1. Overview Correlation Regression -Definition
-Deviation Score Formula, Z score formula
-Hypothesis Test
Regression
Intercept and Slope
Unstandardized Regression Line
Standardized Regression Line
Hypothesis Tests

2. Associations among Continuous Variables
Early developments
Sir Francis Galton was very interested in these issues in the 1880’s
Galton was the cousin of Darwin and thus he became interested in evolution and heredity
Galton had an intuition that the heredity could be understood in terms of deviations from means
He began to measure characteristics of plants and animals including people
Lots of normality
Galton found many characteristics that were normally distributed
Height in humans (by gender)
Weight in humans (by gender)
Length of animal bones
Weights of seeds of plants
Sweat peas
The next step that Galton took was to look at distributions of measurements of parents and offspring together
He first looked at sweet pea plants because the female plants can self fertilize
This makes examining the data easier because only one parent influences the characteristics of the offspring
He plotted mother seed sizes against daughter seed sizes

3. 3 characteristics of a relationship
Direction
Positive(+)
Negative (-)
Degree of association
Between –1 and 1
Absolute values signify strength
Form
Linear
Non-linear

4. Direction Positive Negative Large values of X = large values of Y,
Small values of X = small values of Y.
- e.g. IQ and SAT
Large values of X = small values of Y
Small values of X = large values of Y
-e.g. SPEED and ACCURACY

5. Degree of association
Strong (tight cloud)
Weak (diffuse cloud)

6. Form
Linear
Non- linear

7. Regression & Correlation
Early developments
Sir Francis Galton was very interested in these issues in the 1880’s
Galton was the cousin of Darwin and thus he became interested in evolution and heredity
Galton had an intuition that the heredity could be understood in terms of deviations from means
He began to measure characteristics of plants and animals including people
Lots of normality
Galton found many characteristics that were normally distributed
Height in humans (by gender)
Weight in humans (by gender)
Length of animal bones
Weights of seeds of plants
Sweat peas
The next step that Galton took was to look at distributions of measurements of parents and offspring together
He first looked at sweet pea plants because the female plants can self fertilize
This makes examining the data easier because only one parent influences the characteristics of the offspring
He plotted mother seed sizes against daughter seed sizes

8. What is the best fitting straight line? Regression Equation: Y = a + bX
How closely are the points clustered around the line? Pearson’s R

9. Correlation Early developments
Sir Francis Galton was very interested in these issues in the 1880’s
Galton was the cousin of Darwin and thus he became interested in evolution and heredity
Galton had an intuition that the heredity could be understood in terms of deviations from means
He began to measure characteristics of plants and animals including people
Lots of normality
Galton found many characteristics that were normally distributed
Height in humans (by gender)
Weight in humans (by gender)
Length of animal bones
Weights of seeds of plants
Sweat peas
The next step that Galton took was to look at distributions of measurements of parents and offspring together
He first looked at sweet pea plants because the female plants can self fertilize
This makes examining the data easier because only one parent influences the characteristics of the offspring
He plotted mother seed sizes against daughter seed sizes

10. Correlation - Definition
Correlation: a statistical technique that measures and describes the degree of linear relationship between two variables
Scatterplot
Obs X Y
A 1 1
B 1 3
C 3 2
D 4 5
E 6 4
F 7 5
Dataset Y X

11. Pearson’s r
A value ranging from to 1.00 indicating the strength and direction of the linear relationship.
Absolute value indicates strength
+/- indicates direction

13. The Logic of Correlation
MEAN of X
Below average on X
Above average on X
Above average on Y
Below average on Y
MEAN of Y
For a strong positive association, the cross-products will mostly be positive
Cross-Product =

14. The Logic of Correlation
MEAN of X
Below average on X
Above average on X
Above average on Y
Below average on Y
MEAN of Y
For a strong negative association, the cross-products will mostly be negative
Cross-Product =

15. The Logic of Correlation
MEAN of X
Below average on X
Above average on X
Above average on Y
Below average on Y
MEAN of Y
For a weak association, the cross-products will be mixed
Cross-Product =

16. Pearson’s r Deviation score formula SP (sum of products) =
∑ (X – X)(Y – Y)
Deviation score formula

17. Deviation Score Formula
Femur
Humerus A 38 41 B 56 63 C 59 70 D 64 72 E 74 84
mean
58.2
66.00
SSX
SSY SP

18. Deviation Score Formula
Femur
Humerus A 38 41
-20.2
-25
408.04
625
505 B 56 63
-2.2 -3
4.84 9
6.6 C 59 70
0.8 4
.64 16
3.2 D 64 72
5.8 6
33.64 36
34.8 E 74 84
15.8 18
249.64
324
284.4
mean
58.2
66.00
696.8
1010
834
SSX
SSY SP
= .99

19. Pearson’s r Deviation score formula SP (sum of products) =
Below average on X
Above average on X
Below Average on Y
Above average on Y
Below average on Y
SP (sum of products) =
∑ (X – X)(Y – Y)
Deviation score formula
For a strong positive association, the SP will be a big positive number

20. Pearson’s r Deviation score formula SP (sum of products) =
Below average on X
Above average on X
Below Average on Y
Above average on Y
Below average on Y
Deviation score formula
For a strong negative association, the SP will be a big negative number
∑ (X – X)(Y – Y)
SP (sum of products) =

21. Pearson’s r Deviation score formula SP (sum of products) =
Below average on X
Above average on X
Below Average on Y
Above average on Y
Below average on Y
Deviation score formula
For a weak association, the SP will be a small number (+ and – will cancel each other out)
∑ (X – X)(Y – Y)
SP (sum of products) =

22. Pearson’s r
Z score formula
Standardized cross-products

23. Z-score formula Femur Humerus ZX ZY ZXZY A 38 41 B 56 63 C 59 70 D 6472 E 74 84
mean
58.2
66.00 s
13.20
15.89

24. Z-score formula Femur Humerus ZX ZY ZXZY A 38 41 -1.530 -1.573 B 56 63
-0.167
-0.189 C 59 70
0.061
0.252 D 64 72
0.439
0.378 E 74 84
1.197
1.133
mean
58.2
66.00 s
13.20
15.89

25. Z-score formula Femur Humerus ZX ZY ZXZY A 38 41 -1.530 -1.573 2.408 B56 63
-0.167
-0.189
0.031 C 59 70
0.061
0.252
0.015 D 64 72
0.439
0.378
0.166 E 74 84
1.197
1.133
1.356
mean
58.2
66.00
∑=3.976 s
13.20
15.89
r = .99

26. Formulas for R
Z score formula
Deviations formula

27. Interpretation of R
A measure of strength of association: how closely do the points cluster around a line?
A measure of the direction of association: is it positive or negative?

28. Interpretation of R
r = very small association, not usually reliable
r = small association
r = typical size for personality and social studies
r = moderate association
r = you are a research rock star
r = hmm, are you for real?

29. Interpretation of R-squared
The amount of covariation compared to the amount of total variation
“The percent of total variance that is shared variance”
E.g. “If r = .80, then X explains 64% of the variability in Y” (and vice versa)

30. Hypothesis testing with r
Hypotheses
H0:  = 0
HA :  ≠ 0
Test statistic = r
Or just use table E.2 to find critical values of r

31. Practice alcohol tobacco A 6.47 4.03 B 6.13 3.76 C 6.19 3.77 D 4.89
3.34 E
5.63
3.47
mean
SSX
SSY SP

32. Practice alcohol tobacco A 6.47 4.03 B 6.13 3.76 C 6.19 3.77 D 4.89
3.34 E
5.63
3.47
mean
1.55
.30
.64
SSX
SSY SP

33. Properties of R
A standardized statistic – will not change if you change the units of X or Y. (bc based on z-scores)
The same whether X is correlated with Y or vice versa
Fairly unstable with small n
Vulnerable to outliers
Has a skewed distribution

34. Linear Regression

35. Linear Regression But how do we describe the line?
If two variables are linearly related it is possible to develop a simple equation to predict one variable from the other
The outcome variable is designated the Y variable, and the predictor variable is designated the X variable
E.g. centigrade to Fahrenheit:
F = C
this formula gives a specific straight line

36. The Linear Equation F = 32 + 1.8(C) General form is Y = a + bX
The prediction equation: Y’ = a+ bX
Where
a = intercept
b = slope
X = the predictor
Y = the criterion
a and b are constants
in a given line;
X and Y change

37. The Linear Equation F = 32 + 1.8(C) General form is Y = a + bX
The prediction equation: Y’ = a + bX
Where
a = intercept
b = slope
X = the predictor
Y = the criterion
Different b’s…

38. The Linear Equation F = 32 + 1.8(C) General form is Y = a + bX
The prediction equation: Y’ = a + bX
Where
a = intercept
b = slope
X = the predictor
Y = the criterion
Different a’s…

39. The Linear Equation F = 32 + 1.8(C) General form is Y = a + bX
The prediction equation: Y’ = a + bX
Where
a = intercept
b = slope
X = the predictor
Y = the criterion
Different a’s and b’s …

40. Slope and Intercept Equation of the line
The slope b: the amount of change in y with one unit change in x
The intercept a: the value of y when x is zero

41. Slope and Intercept Equation of the line The slope The intercept
The slope is influenced by r, but is not the same as r

42. When there is no linear association (r = 0), the regression line is horizontal. b=0.
and our best estimate of age is 29.5 at all heights.

43. When the correlation is perfect (r = ± 1
When the correlation is perfect (r = ± 1.00), all the points fall along a straight line with a slope

44. When there is some linear association (0<|r|<1), the regression line fits as close to the points as possible and has a slope

45. Where did this line come from?
It is a straight line which is drawn through a scatterplot, to summarize the relationship between X and Y
It is the line that minimizes the squared deviations (Y’ – Y)2
We call these vertical deviations “residuals”

46. Regression lines
Minimizing the squared vertical distances, or “residuals”

47. Unstandardized Regression Line
Equation of the line
The slope
The intercept

48. Properties of b (slope)
An unstandardized statistic – will change if you change the units of X or Y.
Depends on whether Y is regressed on X or vice versa

49. Standardized Regression Line
Equation of the line
The slope
The intercept
A person 1 stdev above the mean on height would be how many stdevs above the mean on weight?

50. Properties of β (standardized slope)
A standardized statistic – will not change if you change the units of X or Y.
Is equal to r, in simple linear regression

51. Exercise X Y 1 3 4 5 6 9 7 Calculate:
b =
a =
β =
Write the regression equation: Write the standardized equation:
Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5

52. Exercise X Y 1 3 4 5 6 9 7 Calculate:
b = .375
a = 3.125
β = .866
Write the regression equation: Write the standardized equation:
Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5

53. Regression Coefficients Table
Predictor
Unstandardized Coefficient
Standard error
Standardized Coefficient t
sig
Intercept a
SEa -
Variable X b
SEb

54. Summary Correlation: Pearson’s r Unstandardized Regression Line

55. Exercise in Excel X Y 1 -1.5 2 -3 4 -4 7 -2 9 -6 Calculate:2 -3 4 -4 7 -2 9 -6
Calculate:
r =
b =
a =
β =
Write the regression equation: Write the standardized equation:
Sketch the scatterplot and regression line
Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5