2.1 Scatterplots. Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

Transcription

1 2.1 Scatterplots Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

2 Association Between Variables We now consider the situation where we have two variables. Example Let x be the age of a husband, and let y be the age of his wife. Let x be the grip strength of a worker, and let y be his arm strength.

3 Response and Explanatory Variable Example Student volunteers were assigned to drink a certain numbers of cans of beer. Thirty minutes later, a police officer measures their blood alcohol content. Here the number of cans of beer is the explanatory variable, and the blood alcohol content is the response variable. Sometimes, explanatory variables are called independent variables, and response variables are called dependent variables.

4 Scatterplots

5 Example 1: Age of Husband and Wife There is a positive association between the variables, and it is relatively strong (most points are close to a straight line).

6 Example 2: Grip Strength and Arm Strength There is a positive association between the variables, but it is weaker than in Example 1.

7 2.2 Correlation Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

8 Correlation

9 Properties of Correlation Positive r indicates positive association between the variables, and negative r indicates negative association. The correlation r is always between 1 and 1; values near 0 indicate a very weak linear relation. Values near 1 or 1 indicate that most points lie close to a straight line. All points lie exactly on a straight line precisely when r = 1 or r = 1. Correlation measures the strength of only the linear relationship; correlation does not describe curved relationships between variables, no matter how strong they are. Correlation makes no distinction between explanatory and response variables. Correlation only makes sense if both variables are quantitative, it is a sensitive measure, and does not change if a linear change in the variables is applied.

10 Scatterplots and Correlation

11 Examples of Correlation Coefficients The correlation coefficient for the sample of spousal ages is r = 0.97 (left) and r = 0.63 for the strength of workers (right).

12 Example: Diameter and Height of Redwood Trees We have the following data giving the diameter of a redwood tree at breast height (in meters, response variable), together with the height of the tree (in meters, explanatory variable). x: Diameter y: Height x: Diameter y: Height We demonstrate how to compute the correlation coefficient r, using a TI-83/83 Plus/84 calculator.

13 Example: Diameter and Height of Redwood Trees First, make sure the diagnostics are turned on: Press 2ND 0 (CATALOG). Arrow down to DiagnosticOn.

14 Example: Diameter and Height of Redwood Trees Press ENTER and press ENTER again. The calculator is now in DiagnosticOn mode. The previous steps need to be done only once.

15 Example: Diameter and Height of Redwood Trees Now, enter the data by first going to STAT, selecting Edit..., and entering the data into L1 and L1.

16 Example: Diameter and Height of Redwood Trees Go to STAT again, arrow over to CALC, and select LinReg(ax+b).

17 Example: Diameter and Height of Redwood Trees Press ENTER and ENTER again. The correlation coefficient is r =

18 ios App Use Click on Enter Quantitative Data.

19 ios App Use Enter the information for the two variables.

20 ios App Use Return to the main menu and select Two-Variable Statistics.

21 ios App Use Click Select Two Variables from List and select the two variables. Click the Back link.

22 ios App Use The correlation coefficient is r = 0.61.

23 Linear Association It is very important to understand that r is a measure of the linear association of the two variables given. Example. In the following idealized situation, r is zero. However, there is a very pronounced non-linear (quadratic) relationship between the variables.