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r xy

r xy. r xy. When two variables are correlated, we can predict a score on one variable from a score on the other The stronger the correlation, the more accurate our prediction will be. r xy. We need a measure of the “strength” of a correlation. r xy.

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r xy

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  1. rxy

  2. rxy • When two variables are correlated, we can predict a score on one variable from a score on the other • The stronger the correlation, the more accurate our prediction will be

  3. rxy • We need a measure of the “strength” of a correlation

  4. rxy • We need a number that gets bigger when big numbers are paired with big numbers and small numbers are paired with small numbers • We need a number that gets smaller when big numbers are paired with small numbers and small numbers are paired with big numbers

  5. 5’ 5’2 5’4 5’6 5’8 5’10 rxy • Remember the height/weight example: • Big number indicates this (strong positive correlation) f c d a b, e 100 110 120 130 140 150 b c d a e f

  6. 5’ 5’2 5’4 5’6 5’8 5’10 rxy • Remember the height/weight example: • Small number indicates this (strong negative correlation) f c d a b, e 100 110 120 130 140 150 d f e b a c

  7. rxy • Two sets of scores, xi and yi • What could we do?

  8. rxy • What could we do?

  9. rxy • What could we do? • When pairs are multiplied and the products are summed up: • Greatest when big numbers paired with big numbers and small numbers with small numbers • Least when small numbers are paired with big numbers and big numbers are paired with small numbers

  10. rxy • analogy: This gets you most money Pennies Quarters Loonies

  11. rxy • analogy:this gets you the least… Pennies Quarters Loonies

  12. rxy • analogy: Because: 3 x $1 plus 2 x $0.25 plus 1 x $0.01 is more than 1 x $1 plus 2 x $0.25 plus 3 x $0.01

  13. rxy • But there’s a problem Not a good measure because the value ultimately depends on n AND the size of the numbers

  14. rxy • Try this

  15. rxy • Try this Still not so good - doesn’t depend on n anymore, but does depend on size of x’s and y’s

  16. rxy • How about multiply deviation scores • comparing each variable relative to its respective mean

  17. rxy • Multiply deviation scores Now value depends on the spread of the data

  18. rxy • So standardize the scores

  19. rxy • This measures strength of correlation: = rxy =

  20. rxy • rxy ranges from -1.0 indicating a perfect negative correlation to +1.0 indicating a perfect positive correlation • an rxy of zero indicates no correlation whatsoever. Scores are random with respect to each other.

  21. rxy • rxy also has a geometric meaning

  22. rxy • rxy also has a geometric meaning • Recall that the mean of the zx and zy distributions is zero and each z-score is a deviation from the mean

  23. rxy • Each point lands in one of four quadrants point zx,zy zy zx

  24. rxy • notice that: rxy = both zx and zy are positive

  25. rxy • notice that: rxy = zx is negative and zy is positive

  26. rxy • notice that: rxy = zx is negative and zy is negative

  27. rxy • notice that: rxy = zx is positive and zy is negative

  28. rxy • So Thus if most points tend to fall around a line with a positive (45 degree) slope (I and III), the cross-products will tend to be positive I II IV III

  29. rxy • So Thus if most points tend to fall around a line with a positive (45 degree) slope (I and III), the cross-products will tend to be positive I II If most points tend to fall around a line with a negative slope (II and IV), the cross products will tend to be negative IV III

  30. rxy • So If the points were randomly scattered about, the negative and positive cross-products cancel

  31. Covariance • a related measure of the relationship between scores on two different variables is the covariance

  32. Covariance • notice that the variance (S2x) is the covariance between a variable and itself !

  33. Regression • If two variables are perfectly correlated (r = + or - 1.0) then one can exactly predict a score on one variable given a score on another

  34. Regression • For example: a university charges $250 registration fee plus $100 / credit

  35. Regression • tuition = $100(X) + $250 • where X is the number of credits • Notice this is a linear relationship (an equation of the form y = ax + b • a = $100/credit • b = $250 • x = number of credits

  36. Regression • Tuition as a function of credit hours is a straight line • There is a perfect correlation between credit hours and tuition • You could predict perfectly the tuition required given the number of credit hours

  37. Next Time • Regression - read chapter 8

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