Spearman’s rank order correlation coefficient test (Spearman’s rho) is a way to quantify correlations between two variables (where the data is in a the ordinal format).
Note: In the psychology A level exam, you won’t have to actually calculate the value of Spearman’s rho itself. Instead, you will be given the value of Spearman’s rho and will then have to say whether this value is statistically significant. So, to skip to this part, click here.
An example of how to calculate Spearman’s rho
Let’s say we conducted a study of 10 students to see if there was a correlation between the following two variables:
- X: Number of hours spent studying for an exam (independent variable)
- Y: Exam scores achieved by student (dependent variable)
Our experimental hypothesis is that X will be positively correlated with Y.
Here are our results:
| X | 5 | 10 | 8 | 3 | 7 | 6 | 9 | 4 | 1 | 1 |
| Y | 60 | 80 | 75 | 50 | 70 | 65 | 85 | 55 | 45 | 40 |
Step 1: Rank the data
Rank each set of data from 1 to n, considering ties by assigning the average rank:
| X | 5 | 10 | 8 | 3 | 7 | 6 | 9 | 4 | 1 | 1 |
| X rank | 5 | 10 | 8 | 3 | 7 | 6 | 9 | 4 | 1.5 | 1.5 |
| Y | 60 | 80 | 75 | 50 | 70 | 65 | 85 | 55 | 45 | 40 |
| Y rank | 5 | 9 | 8 | 3 | 7 | 6 | 10 | 4 | 2 | 1 |
Step 2: Calculate the differences between ranks
Calculate the differences between the ranks for each pair of observations:
| X rank |
5 | 10 | 8 | 3 | 7 | 6 | 9 | 4 | 1.5 | 1.5 |
| Y rank |
5 | 9 | 8 | 3 | 7 | 6 | 10 | 4 | 2 | 1 |
| Difference | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | -0.5 | 0.5 |
Step 3: Square the differences
Square each difference to eliminate the negative signs:
| Difference | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | -0.5 | 0.5 |
| d^2 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0.25 | 0.25 |
Step 4: Sum up the squared differences
Add together all the numbers from step 3:
∑d^2 = 0 + 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0.25 + 0.25 = 2.5
Step 5: Calculate Spearman’s rho (ρ)
Now, use the formula for Spearman’s rho to calculate the correlation coefficient:
1 – (6 * ∑d^2) / (n * (n^2 – 1))
In this example:
- n = 10 (because we studied 10 students)
- ∑d^2 = 2.5 (see step 4 above)
So, we substitute these values into the formula:
ρ = 1 – (6 * 2.5) / (10 * (10^2 – 1))
= 1 – (15) / (10 * 99)
= 1 – 15 / 990
= 1 – 0.015
= 0.985
The observed Spearman’s rho for this example is 0.985, which indicates a strong positive correlation between the number of hours spent studying for the exam and the exam scores achieved by students.
Statistical significance and Spearman’s rho
Step 6: Determine the critical value and compare with the observed value
To determine statistical significance, we compare the observed value of Spearman’s rho with a critical value. To do this, we need to know 3 things:
- Whether our hypothesis is one-tailed or two-tailed
- Our hypothesis is one-tailed: We expect studying to be positively, not negatively, correlated with exam scores.
- The sample size (n)
- Our sample size is 10 because we studied 10 students.
- The significance level (p)
- The significance level represents the threshold below which we consider the results to be statistically significant. In psychology, the most commonly used significance level is 0.05 (5%), so we’ll use it for our example.
Using this, we can look up the critical value in a critical values table like the one below (for a more extensive critical values table of Spearman’s rho, click here).
| One-tailed test | |||
|---|---|---|---|
| p = 0.1 | p = 0.05 | p = 0.025 | |
| Two-tailed test | |||
| n | p = 0.2 | p = 0.1 | p = 0.05 |
| 9 | 0.483 | 0.6 | 0.783 |
| 10 | 0.455 | 0.564 | 0.745 |
| 11 | 0.427 | 0.563 | 0.709 |
We can see from the table here that our critical value is 0.564.
Step 7: Assess statistical significance
Next, we compare the observed value of Spearman’s rho from step 5 (0.985) with the critical value from step 6 (0.564).
For Spearman’s rho, results are considered statistically significant if the calculated value of Spearman’s rho is equal to or greater than the critical value.
Since 0.985 is greater than 0.564, we can say that the results in our example are statistically significant at the chosen significance level (p = 0.05).
Step 8: Interpret the results
Based on the assessment of statistical significance, we can conclude that there is a significant positive relationship between the number of hours spent studying and exam scores. The higher the number of study hours, the higher the exam scores.
In other words, there is evidence to suggest that there is a significant relationship between the number of hours spent studying and the exam scores achieved by students in this example.
So, at the significance level of p = 0.05, we can reject the null hypothesis and accept our experimental hypothesis.
