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Spearman’s Rank correlation coefficient

Two things correlate when they vary together. For example, we expect land values to fall with distance from the city centre.

• Positive correlation - as one variable increases in value so does the other.
• Negative correlation - as one variable goes up, the other goes down.

A correlation can easily be drawn as a scattergraph, but the most precise way to compare several pairs of data is to use a statistical test - this establishes whether the correlation is really significant or if it could have been the result of chance alone.

Spearman’s Rank correlation coefficient is a technique which can be used to summarise the strength and direction (negative or positive) of a relationship between two variables.

The result will always be between 1 and minus 1.

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Method - calculating the coefficient

1. Create a table from your data. Rank the two data sets (highest = rank 1).
2. Find the difference between the ranks of each of the pairs (d). Square the differences (d²) and then sum them (x d²).
3. Calculate the coefficient (rs) using the formula provided. The answer will always be between 1.0 (a perfect positive correlation) and -1.0 (a perfect negative correlation).

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Significance testing

To check whether your answer could be the result of chance, test the significance of the relationship. This can be done using a graph (Waugh p637). Sometimes in exam questions you will be given a table instead.

1. Work out the ‘degrees of freedom’ you need to use. This is the number of pairs in your sample minus 2 (n-2).
2. Now ‘plot’ your result on the graph.
   1. If it is below the line marked 5%, then it is possible your result was the product of chance and you must reject it.
   2. If it is above the 0.1% significance level, then we can be 99.9% confident the correlation has not occurred by chance.
   3. If it is above 1%, but below 0.1%, you can say you are 99% confident.
   4. If it is above 5%, but below 1%, you can say you are 95% confident (ie statistically there is a 5% likelihood the result occurred by chance).
3. State your result clearly.

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A word of warning

• The fact two variables correlate cannot prove anything - only further research can actually prove that one thing affects the other.
• You must always try to find out what the real story behind the statistic is (the processes underlying the correlation).

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Spearman’s Rank correlation coefficient - Try it!

The following exercise is based on the worked example which appears in B Lenon & P Cleves (1994) Fieldwork Techniques and Projects in Geography pub. Collins Educational (pp139-140).

In this exercise we have 12 settlements ranging in size from 220 inhabitants to over 15,000. We would expect that larger villages and towns would have more services than smaller ones. We are going to see if there is a correlation in our sample.

Hypothesis: The larger a settlement, the greater the number of services it has.

• We would expect that as population rises so does the number of services a place has. As one variable increases, so does the other. We expect a strong positive correlation. Is this what you found? (You can plot your result on the line above as a rough indication.)
• You should test the significance of your result using the graph in Waugh (p637). Does your result hold true at the 0.1%, 1% and 5% significance levels?
• Remember if the result is significant at the 5% level this means you can be 95% confident of your result (5% due to chance).

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