Jaymie Ticknor Quantitative Methods 2317 Sect 001 11 16 and 18 April 2014 Chapter 11 Correlation Lecture 11 Relationship or association strong or weak between two sets or groups of scores Positive Correlation two variables go in the same direction up up down Negative Correlation two variables go in opposite directions up down down Scatter diagram determine the axes the range of values to use for each variable and mark them on the axes and mark a dot for each pair of scores Patterns of Correlation Linear Correlation pattern in scatter diagram follows a straight line Curvilinear Correlation pattern in scatter diagram does not follow a straight down up line No Correlation no systematic relationship between two variables r 0 Strength of Correlation how closely are the scores associated with each other Dots closer to line stronger relationship Correlation Coefficient goal is to determine the extent to which high scores on one variable are associated with high or low scores on the second variable use deviation scores to determine whether a score is high or low X Mx and Y My High scores will have positive deviation scores low scores will have negative Multiply the deviation scores together to get a product of deviation scores Positive Correlation High scores with high scores r Low scores with low scores Multiply a positive deviation score by a positive deviation score you will get a deviations scores positive product Multiply a negative deviation score by a negative deviation score you will also get a positive product If you sum these products you will get a big positive number Negative Correlation High scores with low scores r Low scores with high scores If you multiply a positive deviation score by a negative deviation score you will If you multiply a negative deviation score by a positive deviation score you will get a negative product also get a negative product If you sum these products you will get a big negative number No Correlation for some high scores go with high scores and low scores go with low scores For others high scores go with low scores and low scores with high scores Some positive products some negative products If you sum these products you will get about zero In general the larger the sum of the products is the stronger the correlation however the more people scores the larger the number will be the more spread out the scale is the larger the number will be Need some way to standardize the correlation coefficient Solution divide the sum of the products of the deviation by a correction number SSX SSY This corrects for both the number of people and how spread out the scale is r X MX Y MY SSX SSY Correlation coefficient ranges from 1 to 1 Positive correlation coefficient positive correlation Negative correlation coefficient negative correlation Correlation coefficient of 0 no correlation Value of correlation coefficient strength of linear correlation 85 is larger than 40 68 is larger than 34 Finding Correlation Coefficient Step 1 Change the scores for each variable to deviation scores X Mx and Y Step 2 Figure the product of the deviation scores for each pair of scores X Mx Step 3 Sum the products of the deviation scores X Mx Y My Step 4 For each variable square each deviation score X Mx 2 and Y My 2 Step 5 Sum the squared deviation scores for each variable X Mx 2 and Y Step 6 Multiply the two sums of squared deviations and take the square root to get correction number Step 7 Divide the sum of the products of deviation scores by the correction My Y My My 2 number Significance of Correlation Coefficient procedure is similar to the t test do 5 steps calculate t score cut off t score etc df N 2 t r 1 r2 N 2 Assumptions of the Correlation Coefficient sampling distribution is normally distributed equal symmetric distribution of each variable at each point of the other variable Y values are equally far away from mean at each X value Correlation does NOT imply causation Correlation and Causation if X and Y are correlated there are 3 possible directions of causality X could be causing Y Y could be causing X some third variable could be causing both X and Y Sometimes you can rule out certain directions of causation the presumed cause must precede the presumed effect in time design a true experiment Factors that can Distort a Correlation Coefficient Restriction of Range only have a limited range of values on one of the variables Unreliability of Measurement the less reliable a measure is the lower the correlation coefficient will be increased error variance Outliers scores that are very different from the rest of the data can have an especially large influence on the correlation coefficient Effect Size for Correlation correlation coefficient is itself a measure of effect size Small Medium Large r2 percentage measure of proportion of variance accounted for example the r 10 r 30 r 50 variance in quiz grades is accounted for by the amount of time one spends studying Controversy What is a Large Correlation it depends a small difference can be meaningful