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Course 2, Unit 4 - Regression and Correlation

In the Regression and Correlation unit, students learn to describe the association between two variables by interpreting a scatterplot, to interpret a correlation coefficient, and to understand the limitations of correlation coefficients. They learn to know when it is appropriate to make predictions from the regression line, and to understand the effects of outliers and influential points on the correlation coefficient and on the regression line. Students compute and interpret Pearson's correlation and come to understand that a strong correlation does not imply that one variable causes the other.

Key Ideas from Course 2, Unit 4

  • Least squares regression line: a line that fits a set of data with slope and intercept chosen to minimize the sum of the squared errors (SSE), also called the squared residuals. This line contains the point with x-coordinate the mean of the x values of the data (x-bar) and y-coordinate the mean of the y values of the data (y-bar): (x-bar, y-bar). (See pp. 280-285.)

  • Residual: The difference between the observed value (the value used in calculating the regression equation) and the predicted value is called the residual. These differences are visualized as the vertical gaps between given data points and the regression line. (See p. 283.)

  • Pearson's correlation r: a single number that gives a measure of the strength of the linear association between variables. Perfectly linear data will have an r value of 1 (for a positive association), or an r value of -1 (for a negative association). A set of data that is quite linear, and for which the y values increase as the x values increase, is said to have a strong positive association. If the points appear to fit a linear trend but the y values decrease as x increases, the association is negative. A set of data that is quite random, not at all linear, will have an r value close to zero. (See pp. 258-268 and 291-298.)

  • Association and causation: Two variables may have a strong correlation without there being any reason to suspect that when one variable changes, it "causes" a change in the other variable. The example used in the text to show this idea is data on ice cream consumption and recorded crimes in 12 countries. (See pp. 300-303.)

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