Measures of the relationship between 2 variables Correlation

Measures of the relationship between 2 variables: Correlation Chapter 16

Correlation Coefficient Descriptive statistic l degree of relationship between 2 variables l 2 dependent variables l if we know value of 1 variable… how well can we predict value of other n Values of correlation coefficient l between -1 and +1 l 0 = no relationship ~ n

Correlation Coefficient

Scatter Diagrams Also called scatter plots l 1 variable: Y axis; other X axis l makes no difference which way l plot point at intersection of values l look for trends n e. g. , height vs shoe size ~ n

Scatter Diagrams 84 78 Height 72 66 60 6 7 8 9 Shoe size 10 11 12

Correlation Coefficient Values Sample statistic: r l population parameter: r (rho) l values: -1 < r < +1 n Scatter diagram characteristics l slope & width l determines value of r ~ n

Slope & value of r Determines sign l positive or negative n From lower left to upper right l positive ~ n

Slope & value of r n From upper left to lower right l negative ~

Width & value of r Magnitude of r l draw imaginary ellipse around most points n Narrow: r near -1 or +1 l strong relationship between variables n Wide: r near 0 l little or no relationship between variables ~ n

Width & value of r Weak relationship Strong negative relationship r near 0 r near -1 Weight 300 250 Weight 200 150 100 3 6 9 12 Chin ups 15 18 21

Measures of Correlation Several different measures l depends on level of measurement n Pearson’s r l interval/ratio n Spearman’s rs l ordinal and interval/ratio n Others for nominal and different combinations of levels of measurement ~ n

Factors that affect size of r n Nonlinear relationships l Pearson’s r does not detect more complex relationships l r near 0 ~ Y X

Factors that affect size of r n Range restriction l eliminate values from 1 or both variable l r is reduced l e. g. eliminate people under 72 inches ~

Correlation and Causation requires correlation, but. . . n Correlation does not imply causation! l Does not mean 1 variable causes changes in the other n e. g. # of household appliances negatively correlated with family size l appliances as effective birth control? n Changes may be caused by a third unknown variable ~ n