# Chapter 2. Looking at Data: Relationships. Introduction to the Practice of STATISTICS SEVENTH. Moore / McCabe / Craig. Lecture Presentation Slides

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1 Chapter 2 Looking at Data: Relationships Introduction to the Practice of STATISTICS SEVENTH EDITION Moore / McCabe / Craig Lecture Presentation Slides

2 Chapter 2 Looking at Data: Relationships 2.1 Scatterplots 2.2 Correlation 2.3 Least-Squares Regression 2.4 Cautions about Correlation and Regression 2.5 Data Analysis for Two-Way Tables 2

3 2.1 Scatterplots Scatterplots Explanatory and response variables Interpreting scatterplots Categorical variables in scatterplots 3

4 Associations Between Variables Many interesting examples of the use of statistics involve relationships between pairs of variables. Two variables measured on the same cases are associated if knowing the value of one of the variables tells you something about the values of the other variable that you would not know without this information. When you examine the relationship between two variables, a new question becomes important: Is your purpose simply to explore the nature of the relationship, or do you wish to show that one of the variables can explain variation in the other? A response variable measures an outcome of a study. An explanatory variable explains or causes changes in the response variable. 4

5 Scatterplot The most useful graph for displaying the relationship between two quantitative variables is a scatterplot. A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph. 1. Decide which variable should go on each axis. If a distinction exists, plot the explanatory variable on the x- axis and the response variable on the y-axis. 2. Label and scale your axes. 3. Plot individual data values. How to Make a Scatterplot 5

6 Scatterplot Example: Make a scatterplot of the relationship between body weight and backpack weight for a group of hikers. Body weight (lb) Backpack weight (lb)

7 Interpreting Scatterplots To interpret a scatterplot, follow the basic strategy of data analysis from Chapter 1. Look for patterns and important departures from those patterns. How to Examine a Scatterplot As in any graph of data, look for the overall pattern and for striking deviations from that pattern. You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship. An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship. 7

8 Interpreting Scatterplots Two variables are positively associated when above-average values of one tend to accompany above-average values of the other, and when below-average values also tend to occur together. Two variables are negatively associated when above-average values of one tend to accompany below-average values of the other, and vice-versa. 8

9 Interpreting Scatterplots There is one possible outlier the hiker with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members. Strength Direction Form There is a moderately strong, positive, linear relationship between body weight and backpack weight. It appears that lighter hikers are carrying lighter backpacks. 9

10 Adding Categorical Variables Consider the relationship between mean SAT verbal score and percent of high school grads taking the SAT for each state. To add a categorical variable, use a different plot color or symbol for each category. Southern states highlighted 10

11 2.2 Correlation The correlation coefficient r Properties of r Influential points 11

12 Measuring Linear Association A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. Linear relations are important because a straight line is a simple pattern that is quite common. Our eyes are not good judges of how strong a relationship is. Therefore, we use a numerical measure to supplement our scatterplot and help us interpret the strength of the linear relationship. The correlation r measures the strength of the linear relationship between two quantitative variables. r = 1 n 1 x i x s x y i y s y

13 Measuring Linear Association We say a linear relationship is strong if the points lie close to a straight line and weak if they are widely scattered about a line. The following facts about r help us further interpret the strength of the linear relationship. Properties of Correlation r is always a number between 1 and 1. r > 0 indicates a positive association. r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 toward 1 or 1. The extreme values r = 1 and r = 1 occur only in the case of a perfect linear relationship.

14 Correlation

15 Properties of Correlation 1. Correlation makes no distinction between explanatory and response variables. 2. r has no units and does not change when we change the units of measurement of x, y, or both. 3. Positive r indicates positive association between the variables, and negative r indicates negative association. 4. The correlation r is always a number between 1 and 1. Cautions: Correlation requires that both variables be quantitative. Correlation does not describe curved relationships between variables, no matter how strong the relationship is. Correlation is not resistant. r is strongly affected by a few outlying observations. Correlation is not a complete summary of two-variable data.

16 Correlation Examples For each graph, estimate the correlation r and interpret it in context.

17 2.3 Least-Squares Regression Regression lines Least-square Regression Line Facts about Least-Squares Regression Correlation and Regression 17

18 Regression Line A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We can use a regression line to predict the value of y for a given value of x. Example: Predict the number of new adult birds that join the colony based on the percent of adult birds that return to the colony from the previous year. If 60% of adults return, how many new birds are predicted? 18

19 Regression Line When a scatterplot displays a linear pattern, we can describe the overall pattern by drawing a straight line through the points. Fitting a line to data means drawing a line that comes as close as possible to the points. Regression equation: y = b 0 + b 1 x x is the value of the explanatory variable. y-hat is the predicted value of the response variable for a given value of x. b 1 is the slope, the amount by which y changes for each oneunit increase in x. b 0 is the intercept, the value of y when x = 0. 19

20 Least-Squares Regression Line Since we are trying to predict y, we want the regression line to be as close as possible to the data points in the vertical (y) direction. Least-Squares Regression Line (LSRL): The least-squares regression line of y on x is the line that minimizes the sum of the squares of the vertical distances of the data points from the line. If we have data on an explanatory variable x and a response variable y, the equation of the least-squares regression line is: y = b 0 + b 1 x 20

21 Facts About Least-Squares Regression Regression is one of the most common statistical settings, and leastsquares is the most common method for fitting a regression line to data. Here are some facts about least-squares regression lines. Fact 1: A change of one standard deviation in x corresponds to a change of r standard deviations in y. Fact 2: The LSRL always passes through (x-bar, y-bar). Fact 3: The distinction between explanatory and response variables is essential. 21

22 Correlation and Regression Least-squares regression looks at the distances of the data points from the line only in the y direction. The variables x and y play different roles in regression. Even though correlation r ignores the distinction between x and y, there is a close connection between correlation and regression. The square of the correlation, r 2, is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x. r 2 is called the coefficient of determination. 22

23 2.4 Cautions About Correlation and Regression Predictions Residuals and Residual Plots Outliers and Influential Observations Lurking Variables Correlation and Causation 23

24 Predictions Via Regression Line For the returning birds example, the LSRL is: y-hat = x y-hat is the predicted number of new birds for colonies with x percent of adults returning. Suppose we know that an individual colony has 60% returning. What would we predict the number of new birds to be for just that colony? For colonies with 60% returning, we predict the average number of new birds to be: (0.3040)(60) = birds 24

25 Residuals A regression line describes the overall pattern of a linear relationship between an explanatory variable and a response variable. Deviations from the overall pattern are also important. The vertical distances between the points and the least-squares regression line are called residuals. A residual is the difference between an observed value of the response variable and the value predicted by the regression line: residual = observed y predicted y = y yˆ 25

26 Residual Plots A residual plot is a scatterplot of the regression residuals against the explanatory variable. Residual plots help us assess the fit of a regression line. Look for a random scatter around zero. Residual patterns suggest deviations from a linear relationship. Gesell Adaptive Score and Age at First Word 26

27 Outliers and Influential Points An outlier is an observation that lies outside the overall pattern of the other observations. Outliers in the y direction have large residuals Outliers in the x direction are often influential for the least-squares regression line, meaning that the removal of such points would markedly change the equation of the line. 27

28 Outliers and Influential Points Gesell Adaptive Score and Age at First Word After removing child 18 r 2 = 11% From all of the data r 2 = 41% Chapter 5 28

29 Extrapolation Sarah s height was plotted against her age. Can you predict her height at age 42 months? Can you predict her height at age 30 years (360 months)? 29

30 Extrapolation Regression line: y-hat = x Height at age 42 months? y-hat = 88 Height at age 30 years? y-hat = She is predicted to be at age 30! 30

31 Cautions About Correlation and Regression Both describe linear relationships. Both are affected by outliers. Always plot the data before interpreting. Beware of extrapolation. predicting outside of the range of x Beware of lurking variables. These have an important effect on the relationship among the variables in a study, but are not included in the study. Correlation does not imply causation! 31

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