Spearman Rank Correlation Co-efficient
A method for calculating the strength of the relationship between two variables.
When looking at correlations, a correlation coefficient is calculated to determine how close the data points are to the trend line.
The coefficient is always between 0 and 1; 0 indicating there is no correlation and 1 indicating there is a perfect relationship between the two variables.
The coefficient is also given either a positive or negative sign, indicating whether or not the relationship between the two variables is positively or negatively correlated.
- Ordinal or interval data
- Scatter diagram indicates a possible continuously increasing or decreasing relationship
- At least 5 pairs of measurements (preferably 10 or more)
Example: Moisture content of soil across the saltmarsh
1. Write a null hypothesis (note: this is not the same as your hypothesis in your investigation)
There is no positive correlation between the distance from high water and the water moisture content of the soil.
|% Water Content||31.5||28.2||34.9||57.6||60.1||49.4||62.2||60.4||68.5||54.7|
2. Give a rank to each value for each variable
|B||Rank for Distance||1||2||3||4||5||6||7||8||9||10|
|C||% Water Content||31.5||28.2||34.9||57.6||60.1||49.4||62.2||60.4||68.5||54.7|
|D||Rank for water content||2||1||3||6||7||4||9||8||10||5|
3. Work out the difference, d, between each rank for each pair of values (Row B – Row D)
|Difference in Ranks (d)||-1||1||0||-2||-2||2||-2||0||-1||5|
4. Square the differences
5. Add up the above differences
∑d² = 1+1+0+4+4+4+4+0+1+25 = 44
6. Calculate the Correlation Coefficient, rs, using the formula below:
rs = 1 – 6 ∑d²
n(n² – 1)
Where n is the size of sample. In this example, n = 10.
Therefore, rs = 1 – 6 x 44 = 1 – 264 = + 0.733
This value indicates a strong positive correlation…
…but how close to +1 does it have to be for the correlation not to be attributed to natural variability and for the null hypothesis to be rejected?
Compare the calculated value for rs with a critical value for that particular number of samples (n) in a statistical table at the 5% significance level, see below.
You use a one-tailed test if you know which direction the correlation will be, i.e. positive or negative. If you do not know this information, use a two-tailed test.
If the calculated value is greater than or equal to the critical value in the table, the null hypothesis is rejected (there is only 5% probability the null hypothesis is true)
Example: At the 5% level, using a 0ne-tailed test (refer back to the null hypothesis to see direction), the calculated value of +0.733 is greater than the critical value at +0.648, so the null hypothesis is rejected.
Therefore, there is a statistically significant positive correlation between the distance from high water and the % water content in the soil.
Don’t forget to relate this back to your results – think about zonation in this example! And make sure when writing up statistical results in your project you quote the calculated value, the number of samples, whether you used a one- or two-tailed test, the critical value and the significance level.