1. Correlation

2. What is a correlation?
A correlation examines the relationship between two measured variables.
No manipulation by the experimenter/just observed.
E.g., Look at relationship between height and weight.
You can correlate any two variables as long as they are numerical (no nominal variables)
Is there a relationship between the height and weight of the students in this room?
Of course! Taller students tend to weigh more.

3. 1) Strength of Relationships
2 aspects of the relationship: Strength and Direction.
The relationship between any 2 variables is rarely a perfect correlation.
Perfect correlation: OR –1.00
strongest possible relationship
Tough to find.
No correlation: 0.00 (no relationship).
E.g, height and social security #.

4. 2) Direction of the Relationship
Positive relationship – Variables change in the same direction.
As X is increasing, Y is increasing
As X is decreasing, Y is decreasing
E.g., As height increases, so does weight.
Negative relationship – Variables change in opposite directions.
As X is increasing, Y is decreasing
As X is decreasing, Y is increasing
E.g., As TV time increases, grades decrease
Indicated by
sign; (+) or (-).

5. Positive Correlation–as x increases, y increases
Scatter Plots and Types of Correlation
x = SAT score
y = GPA
4.00
3.75
3.50
3.25
GPA
3.00
2.75
2.50
2.25
2.00
1.75
1.50
300
350
400
450
500
550
600
650
700
750
800
Math SAT
Positive Correlation–as x increases, y increases

6. Negative Correlation–as x increases, y decreases
Scatter Plots and Types of Correlation
x = hours of training
y = number of accidents 60 50 40
Accidents 30 20 10 2 4 6 8 10 12 14 16 18 20
Hours of Training
Negative Correlation–as x increases, y decreases

7. Scatter Plots and Types of Correlation
x = height y = IQ
160
150
140
130 IQ
120
110
100 90 80 60 64 68 72 76 80
Height
No linear correlation

8. Correlation Coefficient Interpretation
Range
Strength of
Relationship
Very Low
Low
Moderate
High Moderate
Very High

9. Direction
Positive relationship
r = +.80
Weight
Height

10. Direction Negative relationship r = -.80 TV watching per week
Exam score

11. Interpreting correlations - Summary
Absolute size shows strength of relationship
The higher the absolute number, the stronger the relationship
A correlation of -.80 is reflects as powerful a relationship as one of +.80
A correlation of 0.00 means no relationship
E.g., Can’t predict GPA from ID number
All correlations range from to +1.00

12. Strength of relationship
Perfect Correlation
r = -1.0
TV watching per
week
Exam score

13. Strength of relationship
Strong Correlation
r = + 0.8
Quality of Breakfast
Exam score

14. Strength of relationship
Moderate Correlation
r = + 0.4
Shoe Size
Weight

15. Strength of relationship
Weak Correlation (negative)
r = - 0.2
Shoe Size
Weight

16. Strength of relationship
No Correlation (horizontal line)
r = 0.0 IQ
Height

17. One more example Exam Grade +.80 Amount of Study Time -.60 .00
# of classes missed
Social Security Number

18. More examples Positive relationships: Negative relationships:
water consumption and temperature.
study time and grades.
time spent in jail to severity of offense.
What else??
Negative relationships:
alcohol consumption and driving ability.
# of hateful remarks and # of friends.
What else??
Why used: 1) Prediction; 2) Validity (does something
measure what it’s suppose to measure; 3) Reliability
(does something produce a consistent score).
*** Easier to do than experiments ***

19. Pearson correlation coefficient
r = the Pearson coefficient
r measures the amount that the two variables (X and Y) vary together (i.e., covary) taking into account how much they vary apart
Pearson’s r is the most common correlation coefficient; there are others.

20. Computing the Pearson correlation coefficient
To put it another way: Or

21. Sum of Products of Deviations
Measuring X and Y individually (the denominator):
compute the sums of squares for each variable
Measuring X and Y together: Sum of Products
Definitional formula
Computational formula
n is the number of (X, Y) pairs

22. Correlation Coefficent:
the equation for Pearson’s r:
expanded form:

23. Example
What is the correlation between study time and test score:

24. Calculating values to find the SS and SP:

25. Calculating SS and SP

26. Calculating r

27. Limitations of Pearson’s r
Correlation does not mean causation!!
Third Variable problem – there’s always the possibility of a third factor causing the relationship.
E.g., Moderate, positive relationship between viewing violent TV and engaging in aggressive behaviors.

28. Possibilities Tendency to engage Viewing violent
in aggressive behaviors
Viewing violent
television
Tendency to engage
in aggressive behaviors
Viewing violent
television
Tendency to engage
in aggressive behaviors
A third factor;
EX. genetic tendency
to like violence
Viewing violent
television

29. Limitations of Pearson’s r
Correlation does not mean causation
Restriction of range
Restricted range of measured values can lead to inaccurate conclusions about the data

30. Limitations of Pearson’s r
Outliers (extreme scores)
Scores with extreme X and/or Y value can drastically effect Pearson’s r
Ambiguity of the strength of the relationship
Pearson r does not give a directly interpretable strength of the relationship between X and Y
5. Interval or ratio data.

31. Coefficient of Determination
r2 = percentage of variance in Y accounted for by X
Calculated by squaring r (Pearson correlational coefficient)
Ranges from 0 to 1 (positive only)
This number is a meaningful proportion (unlike the Pearson’s r).

32. Coefficient of Determination: An example
What percentage of variance is accounted for in Y by X with a Pearson r = 0.50?
The r2 = (0.50)2 = 0.25 = 25%
The number is always positive