## Spearman Rank Order Correlation Coefficient

When we are dealing with data at the ordinal level, such as ranks,
we must use a measure of correlation that is designed to handle ordinal data.
The Spearman Rank Order Correlation Coefficient was developed by
Spearman to use with this type of data. The Symbol for the
Spearman Rank Order Correlation Coefficient is ,
r sub s, or the Greek letter rho ( ).

The formula for the Spearman Correlation Coefficient is: Where 6 is a constant (it is always used in the formula),

D refers to the difference between a subjects ranks on the two
variables,

and N is the number of subjects.

Let’s take an example (data from Harris, 1995, page 182) where we have
seven subjects whose art projects are being ranked by two judges, and
we wish to know if the two judges rankings are related to one another.
In other words, do the two judges tend to rank the seven art show
contestants in the same way?

The following table contains the data for this problem.

Spearman Rho for Rankings of Two Judges of Art
Projects for Seven Subjects

Subject Judge A’s Ranking Judge B’s Ranking  Alan 1 2 -1 1
Beth 2 1 1 1
Carl 3 4 -1 1
Don 4 3 1 1
Edgar 5 6 -1 1
Frances 6 7 -1 1
Gertrude 7 5 2 4
Sum 10

We have included a column for D, the difference between a subjects
ranks on the two variables and D squared. We can then use the sum of
the D Squared column and the formula to calculate the Spearman r.  Thus we can see that there is a high positive correlation, but not
a perfect correlation between the two judges ratings.