 # 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 an asterisk are defined at MathsIsFun.com.  Also see the Stattrek.com Statistics Dictionary.  Key vocabulary for this domain.  Use with ID-A, ID-B, ID-C:

 bivariate and univariate data* histogram* linear fit regression equation boxplot mean normal distribution* scatterplot box-and-whisker plot median outlier standard deviation* correlation coefficient mode quartiles slope dotplot* interquartile range first quartile stemplot third quartile

Alcula.com:

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.

LearnZillion:

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

National Center for Education Statistics: Create a graph

NCTM Illuminations: Advanced Data Grapher: Use this virtual manipulative to analyze data with box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.  Use with ID-A.1.

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

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:

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

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.

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.

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.

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:

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:

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:

Khan Academy: 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.

Wisc-Online, Learning Objects: Use to address standard ID-A.4:

• The Normal Distribution: Learners read a definition of normal distribution.  In this interactive exercise, they enter values for the mean and the standard deviation of normally distributred data and observe the resulting changes in the shape of the normal curve.
• The Normal Distribution and the Empirical Rule: Students use the Empirical Rule to calculate the percentages of data between two data points.  They also calculate the values corresponding to the given percentages of the data.  Practice problems are included throughout.  Note: The Empirical Rule states that in a normal distribution data set, 68% of data values will fall between the mean plus one standard deviation and the mean minus one standard deviation.
• The Area Under the Standard Normal Distribution: The learner identifies and calculates the area under the normal curve specified by given z-scores.  Note: the area under the curve is used as a method to find the percent of data values between two given values.  A z-score tells us how many standard deviations above or below the mean a score is.  See Statistics Help for Students: What are Z-Scores?

Illustrative Mathematics: Statistics and Probability:

Mathematics Assessment Project: Standards: High School: Statistics & Probability: https://www.map.mathshell.org/stds.php?standardid=1233

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 in the classroom, 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.

Technology-enhanced investigations:

Alcula.com:

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

LearnZillion:

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.

NCTM Illuminations:

• Line of Best Fit: This virtual manipulative allows the user to enter a set of data, plot the data on a coordinate grid, and determine the line of best fit.
• Linear Regression I extends this activity.  Learners plot points on a coordinate grid in a relatively straight line to create a scatter plot, show the regression line (line of best fit), and then experiment with adding adding additional points (e.g., outliers) to the grid and viewing the resulting change in the line of best fit.
• Correlation and the Regression Line: “Interactive computer-based tools provide students with the opportunity to easily investigate the relationship between a set of data points and a curve used to fit the data points. As students work with bivariate data in grades 9-12, they will be able to investigate relationships between the variables using linear, exponential, power, logarithmic, and other functions for curve fitting.”
• Determining Functions Using Regression: “This unit guides students though activities that ask students to collect data. Then, they use technology to find functions that best describe a data collected. After analyzing the data, the student should be able to determine a best type of function to describe the trend.”

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.

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.

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:

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-B.6: Linear regression

Multiple Choice:

• Two-way frequency tables ID.5: 4 multiple choice questions.  Includes related review resources.  Aligns with ID-B.5.
• Scatter plot ID.6: 4 multiple choice questions.  Includes related review resources.  Aligns with ID-B.6.
• Fitting functions to data: 5 questions combine multiple choice and free response.  Includes related review resources.  Aligns with ID-B.6.a.
• Residuals: 5 questions combine multiple choice and free response.  Includes related review resources.  Aligns with ID-B.6.b.

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: 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: 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.

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: https://www.map.mathshell.org/stds.php?standardid=1233

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:

Khan Academy: Correlation and Causality video explanation of the difference. Aligns with ID-C.9.

LearnZillion:

• 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.

NCTM Illuminations: Least Squares Regression: “This Unit Plan consists of lessons in which students interpret the slope and y-intercept of least squares regression lines in the context of real-life data. Students use an e-example applet to plot the data and calculate the correlation coefficient and equation of the least squares regression line.”

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.

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: 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: https://www.map.mathshell.org/stds.php?standardid=1233 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. Back to top Common Core Math: Intro | HS Statistics & Probability Domain: ID  |  IC  |  CP  |  MD  |