### Video Transcript

Which of the following is the
formula of Spearman’s rank correlation coefficient?

And then we have five possible
options to choose from. Remember, Spearman’s rank
correlation coefficient is a test that examines the degree to which two data sets
are correlated, if at all. Rather than just trying to infer
this information from a scatter graph, Spearman’s rank correlation coefficient is a
numerical value which tells us the degree of correlation or, indeed,
noncorrelation.

So let’s suppose we have paired
sets of data. After arranging the data sets in a
relevant table, we begin by ranking each data point. For instance, imagine we had 10
pairs of data. We will take the first element and
rank by assigning each one a number from one to 10. If there are, however, two or more
values that are equal, we average those ranks. We then repeat this process with
the second set of data points. Once we have that, we can find the
difference between the ranks, and we call that 𝑑. Let’s say we’ve ranked the first
data set, and we call that 𝑟 sub one, and then we’ve ranked the second and call
that 𝑟 sub two. The difference is 𝑟 sub one minus
𝑟 sub two.

The next step is to square our
values for 𝑑 and find the sum. And we use ∑ notation to represent
this. We now notice that every single
formula that we’ve been given has the ∑ of 𝑑 squared in it. So which is it? Well, one of the formulae we can
use to calculate Spearman’s rank correlation coefficient 𝑟 for 𝑛 pairs of data
points is one minus six times the ∑ of 𝑑 squared over 𝑛 cubed minus 𝑛. But in fact this formula does not
look like any of the formulae we’ve been given. We can, however, factorize the
denominator of this fraction. Taking out a common factor of 𝑛,
we rewrite it as 𝑛 times 𝑛 squared minus one.

And so the formula is 𝑟 equals one
minus six times the ∑ of 𝑑 squared over 𝑛 times 𝑛 squared minus one. We notice then that this is option
(D).