1. Basic Statistics
Correlation

2. Relationships Associations
Var
Relationships
Associations
Var
Var
Var
Var

3. Independent variables
In Research
Information
Dependent variable X1 ? Y X2
COvary X3
Independent variables

4. The Concept of Correlation
Association or relationship between two variables
Co-relate? r
relation X Y
Covary---Go together

5. Patterns of Covariation
Zero or no correlation X Y
Correlation
Covary
Go together X Y X Y
Negative correlation
Positive correlation

6. Scatter plots allow us to visualize the relationships
The chief purpose of the scatter diagram is to study the nature of the relationship between two variables
Linear/curvilinear relationship
Direction of relationship
Magnitude (size) of relationship
Scatter Plots

7. Scatter Plot A Variable Y Variable X high low low high
Represents both the X and Y scores
Variable Y
Exact value
low
low
high
Variable X
An illustration of a perfect positive correlation

8. An illustration of a positive correlation
Scatter Plot B
high
Variable Y
Estimated Y value
low
low
high
Variable X
An illustration of a positive correlation

9. Scatter Plot C Variable Y Variable X high low low high
Exact value
low
low
high
Variable X
An illustration of a perfect negative correlation

10. An illustration of a negative correlation
Scatter Plot D
high
Variable Y
Estimated Y value
low
low
high
Variable X
An illustration of a negative correlation

11. Scatter Plot E Variable Y Variable X high low low high
An illustration of a zero correlation

12. An illustration of a curvilinear relationship
Scatter Plot F
high
Variable Y
low
low
high
Variable X
An illustration of a curvilinear relationship

13. The Measurement of Correlation
The Correlation Coefficient
The degree of correlation between two variables can be described by such terms as “strong,” ”low,” ”positive,” or “moderate,” but these terms are not very precise.
If a correlation coefficient is computed between two sets of scores, the relationship can be described more accurately.
A statistical summary of the degree and direction of relationship or association between two variables can be computed

14. Pearson’s Product-Moment Correlation Coefficient r
No Relationship
Negative correlation
Positive correlation
Direction of relationship: Sign (+ or –)
Magnitude: 0 through +1 or 0 through -1

15. The Pearson Product-Moment Correlation Coefficient
Recall that the formula for a variance is:
If we replaced the second X that was squared with a second variable, Y, it would be:
This is called a co-variance and is an index of the relationship between X and Y.

16. Conceptual Formula for Pearson r
This formula may be rewritten to reflect the actual method of calculation

17. Calculation of Pearson r
You should notice that this formula is merely the sum of squares for covariance divided by the square root of the product of the sum of squares for X and Y

18. Formulae for Sums of Squares
Therefore, the formula for calculating r may be rewritten as:

19. Calculation of r Using Sums of Squares

20. An Example
Suppose that a college statistics professor is interested in how the number of hours that a student spends studying is related to how many errors students make on the mid-term examination. To determine the relationship the professor collects the following data:

21. The Stats Professor’s Data
Student
Hours Studied (X)
Errors (Y) X2 Y2 XY 1 4 15 16
225 60 2 12
144 48 3 5 9 25 81 45 6 10 36
100 7 8 49 64 56 28 42 18
Total
X = 70
Y = 73
 X2 =546
Y2=695
XY=429

22. The Data Needed to Calculate the Sum of SquaresX Y X2 Y2 XY
Total
X = 70
Y = 73
 X2 =546
Y2=695
XY=429
= /10 = = 56
= /10 = = 162.1
= 429 – (70)(73)/10 = 429 – 511 = -82

23. Calculating the Correlation Coefficient
= -82 / √(56)(162.1) =
Thus, the correlation between hours studied and errors made on the mid-term examination is -0.86; indicating that more time spend studying is related to fewer errors on the mid-term examination. Hopefully an obvious, but now a statistical conclusion!

24. Pearson Product-Moment Correlation Coefficient r
perfect negative correlation
Zero correlation
Perfect positive correlation -1 +1
Negative correlation
Positive correlation

25. Numerical values
.73
- .35
Negative correlation Zero correlation Positive correlation
Perfect Strong Moderate

26. The Pearson r and Marginal Distribution
The marginal distribution of X is simply the distribution of the X’s; the marginal distribution of Y is the frequency distribution of the Y’s. Y
Bivariate relationship
Bivariate Normal Distribution X

27. Marginal distribution of X and Y are precisely the same shape.
Y variable
X variable

28. Interpreting r, the Correlation Coefficient
Recall that r includes two types of information:
The direction of the relationship (+ or -)
The magnitude of the relationship (0 to 1)
However, there is a more precise way to use the correlation coefficient, r, to interpret the magnitude of a relationship. That is, the square of the correlation coefficient or r2.
The square of r tells us what proportion of the variance of Y can be explained by X or vice versa.

29. How does correlation explain variance?
Suppose you wish to estimate Y for a given value of X.
high
How does correlation explain variance?
Explained
Variable Y
Free to Vary
49% of variance is explained
Explained
low
low
high
Variable X
An illustration of how the squared correlation accounts for variance in X, r = .7, r2 = .49

30. Now, let’s look at some correlation coefficients and their corresponding scatter plots.

31. What is your estimate of r?

32. Y X
What is your estimate of r?
r = -1.00
r2 = 1.00 = 100%

33. Y X
What is your estimate of r?
r = +1.00
r2 = 1.00 = 100%

34. What is your estimate of r?

35. What is your estimate of r?