Calculate and interpret Spearman's rank correlation coefficient to test for a relationship between two variables, and assess its statistical significance

What this dot point is asking

SEAB wants you to calculate Spearman's rank correlation coefficient, interpret what it says about the relationship between two variables, and test whether that relationship is statistically significant. The central insight is that Spearman's rank turns a scatter of paired field measurements into a single, objective number between minus one and plus one, and a significance test then tells you whether the pattern is strong enough to be more than chance, which is exactly what a hypothesis-driven investigation needs.

The answer

What Spearman's rank tests

Spearman's rank correlation coefficient, written $r_s$, measures the strength and direction of the relationship between two variables by comparing their rank orders rather than their raw values. Because it works on ranks, it copes well with data that are not normally distributed and with ordinal data, which is common in geography. It is the natural partner to a scatter graph and a hypothesis about whether two things vary together.

The formula

The coefficient is calculated as:

where $d$ is the difference between the two ranks for each pair of observations, $\sum d^2$ is the sum of the squared rank differences, and $n$ is the number of pairs.

The method, step by step

1. State the hypothesis and null. For example, "pebble size decreases with distance downstream" (a negative relationship) and a null of no relationship.
2. Collect paired data at a sample of sites, ideally at least about ten pairs for a meaningful test.
3. Rank each variable separately, from highest to lowest (handle ties by sharing the average rank).
4. Find $d$, the difference between the two ranks for each site, then square it to get $d^2$.
5. Sum the squares to get $\sum d^2$.
6. Apply the formula to obtain $r_s$.

Interpreting the coefficient

The value of $r_s$ always lies between minus one and plus one:

• $r_s = +1$: a perfect positive relationship (as one rises, so does the other).
• $r_s = 0$: no relationship.
• $r_s = -1$: a perfect negative relationship (as one rises, the other falls).

The sign gives the direction; the magnitude gives the strength. As a rough guide, values around $\pm 0.7$ to $\pm 0.9$ indicate a strong relationship, around $\pm 0.4$ to $\pm 0.6$ a moderate one, and near $0$ a weak or absent one.

Testing significance

A coefficient on its own could arise by chance, so we test it. Compare the calculated $r_s$ against the critical value for your sample size $n$ at a chosen significance level (commonly the 0.05, or 5 percent, level):

• If $r_s$ is greater than or equal to the critical value, the result is statistically significant: reject the null hypothesis, concluding the relationship is unlikely to be due to chance.
• If $r_s$ is below the critical value, fail to reject the null: the evidence is too weak to claim a real relationship.

The 0.05 level means there is a 5 percent probability of wrongly rejecting a true null hypothesis.

Correlation is not causation

A significant $r_s$ shows the variables are associated, not that one causes the other. The link could be coincidental, reversed, or driven by a confounding variable that affects both. A causal claim needs a credible process mechanism and, ideally, control of other variables, not just a high coefficient.

Examples in context

Example 1. Distance from a city centre and land value in Singapore. A geographer hypothesising that land value falls with distance from the Central Business District collects paired data along a transect, ranks both variables, and computes Spearman's rank, expecting a strong negative $r_s$. Testing it against the critical value confirms whether the classic bid-rent pattern holds significantly here. It shows the test applied to a human-geography relationship, with a clear process (accessibility and competition for central land) underpinning any causal reading.

Example 2. Vegetation cover and slope angle on a hillside. Investigating whether vegetation cover decreases as slope steepens, a student ranks cover and slope at sampled quadrats and calculates $r_s$. A moderate negative coefficient, tested against the critical value, indicates whether steeper slopes significantly support less vegetation. The example highlights the correlation-causation caution: soil depth or aspect could be a confounding variable influencing both, so a mechanism must support the conclusion.

Try this

Q1. State the Spearman's rank formula and explain what $d$ represents. [2 marks]

• Cue. $r_s = 1 - \dfrac{6\sum d^2}{n(n^2-1)}$, where $d$ is the difference between the two ranks given to each paired observation (one rank per variable) and $n$ is the number of pairs.

Q2. A geographer calculates $r_s = -0.76$. Interpret this value. [2 marks]

• Cue. It is a strong negative relationship: as one variable increases the other tends to decrease, and the magnitude ($0.76$) shows the association is strong; whether it is significant depends on comparing it with the critical value for the sample size.

Q3. Explain why a result is declared "significant at the 0.05 level" and what rejecting the null hypothesis means. [3 marks]

• Cue. Significant at 0.05 means there is only a 5 percent probability the result arose by chance under a true null; if the calculated coefficient meets or exceeds the critical value, we reject the null hypothesis, concluding the observed relationship is unlikely to be due to chance and a real association probably exists.