4.10 Spearman's rank correlationIB Maths: Applications and Interpretation HL: Revision notes

Pearson's coefficient $r$ measures how close data lie to a straight line. Spearman's rank correlation coefficient $r_s$ instead measures how well the ranks of two variables agree, so it tests for a monotonic relationship: one that is always increasing or always decreasing. To find $r_s$:

• rank each variable separately (rank $1$ for the highest or lowest value, but be consistent for both variables);
• on your GDC, find Pearson's coefficient of the two lists of ranks. In examinations $r_s$ is found using technology; you do not need to prove the formula. The value satisfies $-1\leq r_s\leq1$. Close to $1$ means a strong positive monotonic association, close to $-1$ a strong negative one and near $0$ little or no monotonic association. Example: maths marks $45,60,72,38,55,80$ have ranks $5,3,2,6,4,1$ and physics marks $52,58,80,40,61,75$ have ranks $5,4,1,6,3,2$. Pearson's coefficient of these ranks gives $r_s=0.886$: students who rank highly in maths also tend to rank highly in physics.

When two or more data items are equal, they share the ranks they would have taken, and each receives the average of those ranks. Example: scores $9,8,7,6,5,5,4$ ranked from the highest. The two $5$s would take ranks $5$ and $6$, so each gets $\frac{5+6}{2}=5.5$. The next value, $4$, gets rank $7$, because ranks $5$ and $6$ have been used. Three equal values taking ranks $2,3,4$ would each get $3$. Then use your GDC on the two lists of ranks as before. For judges $X$ and $Y$ scoring seven dishes with several ties, this gives $r_s=0.791$, a fairly strong positive agreement between the judges.

The two coefficients answer different questions.

• Pearson's $r$ tests only for linearity. It is suitable when the scatter diagram is close to a straight line.
• Spearman's $r_s$ tests for any monotonic relationship, linear or curved. Example: bacteria counts $2,4,9,17,31,52,85,140$ at hours $1$ to $8$ always rise. The ranks of both variables are identical, so $r_s=1$, a perfect monotonic relationship. But the growth is curved, so Pearson's gives only $r=0.911$. So $r<1$ does not mean the variables are unrelated; it may mean the relationship is not linear. Always look at the scatter diagram first. Neither coefficient is suitable for a relationship that rises then falls, which is not monotonic.

An outlier is a data point far from the others. Pearson's $r$ uses the actual values, so one outlier can change it greatly. Spearman's $r_s$ uses only ranks, so an outlier counts only as the highest or lowest rank. This is why Spearman's is less sensitive to outliers. Example: eight students' revision hours $2,4,5,6,8,9,10,30$ and scores $40,52,50,58,64,70,75,45$. The student with $30$ hours is an outlier. For all eight, $r=-0.0948$ and $r_s=0.476$. Without that student, $r=0.987$ and $r_s=0.964$. Pearson's changed by about $1.08$ and Spearman's by about $0.488$. Even Spearman's is not immune. Here it is still pulled down by the outlier. Investigate outliers; do not just delete them without a reason.

When you interpret $r_s$, state the direction and strength, and give your conclusion in context: 'students who rank highly in maths tend to rank highly in physics'.

• Correlation does not prove that one variable causes the other.
• $r_s$ only describes monotonic relationships; check the scatter diagram.
• Small samples can give misleading values.
• Many tied ranks reduce how reliable $r_s$ is.
• Spearman's coefficient ignores how far apart the values are, only their order, so it loses some information that Pearson's uses. Use Pearson's when the data are roughly linear and have no outliers, and Spearman's when the relationship is curved but monotonic or when outliers are present.