Spearman Rank Correlation Coefficient (AQA A-Level Geography): Revision Notes

Spearman Rank Correlation Coefficient

What is the Spearman rank correlation coefficient?

The Spearman rank correlation coefficient is a statistical test that measures the strength and direction of correlation between two sets of data (also called variables). It tells us whether there is a relationship between two variables and how strong that relationship is.

This test provides a numerical value that summarises the degree of correlation. It is an objective indicator, meaning the result can be tested statistically to see how meaningful it is. Once you calculate the coefficient, you must test it against critical values to determine whether the result is significant or could have occurred by chance.

When to use this test

The Spearman rank correlation coefficient can be used with:

• Raw numerical figures
• Percentages
• Index values
• Any data that can be ranked in order

The key requirement is that your data must be capable of being ranked from highest to lowest (or vice versa).

The formula

The Spearman rank correlation coefficient uses the following formula:

$R_s = 1 - \frac{6\sum d^2}{n^3 - n}$

Where:

• $R_s$ = the Spearman rank correlation coefficient
• $d$ = the difference in ranking between the two sets of paired data
• $n$ = the number of sets of paired data
• $\sum$ = sum of (add together all values)

Step-by-step calculation method

Follow these steps carefully to calculate the Spearman rank correlation coefficient:

Step 1: Rank the first data set

Rank one set of data from highest to lowest. The highest value receives rank 1, the second highest receives rank 2, and so on.

Step 2: Rank the second data set

Rank the other set of data in exactly the same way (highest to lowest).

Step 3: Deal with tied ranks

If you have tied values (numbers that are the same), you need to allocate an average rank:

Step 4: Calculate the difference in ranks

For each pair of data, calculate the difference between the two ranks. This is your $d$ value.

Step 5: Square each difference

Square each $d$ value to get $d^2$.

Step 6: Add the squared differences

Add all the $d^2$ values together. This gives you $\sum d^2$ (the sum of squared differences).

Step 7: Multiply by 6

Multiply your $\sum d^2$ value by 6. This gives you the numerator: $6\sum d^2$.

Step 8: Calculate n³ - n

Calculate the value of $n^3 - n$, where $n$ is the number of pairs of data. This is your denominator.

Step 9: Divide and subtract from 1

Divide the result from Step 7 by the result from Step 8, then take this answer away from 1.

The final answer should be a value between +1.0 (perfect positive correlation) and -1.0 (perfect negative correlation).

Worked example

Let's examine a study comparing the number of doctors per 1,000 population against confirmed malaria cases per 1,000 population in Asia-Pacific countries.

Second example: COVID-19 deaths and healthcare spending

Another example examines the relationship between COVID-19 deaths (up to 12 March 2020) and the percentage of GDP spent on healthcare.

Interpreting your results

Direction of the relationship

The sign (positive or negative) of your coefficient tells you the direction of the relationship:

Positive correlation: If the calculation produces a positive value (e.g., $+0.55$), the relationship is positive or direct. As one variable increases, so does the other.

Negative correlation: If the calculation produces a negative value (e.g., $-0.55$), the relationship is negative or inverse. As one variable increases, the other decreases.

Testing for statistical significance

Simply calculating a correlation coefficient is not enough. You must test whether the relationship is statistically significant or could have occurred by chance.

To test significance, compare your calculated $R_s$ value against critical values in a significance table. The critical values depend on:

• Your sample size ($n$)
• The significance level you are testing (usually 0.05 or 0.01)

How to use the critical values table:

1. Find the row for your sample size ($n$)
2. Look at the critical values for both the 0.05 (5%) and 0.01 (1%) significance levels
3. Compare your calculated $R_s$ value with these critical values (ignore whether your value is positive or negative - just compare the absolute size)
4. If your coefficient is greater than the critical value, the correlation is significant at that level

Understanding significance levels

0.05 (5%) significance level: This means there is a greater than 5% possibility of the relationship occurring by chance. If your result is significant at this level, the relationship could have occurred by chance more than 5 times in 100, which is considered an unacceptable level of chance. Therefore, the relationship is not significant.

0.01 (1%) significance level: If there is a less than 5% possibility, the relationship is significant and therefore meaningful. The stricter 0.01 (1%) level means there is less than a 1% chance the relationship occurred by chance.

Less than 1% significance: If there is less than a 1% possibility of the relationship occurring by chance, the relationship is very significant. The result could only have occurred by chance 1 in 100 times, which is very unlikely.

Applying this to our example

In the malaria example, our $R_s$ value was $-0.55$ from 13 sets of paired data.

Important requirements and warnings

Sample size requirements

Dealing with tied ranks

Too many tied ranks can interfere with the statistical validity of the exercise. Although it is understood that the real data collected may have tied values, there is little you can do about this. Be aware that excessive tied ranks can affect the reliability of your results.

Choosing variables carefully

Remember!