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This article explores the concepts of covariance and correlation, which measure the relationship between two variables. It explains how covariance can vary in sign based on the movement of the variables in relation to each other. The article includes practical calculations for Pearson’s correlation coefficient using sample data such as IQ and GPA to illustrate the relationship between two quantitative variables. Additionally, it discusses significance testing for correlation and how to determine if a correlation is meaningful through hypothesis testing and confidence intervals.
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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
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:
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
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 65 34 75 45 72 40 61 37 68 39 • = 341 195 • ()2 = 23379 7671
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
Significance test for correlation • A correlation is an effect size – i.e., standardized measure of amount of covariation • R2 = 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
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)
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)
Transformation of r to r' • Fisher’s transformation will change r into one that is approximately normally distributed • With standard error (sr‘)
Now we can calculate a z value (not t because our standard error is not estimated by the sample statistic)
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
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) • rec = .10 • ric = .59 • rei = .05 • In other words, I’m testing to see if there is a difference between rec and ric
Solution • Yikes! For N = 30, t = 2.21 • tcv (27)= 2.052 Reject H0
Confidence interval on • 'L < ' < 'U • Then convert upper and lower ' values to
