What Is Spearman's Rank Correlation Coefficient?

Spearman's Rank Correlation Coefficient: Denoted $r_{s}$ , this is a non-parametric measure of the strength and direction of a monotonic relationship between two variables. It works by comparing the ranks of data values rather than the values themselves.

Because $r_{s}$ uses ranks rather than raw data, it makes no assumptions about the underlying frequency distribution of the variables. This makes it much more flexible than Pearson's correlation coefficient.

Monotonic Relationship: A relationship where, as one variable increases, the other variable either consistently increases (monotonically increasing) or consistently decreases (monotonically decreasing) , but not necessarily at a constant rate.

Analogy

Think of a monotonic relationship like climbing a hill , you might go fast, then slow, then fast again, but you're always moving upward. Pearson's requires a straight climb at a constant rate; Spearman's just needs a consistent direction.

The value of $r_{s}$ always lies between $- 1$ and $+ 1$ :

$r_{s} = + 1$ : perfect positive monotonic relationship
$r_{s} = - 1$ : perfect negative monotonic relationship
$r_{s} = 0$ : no monotonic relationship
Values close to $\pm 1$ indicate a strong relationship; values close to $0$ indicate a weak relationship

Exam Tip

The absolute value $∣ r_{s} ∣$ tells you the strength of the relationship; the sign tells you the direction. A value of $r_{s} = - 0.9$ is just as strong as $r_{s} = + 0.9$ , it just goes in the opposite direction.

The Formula for $r_s$

The Spearman's rank correlation coefficient is calculated using:

$r_{s} = 1 - \frac{6 \sum d _{i}^{2}}{n ( n ^{2} - 1 )}$

Where:

$d_{i}$ is the difference between the ranks of each corresponding pair of values
$n$ is the number of pairs of data values
$\sum d_{i}^{2}$ means you square each difference and add them all up

Note

In the IB exam, you are expected to use technology (GDC or statistical software) to calculate $r_{s}$ rather than applying this formula by hand. However, understanding the formula helps you appreciate what the calculation is doing , comparing how similarly the two variables rank their data points.

The process, conceptually, is:

Rank each dataset separately (1 = smallest, $n$ = largest)
Find the difference $d_{i}$ between each pair of ranks
Square each difference and sum them
Substitute into the formula

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What Is Spearman's Rank Correlation Coefficient?

Analogy

The value of $r_{s}$ always lies between $- 1$ and $+ 1$ :

$r_{s} = + 1$ : perfect positive monotonic relationship
$r_{s} = - 1$ : perfect negative monotonic relationship
$r_{s} = 0$ : no monotonic relationship
Values close to $\pm 1$ indicate a strong relationship; values close to $0$ indicate a weak relationship

Exam Tip

The Formula for $r_s$

The Spearman's rank correlation coefficient is calculated using:

$r_{s} = 1 - \frac{6 \sum d _{i}^{2}}{n ( n ^{2} - 1 )}$

Where:

$d_{i}$ is the difference between the ranks of each corresponding pair of values
$n$ is the number of pairs of data values
$\sum d_{i}^{2}$ means you square each difference and add them all up

Note

The process, conceptually, is:

Rank each dataset separately (1 = smallest, $n$ = largest)
Find the difference $d_{i}$ between each pair of ranks
Square each difference and sum them
Substitute into the formula

Free preview

SL 4.10—Spearman’s rank correlation coefficient

What Is Spearman's Rank Correlation Coefficient?

The Formula for $r_s$

9 more sections in this topic

SL 4.10—Spearman’s rank correlation coefficient

What Is Spearman's Rank Correlation Coefficient?

The Formula for $r_s$

9 more sections in this topic