Statistics · Topic 6 of 13
Correlation
Theory
Correlation is used when we want to explore the relationship between two numerical variables (bivariate data). At this stage, no variables are being manipulated; we are simply observing how they interact.
1. Visualising the Relationship (Scatterplots)
A scatterplot is the most useful graphical display for determining if a relationship exists between two numerical variables.
- When plotting a scatterplot, the independent variable (the one that causes the change) goes on the x-axis, and the dependent variable goes on the y-axis.
2. Pearson's Correlation Coefficient (r)
We use the statistical software code cor.test(X,Y) to calculate the correlation coefficient, often denoted as r. This calculation gives us a number exactly between -1 and 1, which tells us the strength and direction of the linear relationship.
- r = 1: A perfect positive linear relationship (as X increases, Y increases perfectly).
- r = -1: A perfect negative linear relationship (as X increases, Y decreases perfectly).
- r = 0: No linear relationship.
Exam Rule:
When interpreting an r value, you must state both the strength (weak, moderate, or strong) and the direction (positive or negative) in the context of the question.
3. Correlation vs. Causation (The Exam Trap)
This is one of the most frequently tested concepts in the Higher Applications course.
- Correlation does not imply causation. Just because two variables have a strong mathematical correlation does not prove that a change in one directly causes a change in the other.
- There may be an outside influence, known as a confounding variable, that is actually causing both variables to change simultaneously.
Worked examples
Example 1
Example 1: Interpreting the Correlation Coefficient (r)
A coach tracks the number of hours their athletes spend training each week and the time it takes them to run a 100-metre sprint. The correlation coefficient is calculated as r = -0.89.
Interpret this correlation coefficient in the context of the data.
There is a strong negative correlation between the number of hours spent training and the time taken to run a 100-metre sprint.
(As training hours increase, sprint times decrease).
Example 2
Example 2: The Correlation vs. Causation Trap
A tourist board gathers data over a 12-month period. They run a statistical analysis and find a strong positive correlation (r = 0.94) between the daily number of ice cream sales and the daily number of sunburn cases at a local hospital.
Based on this, a local newspaper publishes an article claiming that eating ice cream causes sunburn. Explain why this claim is invalid.
Correlation does not imply causation. The strong mathematical relationship does not prove that eating ice cream causes sunburn.
It is highly likely that a confounding variable, such as hot, sunny weather, is causing both ice cream sales and sunburn cases to increase at the exact same time.
Example 3
Example 3: Extracting r from Statistical Software
A researcher is investigating the relationship between the number of hours a student spends playing video games per week and their score on a concentration test. They use statistical software and generate the following output:
Pearson's product-moment correlation
data: Gaming.Hours and Test.Score
t = -4.102, df = 48, p-value = 0.000155
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.7245 -0.2819
sample estimates:
cor
-0.5082
- (a) State the correlation coefficient for this data.
- (b) Make an appropriate comment on the relationship between Gaming Hours and Test Score.
(a) -0.5082
(This must be stated explicitly as a number, pulled from the 'cor' value at the bottom of the software output).
(b) There is a moderate negative correlation between the number of hours spent gaming and the concentration test score.