1. Correlation Mechanics

2. Covariance The variance shared by two variables When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg) – cov (x,y) = pos. When X and Y move in opposite directions – cov (x,y) = neg. When no constant relationship – cov (x,y) = 0

3. Covariance Covariance is not easily interpreted on its own and cannot be compared across different scales of measurement Solution: standardize this measure Pearson’s r:

4. Covariance and correlation

5. Computational formula

6. Example IQ and GPA for 12 students IQ = 110,112,118,119,122,125,127,130,132,134, 136,138 GPA = 3.0,1.7,2.0,2.5,3.9,3.5,3.7,3.8,2.2,3.7,3.8,4.0 Sum of IQ = 1503 Sum of GPA = 37.8 Sum of IQ*GPA = 4786.1 Sd IQ = 9.226 Sd GPA =.833

7. Calculation

8. Calculate r Calculate the Pearson product moment correlation for the following data to see if there is a relationship between how fast one drives on the highway and scores on a measure of type A personality. Speed Test Score 6534 7545 7240 6137 68 39 å= 341 195 å() 2 = 23379 7671

9. Solution r =.833 Interpretation? Strong relationship but… p =.08 The observed p-value is really best used as a check on r in terms of a sampling distribution, rather than a determination of significance

10. Significance test for correlation A correlation is an effect size – i.e., standardized measure of amount of covariation R 2 = amount of variability seen in y that can be explained by the variability seen in x A 1- or 2-tailed significance test can be done in an effort to infer to a population – Typically though, correlations are considered descriptive The sig. test result will depend on the power of study (i.e., higher N, more likely to be sig) Alternatively look up r tables with df = N - 2

11. Significance test for correlation We will use the t-distribution as we have to use our sample data as estimates of the population parameters (pop. sd not known)

12. Test of the difference between two rs Since r has limits of +1, the larger the value of r, the more skewed its sampling distribution about the population  (rho)

13. Transformation of r to r' Fisher’s transformation will change r into one that is approximately normally distributed With standard error (s r‘ )

14. Now we can calculate a z value (not t because our standard error is not estimated by the sample statistic)

15. Test that r equals some specified value Now we’re talking about an interesting hypothesis test This is the equivalent to the one-sample z-test for a mean except now we are testing our sample r vs. some specified  Again we transform our r, and this time we will transform  also

16. Test the difference of two dependent rs Suppose we wanted to test to see if there is a difference between creativity and a person’s motivation scores (external and internal) r ec =.10 r ic =.59 r ei =.05 In other words, I’m testing to see if there is a difference between r ec and r ic

17. Solution Yikes! For N = 30, t = 2.21 t cv (27)= 2.052  Reject H 0

18. Confidence interval on    ' L <  ' <  ' U Then convert upper and lower  ' values to 