Looking at data: relationships - Correlation Lecture Unit 7 Objectives Correlation The correlation coefficient “r” r does not distinguish x and y r has no units r ranges from -1 to +1 Influential points The correlation coefficient "r" The correlation coefficient is a measure of the direction and strength of the linear relationship between 2 quantitative variables. It is calculated using the mean and the standard deviation of both the x and y variables. Time to swim: x = 35, sx = 0.7 Pulse rate: y = 140 sy = 9.5 Correlation can only be used to describe quantitative variables. Categorical variables don’t have means and standard deviations. Example: calculating correlation (x1, y1), (x2, y2), (x3, y3) (1, 3) (1.5, 6) (2.5, 8) x 1.67, y 5.67, sx .76, s y 2.52 r 11.67 35.67 1.51.67 65.67 2.51.67 85.67 (31)(.76)(2.52) .9538 Properties of Correlation r is a measure of the strength of the linear relationship between x and y. No units [like demand elasticity in economics (-infinity, 0)] -1 < r < 1 Values of r and scatterplots r near +1 r near -1 y r near 0 r near 0 y x x Properties (cont.) r has no unit Changing the units of variables does not change the correlation coefficient "r", because we get rid of all our units when we standardize (get z-scores). r = -0.75 z-score plot is the same for both plots r = -0.75 Properties (cont.) r ranges from -1 to+1 "r" quantifies the strength and direction of a linear relationship between 2 quantitative variables. Strength: how closely the points follow a straight line. Direction: is positive when individuals with higher X values tend to have higher values of Y. Properties of Correlation (cont.) r = -1 only if y = a + bx with slope b<0 r = +1 only if y = a + bx with slope b>0 10 20 y = 11 - x 8 y = 1 + 2x r=1 r = -1 15 6 Y 10 y 4 5 2 0 0 0 2 4 6 x 8 10 0 2 4 6 X 8 10 Properties (cont.) High correlation does not imply cause and effect CARROTS: Hidden terror in the produce department at your neighborhood grocery Everyone who ate carrots in 1920, if they are still alive, has severely wrinkled skin!!! Everyone who ate carrots in 1865 is now dead!!! 45 of 50 17 yr olds arrested in Raleigh for juvenile delinquency had eaten carrots in the 2 weeks prior to their arrest !!! Properties (cont.) Cause and Effect There is a strong positive correlation between the monetary damage caused by structural fires and the number of firemen present at the fire. (More firemen-more damage) Improper training? Will no firemen present result in the least amount of damage? Properties (cont.) Cause and Effect (1,2) (24,75) (1,0) (18,59) (9,9) (3,7) (5,35) (20,46) (1,0) (3,2) (22,57) x = fouls committed by player; y = points scored by same player The correlation is due to a third “lurking” variable – playing time r measures the strength of the linear relationship between x and y; it does not indicate cause and effect correlation r = .935 (x, y) = (fouls, points) Points 80 70 60 50 40 30 20 10 0 0 5 10 15 Fouls 20 25 30 Properties (cont.) r does not distinguish x & y The correlation coefficient, r, treats x and y symmetrically. r = -0.75 r = -0.75 "Time to swim" is the explanatory variable here, and belongs on the x axis. However, in either plot r is the same (r=-0.75). Outliers Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers. Just moving one point away from the general trend here decreases the correlation from -0.91 to -0.75 PROPERTIES (Summary) r is a measure of the strength of the linear relationship between x and y. No units [like demand elasticity in economics (-infinity, 0)] -1 < r < 1 r = -1 only if y = a + bx with slope b<0 r = +1 only if y = a + bx with slope b>0 correlation does not imply causation r does not distinguish between x and y r can be affected by outliers Correlation: Fuel Consumption vs Car Weight FUEL CONSUMP. (gal/100 miles) FUEL CONSUMPTION vs CAR WEIGHT r = .9766 7 6 5 4 3 2 1.5 2.5 3.5 WEIGHT (1000 lbs) 4.5 SAT Score vs Proportion of Seniors Taking SAT 88-89 SAT vs % Seniors Taking SAT r = -.868 88-89 SAT State Avg. IW ND 1075 1025 975 88-89 SAT 925 875 SC 825 0 20 40 DC NC 60 % Seniors that Took SAT 80 Extra Slides Part of the calculation involves finding z, the standardized score we used when working with the normal distribution. You DON'T want to do this by hand. Make sure you learn how to use your calculator! Standardization: Allows us to compare correlations between data sets where variables are measured in different units or when variables are different. For instance, we might want to compare the correlation between [swim time and pulse], with the correlation between [swim time and breathing rate]. When variability in one or both variables decreases, the correlation coefficient gets stronger ( closer to +1 or -1). Correlation only describes linear relationships No matter how strong the association, r does not describe curved relationships. Note: You can sometimes transform a non-linear association to a linear form, for instance by taking the logarithm. You can then calculate a correlation using the transformed data. Review examples 1) What is the explanatory variable? Describe the form, direction and strength of the relationship? Estimate r. r = 0.94 (in 1000’s) 2) If women always marry men 2 years older than themselves, what is the correlation of the ages between husband and wife? r=1 ageman = agewoman + 2 equation for a straight line Thought quiz on correlation 1. Why is there no distinction between explanatory and response variable in correlation? 2. Why do both variables have to be quantitative? 3. How does changing the units of one variable affect a correlation? 4. What is the effect of outliers on correlations? 5. Why doesn’t a tight fit to a horizontal line imply a strong correlation?