Stat 1510: Statistical Thinking and Concepts Scatterplots and Correlation

1. Stat 1510:Statistical Thinking and Concepts Scatterplots and Correlation

2. Agenda 2 • Explanatory and Response Variables • Displaying Relationships: Scatterplots • Interpreting Scatterplots • Adding Categorical Variables to Scatterplots • Measuring Linear Association: Correlation • Facts About Correlation

3. Explanatory and Response Variables 3 • Example:- A medical study finds that short women are • More likely to have heart attacks than women of average • height, while tall women the fewer heart attacks. • Here this is just a study which looks at the relationship between two variables. We ignored some other variables like weight and exercise habits. • We assume that height of a woman has influence in • heart attack chances.

4. Explanatory and Response Variables 4 A response variable measures an outcome of a study. It is also called dependent variable. An explanatory variable may explain or influence changes in a response variable. You will often find explanatory variables called independent variablesorpredictor variables. For the previous example; Response Variable:- A woman's Chance to have heart attack. Explanatory Variable:- Height of a woman.

5. Scatterplot 5 The most useful graph for displaying the relationship between two quantitative variables is a scatterplot. A scatterplotshows 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. How to Make a Scatterplot 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. Label and scale your axes. Plot individual data values. 5

6. Scatterplot 6 Example:Make a scatterplot of the relationship between body weight and pack weight for a group of hikers.

7. Interpreting Scatterplots 7 To interpret a scatterplot, follow the basic strategy of data analysis discussed earlier. 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 departures 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.

8. 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 Interpreting Scatterplots 8 Two variables have a positiveassociationwhen 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 have a negativeassociationwhen above-average values of one tend to accompany below-average values of the other. • There is a moderately strong, positive, linear relationship between body weight and pack weight. • It appears that lighter hikers are carrying lighter backpacks.

9. Adding Categorical Variables 9 • Consider the relationship between mean SAT verbal score and percent of high-school grads taking SAT for each state. Southern states highlighted To add a categorical variable, use a different plot color or symbol for each category. 9

10. Measuring Linear Association 10 • A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. • How can we understand the relation is linear from a scatterplot? • A linear relation is strong if the points lie close to a arbitrary straight line, and its weak if they are widely scattered about a line. • Make sure you are not fooled by scaling difference in the plot. • If there is a linear relationship, you can use correlation to measure the direction and strength of the relation.

11. Correlation 11 The correlation rmeasures the direction and strength of the linear relationship between two quantitative variables. • 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.

12. Correlation 12

13. Facts About Correlation 13 Correlation makes no distinction between explanatory and response variables. r has no units and does not change when we change the units of measurement of x, y, or both. Positive r indicates positive association between the variables, and negative r indicates negative association. 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.

14. Correlation Practice 14 For each graph, estimate the correlation r and interpret it in context.

15. Case Study 15 Per Capita Gross Domestic Product and Average Life Expectancy for Countries in Western Europe

16. Case Study 17

17. x y 21.4 77.48 -0.078 -0.345 0.027 23.2 77.53 1.097 -0.282 -0.309 20.0 77.32 -0.992 -0.546 0.542 22.7 78.63 0.770 1.102 0.849 20.8 77.17 -0.470 -0.735 0.345 18.6 76.39 -1.906 -1.716 3.271 21.5 78.51 -0.013 0.951 -0.012 22.0 78.15 0.313 0.498 0.156 23.8 78.99 1.489 1.555 2.315 21.2 77.37 -0.209 -0.483 0.101 = 21.52 = 77.754 sum = 7.285 sx =1.532 sy =0.795 Case Study 17

18. Case Study 18

19. Correlation simplified formulas