X & Y relationships Until now, X and Y have been looked at separately (e.g., in a t-test, the IV is manipulation or varied, and variation in the DV is examined) Now, each X and Y pair can be examined to the see to what extent they vary together. This is called “covariation.”

Covariation Examples of strong covariation –Height and weight –Heat and thirst –Fiber and regularity Examples of weak covariation –Height and regularity –Heat and height –Fiber and weight

Scatterplots We were introduced to scatterplots in the beforetime. Just to refresh your memory,... –strong; weak; positive; negative; linear; curvilinear (law of diminishing returns) A regression line summarizes the relationship between X and Y by minimizing the distances between the data points and the line.

Standardization As we learned earlier in the class, by converting raw scores into z-scores, we can compare across samples and populations. So, whereas a 115 on one test is not comparable to a 37 on another, a z- score of +.6 on the first test can be compared to a z-score of +1.2 on the other. So too, to compare X,Y relationships we can convert each raw score to a z-score. By doing this, we can, for example, figure out if the relationship between income and poverty is stronger than the relationship between education and poverty.

Standard scatterplots Now we can replace the raw scores on the x and y axes with z-scores. –For those of you following along in the text, see Figure 11.6 Now, if you divide the scatterplot into quadrants, with the horizontal line at z y =0, and the vertical line at z x =0, you can visually determine how many data points fall into each quadrant.

Z-score products For each X,Y pair, multiply its z-scores. Notice that for z-scores in the upper right and lower left quadrants, products will be positive. For z-score in the upper left and lower right quadrants, products will be negative. Now recall the Pearson correlation: -1 ≤ r ≤ 1

Connection between Pearson r and z-scores So, the further r is from 0, the stronger the relationship, or “correlation” between X,Y. Now look at the difference between the sum of the z-score products for weak vs. strong relationships. Large sum for strong; small for weak. But the sum is affected by the size of the dataset, so divide by n, resulting in:

Formula for Pearson correlation

Pearson r calculation XZ X YZ Y Z X Z Y µ X =3.50µ Y =25.83  (Z X Z Y )=1.31  X =1.71  Y =5.58 n=6 r=.22