1 ES9 Chapters 4 ~ Scatterplots & Correlation

2 ES9 Chapter Goals To be able to present bivariate data in tabular and graphic form To gain an understanding of the distinction between the basic purposes of correlation analysis and regression analysis To become familiar with the ideas of descriptive presentation

3 ES9 Three combinations of variable types: 1.Both variables are qualitative (attribute) 2.One variable is qualitative (attribute) and the other is quantitative (numerical) 3.Both variables are quantitative (both numerical) Bivariate Data Bivariate Data: Consists of the values of two different response variables that are obtained from the same population of interest

4 ES9 Two Quantitative Variables 1.Expressed as ordered pairs: (x, y) 2.x: input variable, independent variable y: output variable, dependent variable Scatter Diagram: A plot of all the ordered pairs of bivariate data on a coordinate axis system. The input variable x is plotted on the horizontal axis, and the output variable y is plotted on the vertical axis. Note:Use scales so that the range of the y-values is equal to or slightly less than the range of the x-values. This creates a window that is approximately square.

5 ES9 Example:In a study involving children’s fear related to being hospitalized, the age and the score each child made on the Child Medical Fear Scale (CMFS) are given in the table below: Example Construct a scatter diagram for this data

6 ES9 age = input variable, CMFS = output variable Solution Child Medical Fear Scale CMFS Age

7 ES9 Linear Correlation Measures the strength of a linear relationship between two variables –As x increases, no definite shift in y: no correlation –As x increases, a definite shift in y: correlation –Positive correlation: x increases, y increases –Negative correlation: x increases, y decreases –If the ordered pairs follow a straight-line path: linear correlation

8 ES9 As x increases, there is no definite shift in y: Example: No Correlation

9 ES9 As x increases, y also increases: Example: Positive Correlation

10 ES9 As x increases, y decreases: Example: Negative Correlation

11 ES9 Please Note  Perfect positive correlation: all the points lie along a line with positive slope  Perfect negative correlation: all the points lie along a line with negative slope  If the points lie along a horizontal or vertical line: no correlation  If the points exhibit some other nonlinear pattern: no linear relationship, no correlation  Need some way to measure correlation

12 ES9 Bivariate Data Coefficient of Linear Correlation: r, measures the strength of the linear relationship between two variables Pearson’s Product Moment Formula: Notes: r = +1: perfect positive correlation r = -1 : perfect negative correlation

13 ES9 Alternate Formula for r SS“sum of squares for()xx”   x x n    2 2 SS“sum of squares for()yy”    y y n    2 2 SS“sum of squares for()xyxy”  xy xy n   

14 ES9 Example:The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear correlation coefficient: Example

15 ES9 Completing the Calculation for r r xy xy    SS () ()(). (.)(.)

16 ES9 Please Note  r is usually rounded to the nearest hundredth  r close to 0: little or no linear correlation  As the magnitude of r increases, towards -1 or +1, there is an increasingly stronger linear correlation between the two variables  Method of estimating r based on the scatter diagram. Window should be approximately square. Useful for checking calculations.