Correlation Coefficients

Correlation coefficient, symbolized as r, is a numerical summary of a bivariate relationship and can range from Ė1.00 to +1.00. Any r that is positive indicates a direct or positive relationship between two measured variables. Negative r indicates indirect or inverse relationship.

Before discussing the correlation coefficient, it is important to screen your data for possible outliers. It is possible to have an outlier when analyzing bivariate relationships, but when examining each variable separately, not have an outlier. Graphing your variables using a scatter diagram will provide a visual examination of your data for outliers.

 

Interpreting the r usually requires knowledge in oneís field. What may be a large r in one field may be interpreted as low in another. For beginning researchers it is helpful to have a starting point for making judgments about the r. Here is a scale provided by Salkin.

.8 to 1.0   or  -.8 to -1.0 (very strong relationship)
.6 to .8 (strong relationship)
.4 to .6 (moderate relationship)
.2 to .4 (weak relationship)
.0 to .2 (weak or no relationship

In the following examples we will be calculating Pearsonís Product Moment Correlation Coefficient. There are many other types of correlation coefficients that should be used when your data meets certain levels of measurement.

 

 

Nominal
(Dichotomous)

Ordinal

Interval
or Ratio

Nominal
(Dichotomous)

Phi

Rank
Biserial

Point
Biserial

Ordinal

Rank
Biserial

Spearman
Rank Corr.

 

Interval
or Ratio

Point
Biserial

 

Pearson's r

Table made by Gene Glass

Pearsonís product-moment correlation coefficient requires each of the variables to be quantitative and continuous in nature. Spearmanís rho (rs or r)is similar to the Pearsonís product-moment in that the two variables are quantitative; however, the variables are measured in a way as being ranks instead of a continuous in nature.

Correlation does not deal with the question of whether two means are similar or different, just examining the rank order of the two variables. Significant correlation coefficients do not indicate causal relationships. Significant correlation coefficients are necessary but not sufficient to indicate causal relationships.

 

Illustrated Example #1: Correlation Coefficients in Text

 A teacher wanted to know if there was a relationship between her students IQ scores and achievement level in reading.

  [Download Data]

IQ

Reading Achievement

99

50

110

57

102

51

85

41

95

50

104

30

122

59

98

54

87

49

111

53

 

SPSS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p value

 

Correlation Coefficient

 

 

Write Up

 

A correlation coefficient was used to examine the relationship between IQ and reading achievement scores. A scatter diagram was examined and one outlier was detected; that is, one student had an average IQ score and a very low reading score. This student was deleted from the correlation analysis. For the nine students in the class, the correlation between IQ and reading achievement was high (r=.87).

 

Illustrated Example #2: Correlation Coefficients in Matrix

A counselor working with a group of caregivers of patients living with a terminal illness is interested in forming a support group to share experiences and therefore reduce the sense of isolation often associated with catastrophic illness. The counselor, working with hospital staff, administers a depression and anxiety inventory to each caregiver who has volunteered for the program. The counselor feels that knowing the levels of depression, anxiety, and stress within the group will help in the design of an effective intervention program. The scores obtained from the administration of the two inventories are given below.

[Download Data]

Sex

Age

Anxiety

Depress

2
2
1
2
2
1
1
1
2
1
1
1
1
1
2
2
1
1
1

32.00
46.00
53.00
62.00
23.00
58.00
47.00
39.00
42.00
59.00
34.00
67.00
61.00
29.00
51.00
34.00
71.00
45.00
54.00

22.00
12.00
68.00
10.00
5.00
53.00
44.00
37.00
.00
21.00
64.00
33.00
55.00
18.00
3.00
4.00
11.00
13.00
7.00

16.00
8.00
33.00
6.00
5.00
24.00
18.00
17.00
2.00
14.00
31.00
17.00
30.00
13.00
3.00
4.00
7.00
9.00
5.00

Note. Sex (1=female) (2=male)

Write Up

There was a large positive correlations between depression and anxiety (r = .98, p <  .01). This would indicate that individuals with high depression scores tend to have high anxiety scores. There was a moderate inverse relationship between depression and physical activity (r=-0.52, p<0.01). This would indicate that individuals with high depression tend to engage in less physical activity. There was no relationship between age and anxiety (r=0.15, p>0.05) and age and depression (r=0.12, p>0.05).