CRIM 483

Chapter 5: Correlation Coefficients

Correlation Coefficients

• Correlation coefficient=numerical index that reflects the linear relationship between two variables for in the dataset– Range -1.00 to +1.00– Known as a bivariate correlation– Statistic often used to measure

correlations=Pearson r correlation (rxy)

– Use with continuous variables

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

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]

Computing the Correlation Coefficient

STEP 1: CALCULATE KEY TERMS

X Y X2 Y2 XY

2 3 4 9 6

4 2 16 4 8

5 6 25 36 30

6 5 36 25 30

4 3 16 9 12

7 6 49 36 42

8 5 64 25 40

5 4 25 16 20

6 4 36 16 24

7 5 49 25 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]= 213.832

STEP 6: 148/213.832= n∑XY-∑X∑Y______

√([n∑X2-(∑X)2][n∑Y2-(∑Y)2]=

0.692

Graphing Data

Perfect Direct or Positive Relationship

Strong Direct or Positive Relationship

Strong Indirect or Negative Relationship

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

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*GPA

2

– 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

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

CRIM 483

Chapter 13: Correlation Coefficients and Statistical Significance

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:

1. State the null hypothesis and research hypothesis• What is the null?• What is the research hypothesis?

2. Set the level of risk for statistical significance:__%3. Select the appropriate test statistic

Deciding What Statistic To Use

Testing Differences/Relationships

4. 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

5. 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

Testing, Continued4. Compare the obtained value to the critical value

• What is the comparison?• Which is a better reflection of this comparison, #7 or #8?

5. If obtained value > critical value, reject the null Observed differences/relationships are not due to chance

8. 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?

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

Figure 13.2. Chapter 13 Data Set 1

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

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

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