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Understanding Scatter Diagrams & Correlation Coefficients

Learn how to draw, interpret scatter diagrams, and compute linear correlation coefficients to describe the relationship between two variables. Understand the difference between correlation & causation.

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Understanding Scatter Diagrams & Correlation Coefficients

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  1. STATISTICSINFORMED DECISIONS USING DATA Fifth Edition Chapter 4 Describing the Relation between Two Variables

  2. 4.1 Scatter Diagrams and CorrelationLearning Objectives 1. Draw and interpret scatter diagrams 2. Describe the properties of the linear correlation coefficient 3. Compute and interpret the linear correlation coefficient 4. Determine whether a linear relation exists between two variables 5. Explain the difference between correlation and causation

  3. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (1 of 6) The response variable is the variable whose value can be explained by the value of the explanatory or predictor variable. A scatter diagram is a graph that shows the relationship between two quantitative variables measured on the same individual. Each individual in the data set is represented by a point in the scatter diagram. The explanatory variable is plotted on the horizontal axis, and the response variable is plotted on the vertical axis.

  4. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (2 of 6) EXAMPLE Drawing and Interpreting a Scatter Diagram The data shown to the right are based on a study for drilling rock. The researchers wanted to determine whether the time it takes to dry drill a distance of 5 feet in rock increases with the depth at which the drilling begins. So, depth at which drilling begins is the explanatory variable, x, and time (in minutes) to drill five feet is the response variable, y. Draw a scatter diagram of the data. Source: Penner, R., and Watts, D.G. “Mining Information.”The American Statistician, Vol. 45, No. 1, Feb. 1991, p. 6.

  5. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (3 of 6)

  6. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (4 of 6) Various Types of Relations in a Scatter Diagram

  7. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (5 of 6) Two variables that are linearly related are positively associated when above-average values of one variable are associated with above-average values of the other variable and below-average values of one variable are associated with below-average values of the other variable. That is, two variables are positively associated if, whenever the value of one variable increases, the value of the other variable also increases.

  8. 4.1 Scatter Diagrams and Correlation4.1.1 Draw and Interpret Scatter Diagrams (6 of 6) Two variables that are linearly related are negatively associated when above-average values of one variable are associated with below-average values of the other variable. That is, two variables are negatively associated if, whenever the value of one variable increases, the value of the other variable decreases.

  9. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(1 of 6) The linear correlation coefficient or Pearson product moment correlation coefficient is a measure of the strength and direction of the linear relation between two quantitative variables. The Greek letter ρ (rho) represents the population correlation coefficient, and r represents the sample correlation coefficient. We present only the formula for the sample correlation coefficient.

  10. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(2 of 6) Sample Linear Correlation Coefficient

  11. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(3 of 6) Properties of the Linear Correlation Coefficient The linear correlation coefficient is always between −1 and 1, inclusive. That is, −1 ≤ r ≤ 1. If r = + 1, then a perfect positive linear relation exists between the two variables. If r = −1, then a perfect negative linear relation exists between the two variables. The closer r is to +1, the stronger the evidence is of a positive association between the two variables. The closer r is to −1, the stronger the evidence is of a negative association between the two variables.

  12. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(4 of 6) • If r is close to 0, then little or no evidence exists of a linear relation between the two variables. So r close to 0 does not imply no relation, just no linear relation. • The linear correlation coefficient is a unitless measure of association. So the unit of measure for x and y plays no role in the interpretation of r. • The correlation coefficient is not resistant. Therefore, an observation that does not follow the overall pattern of the data could affect the value of the linear correlation coefficient.

  13. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(5 of 6)

  14. 4.1 Scatter Diagrams and Correlation4.1.2 Describe the Properties of the Linear Correlation Coefficient(6 of 6)

  15. 4.1 Scatter Diagrams and Correlation4.1.3 Compute and Interpret the Linear Correlation Coefficient(1 of 5) EXAMPLE Determining the Linear Correlation Coefficient Determine the linear correlation coefficient of the drilling data.

  16. 4.1 Scatter Diagrams and Correlation4.1.3 Compute and Interpret the Linear Correlation Coefficient(2 of 5)

  17. 4.1 Scatter Diagrams and Correlation4.1.3 Compute and Interpret the Linear Correlation Coefficient(3 of 5)

  18. 4.1 Scatter Diagrams and Correlation4.1.3 Compute and Interpret the Linear Correlation Coefficient(4 of 5) IN CLASS ACTIVITY Correlation Randomly select six students from the class and have them determine their at-rest pulse rates and then discuss the following: When determining each at-rest pulse rate, would it be better to count beats for 30 seconds and multiply by 2 or count beats for 1 full minute?Explain. What are some other ways to find the at-rest pulse rate?Do any of these methods have an advantage? What effect will physical activity have on pulse rate? Do you think the at-rest pulse rate will have any effect on the pulse rate after physical activity? If so, how? If not, why not? Have the same six students jog in place for 3 minutes and then immediately determine their pulse rates using the same technique as for the at-rest pulse rates.

  19. 4.1 Scatter Diagrams and Correlation4.1.3 Compute and Interpret the Linear Correlation Coefficient(5 of 5) Draw a scatter diagram for the pulse data using the at-rest data as the explanatory variable. Comment on the relationship, if any, between the two variables. Is this consistent with your expectations? Based on the graph, estimate the linear correlation coefficient for the data. Then compute the correlation coefficient and compare it to your estimate.

  20. 4.1 Scatter Diagrams and Correlation4.1.4 Determine whether a Linear Relation Exists between Two Variables(1 of 2) Testing for a Linear Relation Step 1 Determine the absolute value of the correlation coefficient. Step 2 Find the critical value in Table II for the given sample size. Step 3 If the absolute value of the correlation coefficient is greater than the critical value, we say a linear relation exists between the two variables. Otherwise, no linear relation exists.

  21. 4.1 Scatter Diagrams and Correlation4.1.4 Determine whether a Linear Relation Exists between Two Variables(2 of 2) EXAMPLE Does a Linear Relation Exist? Determine whether a linear relation exists between time to drill five feet and depth at which drilling begins. Comment on the type of relation that appears to exist between time to drill five feet and depth at which drilling begins. The correlation between drilling depth and time to drill is 0.773. The critical value for n = 12 observations is 0.576. Since 0.773 > 0.576, there is a positive linear relation between time to drill five feet and depth at which drilling begins.

  22. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (1 of 8) According to data obtained from the Statistical Abstract of the United States, the correlation between the percentage of the female population with a bachelor’s degree and the percentage of births to unmarried mothers since 1990 is 0.940. Does this mean that a higher percentage of females with bachelor’s degrees causes a higher percentage of births to unmarried mothers?

  23. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (2 of 8) Certainly not! The correlation exists only because both percentages have been increasing since 1990. It is this relation that causes the high correlation. In general, time series data (data collected over time) may have high correlations because each variable is moving in a specific direction over time (both going up or down over time; one increasing, while the other is decreasing over time). When data are observational, we cannot claim a causal relation exists between two variables. We can only claim causality when the data are collected through a designed experiment.

  24. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (3 of 8) Another way that two variables can be related even though there is not a causal relation is through a lurking variable. A lurking variable is related to both the explanatory and response variable. For example, ice cream sales and crime rates have a very high correlation. Does this mean that local governments should shut down all ice cream shops? No! The lurking variable is temperature. As air temperatures rise, both ice cream sales and crime rates rise.

  25. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (4 of 8) EXAMPLE Lurking Variables in a Bone Mineral Density Study Because colas tend to replace healthier beverages and colas contain caffeine and phosphoric acid, researchers Katherine L. Tucker and associates wanted to know whether cola consumption is associated with lower bone mineral density in women. The table lists the typical number of cans of cola consumed in a week and the femoral neck bone mineral density for a sample of 15 women. The data were collected through a prospective cohort study.

  26. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (5 of 8) EXAMPLE Lurking Variables in a Bone Mineral Density Study The figure on the next slide shows the scatter diagram of the data. The correlation between number of colas per week and bone mineral density is −0.806.The critical value for correlation with n = 15 from Table II in Appendix A is 0.514. Because |−0.806| > 0.514, we conclude a negative linear relation exists between number of colas consumed and bone mineral density. Can the authors conclude that an increase in the number of colas consumed causes a decrease in bone mineral density? Identify some lurking variables in the study.

  27. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (6 of 8)

  28. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (7 of 8) EXAMPLE Lurking Variables in a Bone Mineral Density Study In prospective cohort studies, data are collected on a group of subjects through questionnaires and surveys over time. Therefore, the data are observational. So the researchers cannot claim that increased cola consumption causes a decrease in bone mineral density. Some lurking variables in the study that could confound the results are: body mass index height smoking alcohol consumption calcium intake physical activity

  29. 4.1 Scatter Diagrams and Correlation4.1.5 Explain the Difference between Correlation and Causation (8 of 8) EXAMPLE Lurking Variables in a Bone Mineral Density Study The authors were careful to say that increased cola consumption is associated with lower bone mineral density because of potential lurking variables. They never stated that increased cola consumption causes lower bone mineral density.

  30. 4.2 Least-squares RegressionLearning Objectives 1. Find the least-squares regression line and use the line to make predictions 2. Interpret the slope and the y-intercept of the least-squares regression line 3. Compute the sum of squared residuals

  31. 4.2 Least-squares RegressionEXAMPLE Finding an Equation that Describes Linearly Relate Data(1 of 2) Using the following sample data:

  32. 4.2 Least-squares RegressionEXAMPLE Finding an Equation that Describes Linearly Relate Data (2 of 2) (b) Graph the equation on the scatter diagram.

  33. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (1 of 7) The difference between the observed value of y and the predicted value of y is the error, or residual. Using the line from the last example, and the predicted value at x = 3: residual = observed y− predicted y = 5.2 − 4.75 = 0.45

  34. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (2 of 7) Least-Squares Regression Criterion

  35. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (3 of 7) The Least-Squares Regression Line The equation of the least-squares regression line is given by

  36. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (4 of 7) The Least-Squares Regression Line

  37. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (5 of 7) EXAMPLE Finding the Least-squares Regression Line Using the drilling data • Find the least-squares regression line. • Predict the drilling time if drilling starts at 130 feet. • Is the observed drilling time at 130 feet above, or below, average. • Draw the least-squares regression line on the scatter diagram of the data.

  38. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (6 of 7) • The observed drilling time is 6.93 seconds. The predicted drilling time is 7.035 seconds. The drilling time of 6.93 seconds is below average.

  39. 4.2 Least-squares Regression4.2.1Find the Least-Squares Regression Line and Use the Line to Make Predictions (7 of 7)

  40. 4.2 Least-squares Regression4.2.2Interpret the Slope and the y-Intercept of the Least-Squares Regression Line (1 of 3) Interpretation of Slope: The slope of the regression line is 0.0116. For each additional foot of depth we start drilling, the time to drill five feet increases by 0.0116 minutes, on average.

  41. 4.2 Least-squares Regression4.2.2Interpret the Slope and the y-Intercept of the Least-Squares Regression Line (2 of 3) Interpretation of the y-Intercept: The y-intercept of the regression line is 5.5273. To interpret the y-intercept, we must first ask two questions: Is 0 a reasonable value for the explanatory variable? Do any observations near x = 0 exist in the data set? A value of 0 is reasonable for the drilling data (this indicates that drilling begins at the surface of Earth. The smallest observation in the data set is x = 35 feet, which is reasonably close to 0. So, interpretation of the y-intercept is reasonable. The time to drill five feet when we begin drilling at the surface of Earth is 5.5273 minutes.

  42. 4.2 Least-squares Regression4.2.2Interpret the Slope and the y-Intercept of the Least-Squares Regression Line (3 of 3) If the least-squares regression line is used to make predictions based on values of the explanatory variable that are much larger or much smaller than the observed values, we say the researcher is working outside the scope of the model. Never use a least-squares regression line to make predictions outside the scope of the model because we can’t be sure the linear relation continues to exist.

  43. 4.2 Least-squares Regression4.2.3 Compute the Sum of Squared Residuals To illustrate the fact that the sum of squared residuals for a least-squares regression line is less than the sum of squared residuals for any other line, use the “regression by eye” applet.

  44. 4.3 Diagnostics on the Least-squares Regression LineLearning Objectives 1. Compute and interpret the coefficient of determination

  45. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(1 of 18) The coefficient of determination, R2, measures the proportion of total variation in the response variable that is explained by the least-squares regression line. The coefficient of determination is a number between 0 and 1, inclusive. That is, 0 <R2< 1. If R2 = 0 the line has no explanatory value If R2 = 1 means the line explains 100% of the variation in the response variable.

  46. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(2 of 18) The data to the right are based on a study for drilling rock. The researchers wanted to determine whether the time it takes to dry drill a distance of 5 feet in rock increases with the depth at which the drilling begins. So, depth at which drilling begins is the predictor variable, x, and time (in minutes) to drill five feet is the response variable, y. Source: Penner, R., and Watts, D.G. “Mining Information.”The American Statistician, Vol. 45, No. 1, Feb. 1991, p. 6.

  47. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(3 of 18)

  48. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(4 of 18) Correlation Between Depth and Time: 0.773 Regression Analysis The regression equation is Time = 5.53 + 0.0116 Depth

  49. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(5 of 18) Suppose we were asked to predict the time to drill an additional 5 feet, but we did not know the current depth of the drill. What would be our best “guess”?

  50. 4.3 Diagnostics on the Least-squares Regression Line4.3.1 Compute and Interpret the Coefficient of Determination(6 of 18) Suppose we were asked to predict the time to drill an additional 5 feet, but we did not know the current depth of the drill. What would be our best “guess”? ANSWER: The mean time to drill an additional 5 feet: 6.99 minutes

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