1. Chapter 5: Correlation Coefficients
CRIM 483
Chapter 5: Correlation Coefficients

2. Correlation Coefficients
Correlation coefficient=numerical index that reflects the linear relationship between two variables for in the dataset
Range to +1.00
Known as a bivariate correlation
Statistic often used to measure correlations=Pearson r correlation (rxy)
Use with continuous variables

3. Descriptions Continued
Correlations can indicate two types of relationships:
Direct/positive correlation: both variables change in the same direction
Indirect/negative: variables change in different directions
Ultimately, the correlation coefficient represents the amount of variability shared between two variables

5. Correlation Coefficient Formula
Formula for Correlation Coefficient
n∑XY-∑X∑Y
√([n∑X2-(∑X)2][n∑Y2-(∑Y)2]
Example from Book (pg. 81)
(10*247)-(54*43)
√ [(10*320)-(54)2] * [(10*201)-(43)2]

6. Computing the Correlation Coefficient
STEP 1: CALCULATE KEY TERMS X Y X2 Y2 XY 2 3 4 9 6 16 8 5 25 36 30 12 7 49 42 64 40 20 24 35 54 43
320
201
247
STEP 2: (10*247)-(54*43)=
n∑XY-∑X∑Y=
148
STEP 3: (10*320)-(54*54)=
n∑X2-(∑X)2=
284
STEP 4: (10*201)-(43*43)=
n∑Y2-(∑Y)2=
161
STEP 5: √284*161=
√([n∑X2-(∑X)2][n∑Y2-(∑Y)2]=
STEP 6: 148/ =
n∑XY-∑X∑Y______
0.692

8. Graphing Data

9. Perfect Direct or Positive Relationship

10. Strong Direct or Positive Relationship

11. Strong Indirect or Negative Relationship

12. Things to Remember
The absolute value of the correlation coefficient indicates strength:
.70 and -.70 are equal in strength, but the relationship is in a different direction
.50 is a weaker correlation than -.70
There will be no correlation in the following cases
When two variables do not share variance
Examining the relationship between education and age when all subjects are the same age (no variance in age)
When the range of one variable is constrained
Examining reading comprehension and grades among high-achieving children

13. Coefficient of Determination
Coefficient of determination (CD)=the percentage of variance in one variable that is accounted for by the variance in the other variable
The more two variables share in common, the more related they will be—they share variability
CD=rstudying*GPA2
rstudying*GPA=.7  rstudying*GPA2 =.49 or 49% of GPA variance is explained by studying time
Conversely, 51% of GPA is not explained by studying time=coefficient of alienation or coefficient of nondetermination…amount of x not explained by y
CD helps to determine the meaningfulness of the relationship

15. Association v. Causation
Be careful when interpreting correlations
Bivariate relationships can lead to spurious conclusions
For example, ice cream sales are correlated highly with crime
Does this mean that increased ice cream consumption causes crime?
Correlations do not account for other variables that may be related to both factors examined
Pearson’s r only one type of correlation statistic—others are found in Table 5.3

17. Chapter 13: Correlation Coefficients and Statistical Significance
CRIM 483
Chapter 13: Correlation Coefficients and Statistical Significance

18. Example
You want to test the relationship between the quality of marriage and the quality of parent-child relationships
Once you have selected the test statistic, follow these steps:
State the null hypothesis and research hypothesis
What is the null?
What is the research hypothesis?
Set the level of risk for statistical significance:__%
Select the appropriate test statistic

19. Deciding What Statistic To Use

20. Testing Differences/Relationships
Compute the test statistic value using the formula on page 81
What is the computed correlation coefficient? The coefficient IS your test statistic.
To determine significance, you will need the Degrees of Freedom, which is DF = n-2
Degrees of freedom represents a measure of the number of independent observations in the sample that can be used to estimate the standard deviation of the parent population
NOTE: A t-test distribution (similar to a z-score) is usually computed—in this case, the text makes it a little easier for you
Determine the critical value—the value needed to reject the null hypothesis
Turn to Table B4 in the appendix
What is the critical value in the this table for .05
Since it is non-directional, you must use the two-tailed figures

21. Testing, Continued Compare the obtained value to the critical value
What is the comparison?
Which is a better reflection of this comparison, #7 or #8?
If obtained value > critical value, reject the null
Observed differences/relationships are not due to chance
If obtained value < critical value, do not reject the null
Observed differences/relationships are due to chance
What is the final answer to your research question using a correlation coefficient?

22. Interpretation: Always Remember
Cause v. Associations
Correlation coefficients are only bivariate
They do not control for any other variables nor do they determine which variable came first
Thus, they are limited in their ability to signify cause
Significance v. Meaningfulness
A test statistic can be significant but it may not be very meaningful
For instance, .393 was significant in this example, but the coefficient of determination shows that only 15.4% of the variance is shared
Thus, the correlation leaves a lot of room for doubt and speculation for what other factors are more important

23. Example #2 Using SPSS: Ch. 13 Data Set 1
Figure Chapter 13 Data Set 1

25. Figure 13.4. SPSS Output for testing the Significance of the Correlation Coefficient

26. Exercise #2, Page 239: Chapter 13 Data Set 2