Lesson Scatter Diagrams and Correlation

Objectives Draw and interpret scatter diagrams Understand the properties of the linear correlation coefficient Compute and interpret the linear correlation coefficient

Vocabulary Response Variable – variable whose value can be explained by the value of the explanatory or predictor variable Predictor Variable – independent variable; explains the response variable variability Lurking Variable – variable that may affect the response variable, but is excluded from the analysis Positively Associated – if predictor variable goes up, then the response variable goes up (or vice versa) Negatively Associated – if predictor variable goes up, then the response variable goes down (or vice versa)

Scatter Diagram Shows 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. Explanatory variable plotted on horizontal axis and the response variable plotted on vertical axis. Do not connect the points when drawing a scatter diagram.

TI-83 Instructions for Scatter Plots Enter explanatory variable in L1 Enter response variable in L2 Press 2 nd y= for StatPlot, select 1: Plot1 Turn plot1 on by highlighting ON and enter Highlight the scatter plot icon and enter Press ZOOM and select 9: ZoomStat

Response Explanatory Response Explanatory Response Explanatory Response Explanatory Response Explanatory Nonlinear No RelationPositively Associated Negatively Associated Scatter Diagrams

Where x is the sample mean of the explanatory variable s x is the sample standard deviation for x y is the sample mean of the response variable s y is the sample standard deviation for y n is the number of individuals in the sample Equivalent Formula for r (x i – x) s x (y i – y) s y n – 1 r = Σ x i y i x i y i – n Σ Σ Σ √ x i x i 2 – n Σ ( Σ ) 2 y i y i 2 – n Σ ( Σ ) 2 = s xy √s xx √s yy Linear Correlation Coefficient, r

Properties of the Linear Correlation Coefficient The linear correlation coefficient is always between -1 and 1 If r = 1, then the variables have a perfect positive linear relation If r = -1, then the variables have a perfect negative linear relation The closer r is to 1, then the stronger the evidence for a positive linear relation The closer r is to -1, then the stronger the evidence for a negative linear relation If r is close to zero, then there is little evidence of a linear relation between the two variables. R close to zero does not mean that there is no relation between the two variables The linear correlation coefficient is a unitless measure of association

TI-83 Instructions for Correlation Coefficient With explanatory variable in L1 and response variable in L2 Turn diagnostics on by –Go to catalog (2 nd 0) –Scroll down and when diagnosticOn is highlighted, hit enter twice Press STAT, highlight CALC and select 4: LinReg (ax + b) and hit enter twice Read r value (last line)

Example Draw a scatter plot of the above data Compute the correlation coefficient x y r = y x

Observational Data If bivariate (two variable) data are observational, then we cannot conclude that any relation between the explanatory and response variable are due to cause and effect Observational versus Experimental Data

Summary and Homework Summary –Correlation between two variables can be described with both visual and numeric methods –Visual methods Scatter diagrams Analogous to histograms for single variables –Numeric methods Linear correlation coefficient Analogous to mean and variance for single variables –Care should be taken in the interpretation of linear correlation (nonlinearity and causation) Homework –pg 203 – 211; 4, 5, 11-16, 27, 38, 42

Homework Answers 12: Linear, negative 14: nonlinear (power function perhaps) 16 a) IV b) III c) I d) II 38: Example problem 42: No linear relationship, but not no releationship. Power function relationship (negative parabola)