1. Theme 7 Correlation

2. Overview
Researchers are often interested in exploring whether or not two variables are associated
This lecture will consider
Scatter plots
Pearson correlation coefficient
Spearman correlation coefficient

3. Scatter Plots
To get a pictorial view of the relationship between two variables you can create a scatter plot
Each data point is marked on a graph
If there is a positive relationship between your variables then there will be a general trend from the bottom left to the top right
If there is a negative relationship between your variables then there will be a general trend from the top left to the bottom right

4. Line of best fit
Imagine drawing a line on your scatter plot which best fits the data
The closer your data points are in relation to this line, the stronger the relationship between your two variables
Similarly, the further your data points are in relation to this line, the weaker the relationship between your two variables
Remember the direction of the line indicates the direction of the relationship

5. Example of a Scatter Plot and Line of Best Fit
Figure A scattergram showing the relationship between maths ability and musical ability

6. Correlation Coefficients
A correlation coefficient indicates the amount of variance shared by two variables
A correlation of 1.0 shows that all of the variance can be explained
A correlation of zero shows that none of the variance can be explained
The closer to 1 (or -1) the stronger the relationship; the closer to 0 the weaker the relationship
Positive values indicate a positive relationship
Negative values indicate a negative relationship

7. Proportion of Variance Explained
One can calculate this by squaring the correlation coefficient and multiplying by 100
A coefficient of 1.0 indicates that 100% of the variance is explained (i.e. 1.0 ×1.0) × 100
A coefficient of 0.6 indicates that 36% of the variance is explained (i.e. 0.6 × 0.6) × 100
A coefficient of -0.6 also indicates that 36% of the variance is explained (i.e × -0.6) × 100

8. Types of Correlation There are two key tests of correlation
Pearson Product Moment Correlation Coefficient – This can be used to explore the linear relationship between two variables
Spearman Rho Correlation Coefficient – This is similar to the PPMCC but uses ranked scores rather than raw data

9. Raw Data
Case
Music score
Maths score 1 2 8 6 3 4 9 7 5 10

10. Scatter Plot

11. Interpretation
You should never report a correlation coefficient without examining the scatter plot for problems such as curved relationships or outliers
We would write of the scatter plot: ‘A scatter plot of the relationship between mathematical ability and musical ability was examined. There was no evidence of a curvilinear relationship or the undue influence of outliers.’

12. Steps on SPSS Data Analysis Output
In Variable View, ‘name’ the variables and enter the data
Analysis
For the correlation, select ‘Analyze’, ‘Correlate’ and ‘Bivariate...’.
Move appropriate variables to the ‘Variables:’ box,
Select the appropriate correlation and then ‘OK’.
For the scatter plot, select ‘Graphs’.
Then select ‘Chart Builder’ ‘OK’, ‘Scatter/Dot’ and move the ‘Simple Scatter’ figure to the box above.
Move appropriate variable names to the vertical and horizontal axes.
Output
The correlation table shows the correlation, its significance level and the sample size.

13. Pearson Output

14. Interpretation The correlation coefficient is −.90
The relationship is negative and statistically significant
In a report, we would write ‘There is a significant negative relationship between musical ability and mathematical ability, r(8) = −.90, p < Children with more musical ability have lower mathematical ability.’

15. Spearman Output

16. Interpretation The correlation coefficient is −.89
Again the relationship is negative and statistically significant
We would report this in the following way: ‘There is a statistically significant negative correlation between musical ability and mathematical ability, ρ(8) = −.89, p < Those with the highest musical ability tend to be those with the lowest mathematical ability and vice versa.’

17. Conclusion
When exploring the association between two variables it is important to first inspect the scatter plot
The Pearson or Spearman correlation coefficient is a statistical indicator of the relationship
They inform you about the strength and direction of the relationship as well as the amount of variance explained