The Squared Correlation r2

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The Squared Correlation r2 What Does It Tell Us?

• Lecture 51
• Sec. 13.9
• Mon, Dec 12, 2005

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Residual Sum of Squares

• Recall that the line of best fit was that line
  with the smallest sum of squared residuals.
• This is also called the residual sum of squares

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Other Sums of Squares

• There are two other sums of squares associated
  with y.
• The regression sum of squares
• The total sum of squares

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Other Sums of Squares

• The regression sum of squares, SSR, measures the
  variability in y that is predicted by the model,
  i.e., the variability in y.
• The total sum of squares, SST, measures the
  observed variability in y.

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Example SST, SSR, and SSE

• Plot the data in Example 13.14, p. 800, with?y.

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Example SST, SSR, and SSE

• The deviations of y from?y (observed).

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Example SST, SSR, and SSE

• The deviations of y from?y (predicted).

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Example SST, SSR, and SSE

• The deviations of y from y (residual deviations).

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The Squared Correlation

• It turns out that
• It also turns out that

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Explaining Variation

• One goal of regression is to explain the
  variation in y.
• For example, if x were height and y were weight,
  how would we explain the variation in weight?
• That is, why do some people weigh more than
  others?
• Or if x were the hours spent studying for a math
  test and y were the score on the test, how would
  we explain the variation in scores?
• That is, why do some people score higher than
  others?

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Explaining Variation

• A certain amount of the variation in y can be
  explained by the variation in x.
• Some people weigh more than others because they
  are taller.
• Some people score higher on math tests because
  they studied more.
• But that is never the full explanation.
• Not all taller people weigh more.
• Not everyone who studies longer scores higher.

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Explaining Variation

• High degree of correlation between x and y ?
  variation in x explains most of the variation in
  y.
• Low degree of correlation between x and y ?
  variation in x explains only a little of the
  variation in y.
• In other words, the amount of variation in y that
  is explained by the variation in x should be
  related to r.

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Explaining Variation

• Statisticians consider the predicted variation
  SSR to be the amount of variation in y (SST) that
  is explained by the model.
• The remaining variation in y, i.e., residual
  variation SSE, is the amount that is not
  explained by the model.

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Explaining Variation
SST SSE SSR
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Explaining Variation
SST SSE SSR
Total variation in y (to be explained)
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Explaining Variation
SST SSE SSR
Total variation in y (to be explained)
Variation in y that is explained by the model
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Explaining Variation
Variation in y that is unexplained by the model
SST SSE SSR
Total variation in y
Variation in y that is explained by the model
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Example SST, SSR, and SSE

• The total (observed) variation in y.

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Example SST, SSR, and SSE

• The variation in y that is explained by the model.

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Example SST, SSR, and SSE

• The variation in y that is not explained by the
  model.

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Explaining Variation

• Therefore,
• r2 is the proportion of variation in y that is
  explained by the model and 1 r2 is the
  proportion that is not explained by the model.

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TI-83 Calculating r2

• To calculate r2 on the TI-83,
• Follow the procedure that produces the regression
  line and r.
• In the same window, the TI-83 reports r2.

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Lets Do It!

• Lets Do It! 13.3, p. 819 Oil-Change Data.
• Do part (b) on the TI-83.
• How much of the variation in repair costs is
  explained by frequency of oil change?