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3.1b Correlation Target Goal: I can determine the strength of a distribution using the correlation. D2 h.w: p 160 – 14 – 18, 21, 26

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Scatterplot Recall: Scatterplot reveals the strength, direction, and form for 2 quantitative variables.

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Two scatterplots of the same data: The straight-line pattern in the lower plot appears stronger because of the surrounding white space. Our eyes are not good judges. We need a numerical measure to supplement graphs.

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Correlation (r) Measures the and of the linear relationship The formula for the correlation r between x and y is: directionstrength between two variables.

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The average of the products of the x and y values for n people. standardized

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Exercise: Classifying Fossils The data gives the lengths of two bones in five fossil specimens of the extinct beast Archaeopteryx. Femur: 38 56 59 64 74 Humerus: 41 63 70 72 84 Enter data into L1 and L2.

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Find the correlation r step-by-step. Find the mean and the standard deviation of the femur and humerus lengths. Then find the five standardized values of each variable by using the formula for r. Use STAT: CALC; 2-VAR Stats L1, L2 to find the following:

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Use the formula to find the correlation r step-by-step. Refer to formula. x bar = 58.2 Sx = 13.2 y bar = 66.0 Sy = 15.89 Calculate r by hand.

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Interpreting Correlation 1.Correlation: makes no distinction between explanatory and response variable. 2.Correlation requires both variables be quantitative. 3.Because r uses the standardized values of the observations: r does not change when we change the unit measure of x,y, or both, r itself has no unit of measure.

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4.Positive r indicates: positive association between variables. Negative r indicates: negative assoc. between variables.

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5. Correlation r is always a number between -1 and 1 values of r near 0 indicate a very weak linear relationship as r moves away from 0 toward either -1 or 1: the strength of the linear relationship increases values of r close to -1 or 1: indicate that the points in a scatterplot lie close to a straight line extreme values of r = -1 and r = 1 occur only in the case of: a perfect linear relationship

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6.Correlation measures the strength of only a linear relationship between two variables (not a curve). 7.Like the mean and standard deviation, the correlation r: is not resistant (use r with caution when outliers appear).

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Remember: correlation is not a complete description of two-variable data. Also include the means and standard deviations of both x and y.

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Pg. 155

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Exercise: More Archaeopterx The data gives the lengths of two bones in five fossil specimens of the extinct beast. You found the correlation r in ex. r = 0.994

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a.Make a scatterplot if you did not so earlier. Explain why the value of r matches the scatterplot. (3 min) r = 0.994 The plot shows a strong positive linear relationship, with little scatter, so we expect that r is close to 1.

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b.The lengths were measured in centimeters. If we changed to inches, how would r change? (There are 2.54 centimeters in an inch.) r would not change – it is computed from standardized values.

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+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 3: Describing Relationships Section 3.1 Scatterplots and Correlation.

+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 3: Describing Relationships Section 3.1 Scatterplots and Correlation.

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