# Bivariate Statistics

• Represent data on two quantitative variables on a scatter plot (S-ID.B.6).
• Describe how two quantitative variables on a scatter plot are related (S-ID.B.6).
• Interpret the slope and the intercept of a linear model in the context of the data (S-ID.B.7).
• Use available technology to find lines of best fit (S-ID.B.6a).
• Assess the goodness of fit of a line to a small data set by plotting and analyzing residuals (S-ID.B.6b).
• Fit a linear function for a scatter plot that suggests a linear association (S-ID.B.6c).
• Use available technology to compute correlation coefficients (S-ID.B.8).
• Understand that the correlation coefficient measures the “tightness” of a line fitted to data (S-ID.B.8).
• Understand that correlation does not necessarily imply causality (S-ID.B.9).

The story before this unit:

From their experiences with linear functions in grade 8, students are familiar with slope and intercept. They gained experience with scatter plots and described patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association (8.SP.A.1). For scatter plots that suggested a linear association, they informally fit a line and informally assessed its fit (8.SP.A.2). They wrote equations for these linear models, and interpreted their slopes and intercepts in the context of the data (8.SP.A.3).

The part of the story happening in this unit:

In this unit, students build on the statistical work they did in grade 8. They work with bivariate data and find the line of best fit by using a graphing calculator or other software. (These lines of best fit are regression lines (or technologically-generated approximations of them) but the Standards do not require students to learn or use the terms “regression,” “regression line,” “regression equation,” or “least squares.”) They assess the fit of a line to data more precisely by plotting and analyzing residuals. Students compare strength of associations between different pairs of variables by interpreting correlation coefficients, which they compute using technology. And, they gain experience in distinguishing between correlation and causality. Modeling is an intrinsic part of the high school statistics and probability standards, and of this unit.

The story after this unit:

In a future statistics unit on Inferences, student build on the techniques used this unit. Plotting residuals suggests that not some data sets may be better modeled non-linear functions than linear ones.

## Sections

A1.3.0 Pre-unit diagnostic assessment

#### Summary

Diagnose students' ability to

• write a linear equation and interpret it in a context (8.F.B.4)
• determine a rate of change and initial value given several data points that exhibit a linear relationship;
• write an equation for a linear relationship;
• use the equation to make predictions;
• interpret the slope and intercept of a linear equation in a context.

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A1.3.1 Preview

#### Summary

• Activate prior experience in grade 8 with informally fitting a line to a scatter plot and informally judging its goodness of fit (8.SP.A.2).
• Model the relationship with an equation for a line and use it to make predictions (8.SP.A.3).
• Represent data on two quantitative variables on a scatter plot (S-ID.B.6).
• Describe how two variables on a scatter plot are related (S-ID.B.6).
• Interpret the slope and the intercept of a linear model in the context of the data (S-ID.C.7).

(Note: 8.SP.A.2 and 8.SP.A.3 are prerequisites, not target standards in this unit. However, they are standards involved in one of the suggested activities.)

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A1.3.2 Lines of best fit and residuals

#### Summary

• Use available technology to find lines of best fit (S-ID.B.6a).
• Quantify the goodness of fit by plotting and analyzing residuals (S-ID.B.6b).

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A1.3.3 Interpreting the correlation coefficient

#### Summary

• Fit a linear function for a scatter plot that suggests a linear association (S-ID.B.6c).
• Use available technology to compute correlation coefficients (S-ID.C.8).
• Understand that the correlation coefficient measures the “tightness” of a line fitted to data (S-ID.C.8).
• Understand the significance of correlation coefficients close to 1 or –1 (S-ID.C.8).
• Interpret the rate of change and constant term of a line fitted to data in the context of the data (S-ID.C.7).

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A1.3.4 Correlation vs causation

#### Summary

Understand that correlation does not necessarily imply causality (S-ID.C.9).

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A1.3.5 Bringing it all together

#### Summary

• Describe how two quantitative variables are related (S-ID.B.6).
• Use technology to create a line of best fit (S-ID.B.6c).
• Fit a line to given data and use it to make predictions (S-ID.B.6a).
• Interpret the coefficients of a line of best fit in the context of the data to which it is fitted (S-ID.B.7).
• Compute and interpret a correlation coefficient (S-ID.B.8).
• Understand that variables with a high correlation do not necessarily have a causal relationship (S-ID.B.9).

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A1.3.6 Summative Assessment

#### Summary

Assess students’ ability to
• create a scatter plot and a line of best fit using technology (S-ID.B.6);
• interpret the coefficients of the line of best fit in the context of the data (S-ID.C.7);
• interpret the correlation coefficient (S-ID.C.8);
• articulate the difference between correlation and causation (S-ID.C.9).

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