Spearman's Rank Correlation Coefficient

Spearman's rank graph

When looking to see if there might be a correlation when using the Spearman's rank, it can help to graph the data. Remember, you are not looking to see if the data in the graph makes a line, you are looking to see if the rankings are the same.

In the graph below, you can see the rankings that two judges gave at a competition. The rankings that Judge A gave the competitors are noted by circles, while the rankings that Judge B gave are noted by crosses.

Fig. 1 - Plot of rankings given by two different judges.

For example, Judge A gave the first competitor a ranking of \(1\), while Judge B gave the competitor a ranking of \(2\). While the data plotted does not form a line, it does appear that both judges gave approximately the same score to all of the competitors, and in three cases, they gave exactly the same score. So you could expect the Spearman's rank correlation coefficient for the rankings here to be closer to \(1\) than to \(0\).

Spearman's rank correlation coefficient formula

Using the formula for the Spearman's rank correlation coefficient requires the data sets to be ranked. It doesn't matter how you rank them (for example, best to worst or worst to best) as long as you rank both sets the same way. Before looking at the formula, let's look at an example of organising the rankings.

To calculate the correlation coefficient, you will need the following values:

\[ S_{xy} = \sum x_iy_i - \frac{1}{n}\sum x_i \sum y_i; \]

\[ S_{xx} = \sum x_i^2 - \frac{1}{n} \left(\sum x_i\right)^2;\]

and

\[S_{yy} = \sum y_i^2 - \frac{1}{n} \left(\sum y_i\right)^2.\]

Then the Spearman's rank correlation coefficient can be found using the formula

\[ r_s = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} .\]

You might find an example where the same score is given to more than one data point. This is called a tied rank.

Let's look at a quick example.

Notice that in the previous example, you are not comparing the ranks of taster \(x\) to the ranks of taster \(y\). You are only comparing the ranks given by a single taster.

If there are more than two tied ranks, then the formula

\[ r_s = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \]

needs to be used. However, if there are two or fewer tied ranks, then you can use the following formula instead:

\[ r_s = 1 - \frac{6}{n(n^2-1)} \sum d^2,\]

where \(n\) is the number of pairs of observations and \(d\) is the difference between the ranks of each observation. The difference formula will give you a good approximation of the Spearman's rank correlation coefficient as long as there aren't tied ranks.

Spearman's rank table

Once you know the Spearman's rank correlation coefficient, you will often use it to do a hypothesis test. While you can use technology to find the critical value, it is helpful to be able to read a Spearman's rank table. Below is a section from a Spearman's rank table.

Table 4. Spearman's rank table

The first column of the table is the sample size \(n\), and the first row of the table gives you the confidence level. Notice that as the sample size increases, the critical value for a given confidence level decreases. Remember that the margin of error depends on the critical value:

margin of error = (critical value)(standard error).

This means that if you increase the sample size, the margin of error will decrease.

Critical value of Spearman's rank correlation coefficient

The critical value of the Spearman's rank correlation coefficient depends on the sample size and the confidence level you are using. The critical value can be found using a table or through statistical software. For example, if you are doing a one-tailed test, with a sample size of \(7\), at the \(0.25\) confidence level, you would use a table of Spearman's coefficients to see that the critical value is \(0.786\). You can find this critical value in the table above.

In other words, for a sample size of \(7\), the critical value of \(r_s\) is significant at the \(0.25\) level on a one-tailed test at \(\pm 0.786\).

Spearman's rank correlation coefficient example

Let's go back to the coffee example and work out what the correlation coefficient is.