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Chapter 3

Chapter 3. Scatterplots and Correlation. Both Ch 3 and Ch 4. Relationships between two quantitative variables X  explanatory variable Y  response variable

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Chapter 3

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  1. Chapter 3 Scatterplots and Correlation Chapter 3

  2. Both Ch 3 and Ch 4 • Relationships between two quantitative variables X  explanatory variable Y response variable • Illustrative Example: What is the relationship between “per capita gross domestic product” (X) and “life expectancy” (Y)?

  3. Data n = 10 “GDP & life expectancy”

  4. Scatterplot: Life_Exp vs. GDP This is the data point for Switzerland (23.8, 78.99)

  5. Interpreting Scatterplots • Form: Straight? Curved? • Outliers: Deviations from overall pattern • Direction of association: • Positive association (upward) • Negative association (downward) • No association (flat) • Strength: Extent to which points adhere to predicted trend line (next slide)

  6. No association Moderate positive assn Strong positive assn Strong negative assn. Weak negative assn. Very strong negative assn.

  7. Interpretation: life expectancy example • Form: linear • Outliers: none • Direction: positive • Strength: hard to tell by eye This is the data point for Switzerland (23.8, 78.99) 10/9/2014 7

  8. Interpretation Form: linear Outliers: none Direction: positive Strength: looks strong Example #2

  9. Form: linear Outliers: none Direction: negative Strength: weak(?) Example #3

  10. Form: linear(?) Outliers: none Direction: negative(?) Strength: weak Example #4 (Age & Health)

  11. Form: U-shaped Outliers: (?) Direction: down then up Strength: (?) Example #5 (Physical & Mental Health)

  12. Strength is Difficult to Judge by Eye Alone • These two figures display the same data set with different axis scaling but the bottom figure looks “stronger” (optical illusion) • To overcome this difficulty: calculate correlation coefficient r

  13. Correlation Coefficient r • Notation: r≡ Pearson’s correlation coefficient • Always between−1 and +1 r = +1  all points on upward sloping line r = -1 all points on downward line r = 0 no line or horizontal line • The closer r is to +1 or –1, the stronger the correlation • Positive or negative sign indicatesdirection of correlation

  14. Guidelines for interpreting “strength” via r • 0.0  | r | < 0.3  “weak” • 0.3  | r | < 0.7  “moderate” • 0.7  | r | < 1.0  “strong” 10/9/2014 14

  15. Examples • Husband’s age / Wife’s age • r = .94 (strong positive correlation) • Husband’s height / Wife’s height • r = .36 (moderate positive correlation) • Distance of golf putt / percent success • r = -.94 (strong negative correlation)

  16. Calculating r by hand • Calculate mean and standard deviation of X • Calculate mean and standard deviation of Y • Turn all X values into “z scores” • Turn all Y values into “z scores” • Calculate r

  17. What is a z score? • z ≡ “standardized value” • Tells you the number of units above or below the mean in standard deviation units Examples: • A z score of 1 indicates the value is 1 standard deviation abovethe mean • A z score of –1 indicates the value is 1 standard deviation below the mean • A z score of 0 indicates the value is equal to the mean

  18. Calculating r by hand (Example)

  19. r by hand can be tedious!(Life expectancy data) 7.285 x-bar= 21.52 sx= 1.532 y-bar= 77.754 sy= 0.795 10/9/2014 19

  20. Example: Calculating r r = .809  strong positive correlation

  21. Calculating r Use your calculator in 2-var mode! TI two-variable calculator 10/9/2014 21

  22. Beware! • r applies to linear relations only • Outliers have large influences on r • Association ≠ causation

  23. Nonlinear relation (mpg vs. speed) Strong non-linear relationships Can show r = 0 r = 0

  24. Outliers Have Undue Influence x Without the outlier, r  .8 With the outlier, r  0

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