Video Transcript
Is it true or false that when
Spearman’s rank correlation coefficient for two groups of data equals one, it means
that the data points perfectly lie on a straight line?
We know that when Spearman’s rank
correlation coefficient is equal to one, we have perfect agreement between the ranks
of the data. And if Spearman’s rank correlation
coefficient is equal to one, then the term containing the sum of the differences
squared must be equal to zero. So let’s look at this final
example. Suppose we have the time in minutes
it took for five students to take a test and their marks as a percentage. And now suppose we rank both our
time and our marks, taking the shortest time and the lowest marks as one and the
highest as five. And now, if we calculate the
difference in ranks, each of the differences are zero because the ranks are in
perfect agreement.
Now, if we square all the
differences, each of these is equal to zero because zero squared is zero. And so the sum of the differences
squared is also zero. And if we put this into our
formula, the sum of the 𝑑 𝑖 squared is equal to zero, so the second term is equal
to zero as we would expect. But now suppose we plot our
original data. We can see from our scatter plot
that although Spearman’s rank is equal to one, the data points themselves do not lie
perfectly on a straight line. And this means that our statement
is false.
