and 5.

Statistics in Schools from the U.S. Census Bureau: Activities: Math: 9-12:

• Commuting to Work: Box Plots, Central Tendency, Outliers.  Students "calculate various measures of central tendency using data on the number of people who bike to work in select states. Students will then create a box plot to represent the data set and answer conceptual questions about the impact of the data set’s outlier."  Aligns with ID-A.1.
• Differences in Earnings Across Sex and Educational Attainment: Comparing Box Plots.  Students "interpret box plots that represent the national median earnings of men and women aged 25 and older whose highest levels of educational attainment are either a high school diploma (or equivalent) or a bachelor’s degree. Students will use the box plots to identify each data set’s median, maximum, minimum, first quartile, third quartile, range, interquartile range, and outliers. They will also compare the box plots to draw conclusions about differences in earnings between the sexes and between levels of educational attainment."  Aligns with ID-A.1, ID-A.2, ID-A.3.
• Describing and Comparing Data Distributions.  Students "use data on the organization, spending, and populations of governments at different levels (city or town, county, and state) to compare and contrast the distributions of these variables in graphs [box plots and histograms], analyzing the shape, center, and spread of each."  Aligns with ID-A.2, ID-A.3.
• Census in Countries: Describing and Comparing Histograms to Understand American Life.  Students "analyze a variety of county-level census data, including on employment, technology, and transportation, in histograms to compare and contrast the shapes of their distributions and to interpret measures of center and spread in context."  Aligns with ID-A.2, ID-A.3.
• The New Normal.  Students "explore distributions of various census data sets to determine whether it can be reasonably assumed that those data follow a normal distribution, based on students’ analysis of either a histogram or a normal probability plot for each data set. They will then discuss their findings with a partner who analyzed the other type of graph for each data set."  Aligns with ID-A.3.
• Over the Hill: Aging on a Normal Curve.  Students "use census data from a sample of 136 U.S. counties and other sample data to make estimates about the U.S. population that is 65 or older in all other counties and about other variables, using normal distribution models."  Aligns with ID-A.4.

### ID-B: Summarize, represent, and interpret data on two categorical and quantitative variables

Standards:

• ID-B.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
• ID-B.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
• ID-B.6.a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
• ID-B.6.b. Informally assess the fit of a function by plotting and analyzing residuals.
• ID-B.6.c. Fit a linear function for a scatter plot that suggests a linear association.

Notes:

Line graphs and histograms are only used with continuous data, which means that theoretically all values are possible (no gaps) in an interval (between any two it is possible to get another).  Examples include height, weight, time to complete your homework, and time to complete a trip.  Line graphs are good to demonstrate trends, such as sales in a company over time or changes in temperature over time.

Bar graphs and circle graphs (often called pie charts) are only used with discrete data, which means that only certain values (gaps) are possible. Examples typically include counting, as in the number of students in a class, the number of crimes reported to the police, or the number of tickets sold to a game.  Note that pie charts are good for presenting percentages.  Bar graphs can be presented vertically or horizontally.  For example, a horizontal bar graph could be used to compare expenses among four departments of a company.

Scatter plots are used for experimental data to determine if there is a relationship between the variables studied.

When creating graphs, note the following 10 common errors, provided by CanTeach (CA):

1. No title on graph
2. Source of data not given
3. Pictograph - no key
4. Scales are interrupted
5. Scales are not labeled
6. Symbols in pictographs not same size or equally spaced on graph
7. Use of 2 or 3 dimensional objects to compare data (area/volume)
8. Scales do not start at zero
9. Numbers on axis (vertical or horizontal) not equally spaced
10. Scale is selected to produce desired result

Technology-enhanced investigations:

Alcula.com:

CK-12: Algebra: Concepts, video, and practice problems for learners.

College Preparatory Math: Student Tutorials: TI-84 Graphing Calculator: Bivariate Data: Videos and screen shots illustrating use of the calculator for setting up a scatter plot, non-linear regressions, least squares regression line, the correlation coefficient, residuals & residual plots.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled."  Additional simulations are included.  Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.

National Library of Virtual Manipulatives: Scatterplots (Java required).

NCTM Illuminations:

Ohio Resource Center on YouTube: Tutorials for High School Mathematics: Lines of Fit: Defining and finding lines of fit using real data.

PhET Interactive Simulations:

Saltire Software: Common Core Nuggets: There are five applets addressing residual plots and least squares, which align with ID-B.6.

Shodor Interactivate:

• Linear Regression and Correlation: Lesson introducing correlations between two variables and line of best fit.
• Univariate and Bivariate Data: Lesson introducing the difference between these two types and how to determine the best graph to use to display the data.
• Regression: A virtual manipulative to plot a bivariate data set, determine the line of best fit for the data, and then check the accuracy of a line of best fit.

Stat-Trek: Tutorials:

Thinking Mathematics: Line of Best Fit: Least Squares Method: Video with explanations on why we calculate the line of best fit the way we do, and the mathematics behind the formula for the best line fitting data on a scatter plot.

U.S. Census Bureau: Applying Correlation Coefficients - Educational Attainment and Unemployment.  Per its description: "Students will use state and regional unemployment data for various education levels to create scatter plots and calculate correlation coefficients [using technology and by hand]. Students will then compare scatter plots with different strengths of linear relationships and will determine the impact of any influential points on the correlation coefficient."  Aligns with ID-B.6, ID-C.8, and math practice standards MP4 (model with math) and MP6 (attend to precision).

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 1 manipulative addressing ID-B.5, 20 manipulatives for ID-B.6.  Among those:

Multiple Choice:

MathsIsFun.com: Scatter Plots: Contains a concise explanation of the scatter plot and its relation to line of fit and correlation.  Following the explanation are nine multiple choice exercises.  Use with ID-B.6.

Khan Academy: Exploring Bivariate Numerical Data unit.  Practice questions with videos.

MathBitsNotebook: Algebra 1: Statistics: Bivariate Data includes lessons, then practice problems on two-way frequency tables, fitting functions to data, residuals, linear regression, correlation and correlation coefficients, slopes and intercepts of linear models.  Use with ID-B.5, ID-B.6, ID-C.7, ID-C.8, and ID-C.9.

Constructed-response:

Khan Academy: HSS-S-ID includes HSS-ID.B.5 and HSS-ID.B.6 multiple choice and constructed-response Common Core aligned problems.

Also see the Summarizing quantitative data unit with practice questions with videos.

The Math You Need, When You Need It: Constructing a line of best fit includes tutorials followed by practice problems focusing on real-life scenarios on this topic that geoscientists would encounter.  The site is by SERC, the Science Education Resource Center at Carleton College.

Zweigmedia.com: Tutorials and examples using fill-in to test understanding.  Learners can also choose to do game versions of some topics.  Use with ID-B.6: Linear regression

Illustrative Mathematics: Statistics and Probability:

Inside Mathematics: MARS Tasks: The following align with ID-B.6:

• Population: Students work with a scatter plot to identify specific information on it, describe main features of a scatter plot and make sense of trends in order to graph a line to represent average density and calculate density relationships in the given situation.
• Snakes: Students work with two scatter plots to make sense of data displayed.  They make sense of a table and look for trends including correlations and lines of best fit, and they make inferences based on data and conclusions about a situation being modeled.

Mathematics Assessment Project: Standards: High School: Statistics & Probability:

Mathematics Vision Project, Secondary 1 Student Edition: Module 8: Modeling Data: This module contains eight classroom tasks.  Module 8 addresses standards ID-A.1, ID-A.2, ID-A.3 and ID-B.5, ID-B.6 and ID-C.7, ID-C.8.  Task 3: After School Activity; and Task 4: Relative Frequency align to standard ID-B.5.  Task 7: Getting Schooled; and Task 8: Rocking the Residuals align to standard ID-B.6.

Statistics in Schools from the U.S. Census Bureau: Activities: Math: 9-12:

• Linear Models – Analyzing Relationships: Marriage, Divorce, and Linear Regression.  Students "create a scatter plot, find a line of best fit, and analyze the relationship between the two variables (i.e., sex and marriage/ divorce rates). They will also use a residual plot, explain the meaning of the slope and of the y-intercept of the line of best fit, and investigate the effect of outliers on this line."  Aligns with ID-B.6, ID-C.7.
• Applying Correlation Coefficients: Educational Attainment and Unemployment.  Students "use state and regional unemployment data for various education levels to create scatter plots and calculate correlation coefficients. Students will then compare scatter plots with different strengths of linear relationships and will determine the impact of any influential points on the correlation coefficient.  Aligns with ID-B.6, ID-C.8.

### ID-C: Interpret linear models

Standards:

• ID-C.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
• ID-C.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
• ID-C.9. Distinguish between correlation and causation.

Technology-enhanced investigations:

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

• Lesson set: Interpret the slope and intercept of a linear function in context: Three video lessons: Distinguish between scatterplots and lines; interpret intercepts and slope using a line of best fit.  Aligns with ID-C.7.
• Lesson set: Interpret the slope and the intercept of a linear model using data: Four video lessons: Interpret the parts of a linear model, find the slope of a linear function by using two points, find the slope and y-intercept of a line by using the slope formula and analyzing linear models, solve real-world problems using slope and y-intercept, determine type of slope by analyzing linear function word problems.  Aligns with ID-C.7.
• Lesson set: Find correlation coefficient of a linear fit: Three video lessons: Understand correlation in terms of the strength of a relationship, find the correlation coefficient by using technology, understand and interpret the slope of a regression line.  Aligns with ID-C.8.
• Lesson set: Compute and interpret the correlation coefficient of a linear fit: Four video lessons: Use the correlation coefficient to assess the strength of a linear fit, assess the appropriateness of a linear correlation model, calculate the correlation coefficient by using a graphing calculator, solve problems using linear regression.  Aligns with ID-C.8.
• Lesson set: Distinguish between correlation and causation: Three video lessons: Understand causation; distinguishing correlation and causation using the reverse causation example and by examining a common causation example.  Aligns with ID-C.9.
• Lesson set: Distinguish between correlation and causation and assess causation: Three video lessons: Differentiate between correlation and causation, evaluate language that confuses correlation and causation, establish causation through experimental design.  Aligns with ID-C.9.

PhET Interactive Simulations: Least-Squares Regression.  Aligns with ID-B.6, ID-B.6.a, ID-B.6.b, ID-B.6.c, and ID-C.8.

Stat Trek: Tutorial: Linear Correlation Coefficient: The tutorial includes a video, and explanation on how to interpret and calculate the correlation coefficient.   Use with ID-C.8.

Social Science Statistics: Pearson Correlation Coefficient Calculator.  Free online calculator.  Use with ID-C.8.

College Preparatory Math: Student Tutorials: TI-84 Graphing Calculator: Bivariate Data: Videos and screen shots illustrating use of the calculator for setting up a scatter plot, non-linear regressions, least squares regression line, the correlation coefficient, residuals & residual plots.

U.S. Census Bureau: Applying Correlation Coefficients - Educational Attainment and Unemployment.  Per its description: "Students will use state and regional unemployment data for various education levels to create scatter plots and calculate correlation coefficients [using technology and by hand]. Students will then compare scatter plots with different strengths of linear relationships and will determine the impact of any influential points on the correlation coefficient."  Aligns with ID-B.6, ID-C.8, and math practice standards MP4 (model with math) and MP6 (attend to precision).

Wolfram Alpha: Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 3 manipulatives for ID-C.8.  Among those:

• Correlation and regression explorer: Aligns with ID-B.6, ID-C.8.
• Anscombe Quartet: The Anscombe Quartet is comprised of four scatterplots that have nearly identical correlations, as well as means and standard deviations, but disparate shapes. These graphs show the crucial role that data visualization plays in developing a sensible statistical model.  Aligns with ID-C.8.

Multiple Choice:

Khan Academy: HSS-S-ID includes HSS-ID.C.7 through HSS-ID.C.9 Common Core aligned problems.

Also see the Exploring Bivariate Numerical Data unit with practice questions with videos.

MathBitsNotebook: Algebra 1: Statistics: Bivariate Data includes lessons, then practice problems on two-way frequency tables, fitting functions to data, residuals, linear regression, correlation and correlation coefficients, slopes and intercepts of linear models.  Use with ID-B.5, ID-B.6, ID-C.7, ID-C.8, and ID-C.9.

MathsIsFun.com: Correlation: Contains a concise explanation of the meaning of correlation, correlation does not mean causation, and how to calculate the correlation coefficient.  Worked examples are provided.  Following the explanation are four multiple choice exercises.  Use with ID-C.8 and ID-C.9.

Constructed-response:

Illustrative Mathematics: Statistics and Probability:

Mathematics Assessment Project: Standards: High School: Statistics & Probability: Task 217: Interpreting Statistics: A Case of Muddying the Waters

Mathematics Vision Project, Secondary 1 Student Edition: Module 8: Modeling Data: This module contains eight classroom tasks.  Module 8 addresses standards ID-A.1, ID-A.2, ID-A.3 and ID-B.5, ID-B.6 and ID-C.7, ID-C.8.  Task 5: Connect the Dots aligns to standard ID-C.8. Task 6: Making More \$; and Task 7: Getting Schooled align to both standards ID-C.7, ID-C.8.

Statistics in Schools from the U.S. Census Bureau: Activities: Math: 9-12:

• Linear Models – Analyzing Relationships: Marriage, Divorce, and Linear Regression.  Students "create a scatter plot, find a line of best fit, and analyze the relationship between the two variables (i.e., sex and marriage/ divorce rates). They will also use a residual plot, explain the meaning of the slope and of the y-intercept of the line of best fit, and investigate the effect of outliers on this line."  Aligns with ID-B.6, ID-C.7.
• Applying Correlation Coefficients: Educational Attainment and Unemployment.  Students "use state and regional unemployment data for various education levels to create scatter plots and calculate correlation coefficients. Students will then compare scatter plots with different strengths of linear relationships and will determine the impact of any influential points on the correlation coefficient.  Aligns with ID-B.6, ID-C.8.
• Educational Attainment and Marriage: Testing a Correlation Coefficient's Significance.  Students "develop, justify, and evaluate conjectures about the relationship between two quantitative variables over time in the United States: the median age (in years) when women first marry and the percentage of women aged 25–34 with a bachelor’s degree or higher. Students will write a regression equation for the data, interpret in context the linear model’s slope and y-intercept, and find the correlation coefficient (r), assessing the strength of the linear relationship and whether a significant relationship exists between the variables. Students will then summarize their conclusions and consider whether correlation implies causation."  Aligns with ID-C.8, ID-C.9, and IC-A.1.
•

Related to the above task is the article: What Is The Aim Of Finding Correlation? Why Is It Used If Correlation Doesn’t Imply Causation? posted at ScienceABC.  Author Argha Sengupta (2022, July 8) delved into Finding Meaning From Random Data: Exploratory Analysis and Correlation May Not Imply Causation, the latter of which might be owing to an influence of a third variable.