IB Mathematics Applications & Interpretation Topic SL…

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

Pearson's $r$ from Topic 4.4 only measures how close the points sit to a straight line, so it can miss a strong relationship that is curved.
Many real relationships are monotonic but not linear: as one variable rises the other always rises (or always falls), just not at a constant rate.
Some data is only ordinal: rankings, ratings or positions where the order is meaningful but the gaps between values are not, so multiplying the raw values (as Pearson's $r$ does) makes no sense.
A single outlier can drag Pearson's $r$ a long way, because $r$ is built from the actual sizes of the values.

Note

Spearman's rank correlation coefficient $r_s$ fixes all three: it replaces each value by its rank and then measures correlation on those ranks.
Working with ranks means $r_s$ detects any monotonic trend, copes with ordinal data, and is far more resistant to outliers.

What $r_s$ Measures: Monotonic, Not Just Linear

A relationship is monotonic if it is either always increasing or always decreasing as you move along one variable, even if the curve bends.
$r_s$ measures how close the relationship is to being perfectly monotonic, not how close it is to a straight line.
Like Pearson's $r$, it always lies in the range $-1 \le r_s \le 1$.
The sign gives the direction (positive means both rise together, negative means one rises as the other falls) and the size $|r_s|$ gives the strength.
It is non-parametric: it makes no assumption that the data is normally distributed, unlike Pearson's $r$.
Because it only uses order, it works directly on ordinal data such as judges' rankings or quality ratings.

Note

A non-zero $r_s$ shows association, not causation: a strong monotonic link may still be driven by a third variable.

How To Rank Paired Data

List the values of the first variable and give the highest value rank $1$, the next highest rank $2$, and so on down to rank $n$.
Rank the second variable the same way, again with rank $1$ for its highest value.
Keep each data point's two ranks paired together, exactly as the original measurements were paired.
The direction you choose (rank $1$ = highest or rank $1$ = lowest) does not change $r_s$, as long as you rank both variables in the same direction.

Common Mistake

Ranking one variable from highest-to-lowest and the other from lowest-to-highest flips the sign of $r_s$ and gives a wrong answer.
Always rank both variables in the same direction, then decide the sign from the trend in the data.

Handling Ties With Average (Mid) Ranks

When two or more values are equal, they must share the ranks they would have occupied, so each tied value gets the average of those ranks.
Suppose four marks are $88, 75, 75, 62$ and you rank with $1$ as the highest.
$88$ takes rank $1$; the two $75$s would have taken ranks $2$ and $3$, so each gets the mid-rank $\tfrac{2+3}{2}=2.5$; $62$ takes rank $4$.
Averaging tied ranks keeps the total of all the ranks unchanged, which is what the calculation needs.

Example question

Six students recorded their weekly revision hours and their exam mark: A (12 h, 74), B (5 h, 52), C (9 h, 64), D (3 h, 48), E (7 h, 66), F (15 h, 80). Build a rank table for both variables, ranking 1 as the highest.

Solution

Rank the hours (highest = 1): F $15\to1$, A $12\to2$, C $9\to3$, E $7\to4$, B $5\to5$, D $3\to6$.
Rank the marks (highest = 1): F $80\to1$, A $74\to2$, E $66\to3$, C $64\to4$, B $52\to5$, D $48\to6$.
Rank table by student (hours rank, mark rank): A $(2,2)$, B $(5,5)$, C $(3,4)$, D $(6,6)$, E $(4,3)$, F $(1,1)$.
Only C and E swap places between the two rankings, so the two orders are almost identical.

Example question

Six employees have years of experience and a performance score: P1 (2 yr, 55), P2 (8 yr, 80), P3 (5 yr, 68), P4 (11 yr, 80), P5 (3 yr, 60), P6 (7 yr, 72). Build the rank table, ranking 1 as the highest.

Solution

Rank experience (highest = 1): P4 $11\to1$, P2 $8\to2$, P6 $7\to3$, P3 $5\to4$, P5 $3\to5$, P1 $2\to6$.
Score has a tie: P2 and P4 both score $80$, the two highest, so they share ranks $1$ and $2$ as the mid-rank $\tfrac{1+2}{2}=1.5$ each.
Remaining scores: P6 $72\to3$, P3 $68\to4$, P5 $60\to5$, P1 $55\to6$.
Rank table (experience rank, score rank): P1 $(6,6)$, P2 $(2,1.5)$, P3 $(4,4)$, P4 $(1,1.5)$, P5 $(5,5)$, P6 $(3,3)$.

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

In AI, $r_s$ is defined as Pearson's correlation coefficient computed on the ranks instead of on the raw data.
Rank both variables (handling any ties with mid-ranks), giving two lists of ranks.
Enter the two rank lists into the GDC as the two data lists and run its linear-regression / correlation command.
The value of $r$ the GDC returns for those ranks is $r_s$; you do not need a separate Spearman key.
$r_s$ is not listed in the formula booklet, so it is a GDC / technique result: the marks are for entering the ranks and reading off $r$.

Exam technique

The method mark is the correct set-up: entering the ranks, not the raw values, into the correlation command.
State clearly that you are finding Pearson's $r$ on the ranks; this shows the examiner you know $r_s$ is $r$ applied to ranks.
Give the final $r_s$ to $3$ significant figures unless it is exact.

The $d^2$ Formula: A No-Ties Aside

For the special case of no ties, there is a hand formula that gives the same value: $$ r_s = 1 - \frac{6\sum d_i^2}{n(n^2-1)} $$
$d_i$ is the difference between the two ranks of the $i$-th pair, and $n$ is the number of pairs.
It is a useful check by hand when there are no ties, but it is not in the AI formula booklet and it gives a biased value when ranks are tied.

Note

Because it breaks down with ties, treat the $d^2$ formula only as an optional cross-check.
The GDC method (Pearson's $r$ on the ranks) is the primary method and works with or without ties.

Example question

For seven cities, an air-pollution index and the local asthma rate (per 1000) are: (20, 3.1), (45, 6.5), (30, 4.2), (60, 7.8), (15, 2.5), (50, 6.0), (38, 5.0). Find $r_s$ and interpret it.

Solution

Pollution ranks (highest = 1): $60\to1, 50\to2, 45\to3, 38\to4, 30\to5, 20\to6, 15\to7$, giving the list $6,3,5,1,7,2,4$.
Asthma ranks (highest = 1): $7.8\to1, 6.5\to2, 6.0\to3, 5.0\to4, 4.2\to5, 3.1\to6, 2.5\to7$, giving the list $6,2,5,1,7,3,4$.
Enter the two rank lists into the GDC and read Pearson's $r$ on the ranks: $r_s = 0.964$ ($3$ s.f.).
Check with the no-ties formula: the only rank differences are at the two swapped cities, so $\sum d_i^2 = 1^2 + (-1)^2 = 2$.
$r_s = 1 - \dfrac{6(2)}{7(7^2-1)} = 1 - \dfrac{12}{336} = 0.964$, matching the GDC.
Interpretation: $r_s$ is close to $+1$, so there is a strong positive monotonic relationship: cities with higher pollution tend to have higher asthma rates.

Interpreting $r_s$: Strength And Direction

$r_s = +1$ means a perfect increasing monotonic relationship: the two rankings are identical.
$r_s = -1$ means a perfect decreasing monotonic relationship: one ranking is the exact reverse of the other.
$r_s = 0$ means no monotonic trend in the ranks.
Judge strength by how close $|r_s|$ is to $1$ (for example $0.8$ strong, $0.5$ moderate, $0.2$ weak) and always describe the result in the context of the two variables.

Spearman's Vs Pearson's: Which To Use

Use Pearson's $r$ when the relationship looks linear and the data is continuous and roughly normal with no strong outliers.
Use Spearman's $r_s$ when the relationship is monotonic but curved, when the data is ordinal, or when outliers are present.
If the data is perfectly monotonic but bent, Pearson's $r$ falls below $1$ while $r_s$ can still be exactly $1$.

Feature	Pearson's r	Spearman's r_s
Detects	linear relationship only	any monotonic relationship
Uses	raw values	ranks of the values
Data type	continuous	continuous or ordinal
Effect of outliers	strongly affected	resistant
Assumes normality	yes (for inference)	no (non-parametric)
In formula booklet	regression is; r itself is not	no (GDC / technique)

Common Mistake

Reporting a low Pearson's $r$ as no relationship when the scatter is clearly a smooth curve: the relationship is monotonic, so $r_s$ is the right measure.
Pearson's $r$ answers how linear, not whether a trend exists.

The Effect Of Outliers

Pearson's $r$ multiplies the actual sizes of the values, so a single extreme point can pull $r$ a long way toward $0$ or $\pm 1$.
Spearman's $r_s$ replaces each value by its rank, so the largest value is only rank $1$ no matter how extreme it is.
This bounds an outlier's influence: pushing the biggest value even higher does not change any rank, so $r_s$ is unchanged.
So when a scatter plot shows one clear outlier, $r_s$ usually gives the more honest summary of the trend.

Example question

A biologist measures a dose and a response for seven samples: (1, 2), (2, 5), (3, 11), (4, 20), (5, 34), (6, 55), (7, 90). Pearson's $r$ for this data is $0.936$. State, with reasons, whether Pearson's $r$ or Spearman's $r_s$ better describes the relationship, and find that coefficient.

Solution

As the dose rises the response always rises, so the relationship is strictly monotonic, but the increases get bigger each step, so it is curved, not linear.
Pearson's $r = 0.936$ is held below $1$ only because the trend is not a straight line, so it understates the true relationship.
Both variables have the same ranking order, so ranking each ($1$ = highest) gives two identical rank lists.
Pearson's $r$ on those ranks gives $r_s = 1.00$ ($3$ s.f.), a perfect positive monotonic relationship.
Spearman's $r_s$ is the appropriate measure here, because the relationship is monotonic but non-linear.

Active recall

What does $r_s$ measure, and how does a monotonic relationship differ from a linear one?
Describe how to rank paired data and how to assign ranks when two values are tied.
In AI, how do you compute $r_s$ on the GDC, and why is entering the ranks (not the raw values) the key step?
Give two situations where Spearman's $r_s$ is more appropriate than Pearson's $r$, and say why.
Why is $r_s$ far less affected by a single extreme outlier than Pearson's $r$ is?

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

Pearson's $r$ from Topic 4.4 only measures how close the points sit to a straight line, so it can miss a strong relationship that is curved.
Many real relationships are monotonic but not linear: as one variable rises the other always rises (or always falls), just not at a constant rate.
Some data is only ordinal: rankings, ratings or positions where the order is meaningful but the gaps between values are not, so multiplying the raw values (as Pearson's $r$ does) makes no sense.
A single outlier can drag Pearson's $r$ a long way, because $r$ is built from the actual sizes of the values.

Note

Spearman's rank correlation coefficient $r_s$ fixes all three: it replaces each value by its rank and then measures correlation on those ranks.
Working with ranks means $r_s$ detects any monotonic trend, copes with ordinal data, and is far more resistant to outliers.

What $r_s$ Measures: Monotonic, Not Just Linear

A relationship is monotonic if it is either always increasing or always decreasing as you move along one variable, even if the curve bends.
$r_s$ measures how close the relationship is to being perfectly monotonic, not how close it is to a straight line.
Like Pearson's $r$, it always lies in the range $-1 \le r_s \le 1$.
The sign gives the direction (positive means both rise together, negative means one rises as the other falls) and the size $|r_s|$ gives the strength.
It is non-parametric: it makes no assumption that the data is normally distributed, unlike Pearson's $r$.
Because it only uses order, it works directly on ordinal data such as judges' rankings or quality ratings.

Note

A non-zero $r_s$ shows association, not causation: a strong monotonic link may still be driven by a third variable.

How To Rank Paired Data

List the values of the first variable and give the highest value rank $1$, the next highest rank $2$, and so on down to rank $n$.
Rank the second variable the same way, again with rank $1$ for its highest value.
Keep each data point's two ranks paired together, exactly as the original measurements were paired.
The direction you choose (rank $1$ = highest or rank $1$ = lowest) does not change $r_s$, as long as you rank both variables in the same direction.

Common Mistake

Ranking one variable from highest-to-lowest and the other from lowest-to-highest flips the sign of $r_s$ and gives a wrong answer.
Always rank both variables in the same direction, then decide the sign from the trend in the data.

Handling Ties With Average (Mid) Ranks

When two or more values are equal, they must share the ranks they would have occupied, so each tied value gets the average of those ranks.
Suppose four marks are $88, 75, 75, 62$ and you rank with $1$ as the highest.
$88$ takes rank $1$; the two $75$s would have taken ranks $2$ and $3$, so each gets the mid-rank $\tfrac{2+3}{2}=2.5$; $62$ takes rank $4$.
Averaging tied ranks keeps the total of all the ranks unchanged, which is what the calculation needs.

Example question

Solution

Rank the hours (highest = 1): F $15\to1$, A $12\to2$, C $9\to3$, E $7\to4$, B $5\to5$, D $3\to6$.
Rank the marks (highest = 1): F $80\to1$, A $74\to2$, E $66\to3$, C $64\to4$, B $52\to5$, D $48\to6$.
Rank table by student (hours rank, mark rank): A $(2,2)$, B $(5,5)$, C $(3,4)$, D $(6,6)$, E $(4,3)$, F $(1,1)$.
Only C and E swap places between the two rankings, so the two orders are almost identical.

Example question

Solution

Rank experience (highest = 1): P4 $11\to1$, P2 $8\to2$, P6 $7\to3$, P3 $5\to4$, P5 $3\to5$, P1 $2\to6$.
Score has a tie: P2 and P4 both score $80$, the two highest, so they share ranks $1$ and $2$ as the mid-rank $\tfrac{1+2}{2}=1.5$ each.
Remaining scores: P6 $72\to3$, P3 $68\to4$, P5 $60\to5$, P1 $55\to6$.
Rank table (experience rank, score rank): P1 $(6,6)$, P2 $(2,1.5)$, P3 $(4,4)$, P4 $(1,1.5)$, P5 $(5,5)$, P6 $(3,3)$.

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

In AI, $r_s$ is defined as Pearson's correlation coefficient computed on the ranks instead of on the raw data.
Rank both variables (handling any ties with mid-ranks), giving two lists of ranks.
Enter the two rank lists into the GDC as the two data lists and run its linear-regression / correlation command.
The value of $r$ the GDC returns for those ranks is $r_s$; you do not need a separate Spearman key.
$r_s$ is not listed in the formula booklet, so it is a GDC / technique result: the marks are for entering the ranks and reading off $r$.

Exam technique

The method mark is the correct set-up: entering the ranks, not the raw values, into the correlation command.
State clearly that you are finding Pearson's $r$ on the ranks; this shows the examiner you know $r_s$ is $r$ applied to ranks.
Give the final $r_s$ to $3$ significant figures unless it is exact.

The $d^2$ Formula: A No-Ties Aside

For the special case of no ties, there is a hand formula that gives the same value: $$ r_s = 1 - \frac{6\sum d_i^2}{n(n^2-1)} $$
$d_i$ is the difference between the two ranks of the $i$-th pair, and $n$ is the number of pairs.
It is a useful check by hand when there are no ties, but it is not in the AI formula booklet and it gives a biased value when ranks are tied.

Note

Because it breaks down with ties, treat the $d^2$ formula only as an optional cross-check.
The GDC method (Pearson's $r$ on the ranks) is the primary method and works with or without ties.

Example question

For seven cities, an air-pollution index and the local asthma rate (per 1000) are: (20, 3.1), (45, 6.5), (30, 4.2), (60, 7.8), (15, 2.5), (50, 6.0), (38, 5.0). Find $r_s$ and interpret it.

Solution

Pollution ranks (highest = 1): $60\to1, 50\to2, 45\to3, 38\to4, 30\to5, 20\to6, 15\to7$, giving the list $6,3,5,1,7,2,4$.
Asthma ranks (highest = 1): $7.8\to1, 6.5\to2, 6.0\to3, 5.0\to4, 4.2\to5, 3.1\to6, 2.5\to7$, giving the list $6,2,5,1,7,3,4$.
Enter the two rank lists into the GDC and read Pearson's $r$ on the ranks: $r_s = 0.964$ ($3$ s.f.).
Check with the no-ties formula: the only rank differences are at the two swapped cities, so $\sum d_i^2 = 1^2 + (-1)^2 = 2$.
$r_s = 1 - \dfrac{6(2)}{7(7^2-1)} = 1 - \dfrac{12}{336} = 0.964$, matching the GDC.
Interpretation: $r_s$ is close to $+1$, so there is a strong positive monotonic relationship: cities with higher pollution tend to have higher asthma rates.

Interpreting $r_s$: Strength And Direction

$r_s = +1$ means a perfect increasing monotonic relationship: the two rankings are identical.
$r_s = -1$ means a perfect decreasing monotonic relationship: one ranking is the exact reverse of the other.
$r_s = 0$ means no monotonic trend in the ranks.
Judge strength by how close $|r_s|$ is to $1$ (for example $0.8$ strong, $0.5$ moderate, $0.2$ weak) and always describe the result in the context of the two variables.

Spearman's Vs Pearson's: Which To Use

Use Pearson's $r$ when the relationship looks linear and the data is continuous and roughly normal with no strong outliers.
Use Spearman's $r_s$ when the relationship is monotonic but curved, when the data is ordinal, or when outliers are present.
If the data is perfectly monotonic but bent, Pearson's $r$ falls below $1$ while $r_s$ can still be exactly $1$.

Feature	Pearson's r	Spearman's r_s
Detects	linear relationship only	any monotonic relationship
Uses	raw values	ranks of the values
Data type	continuous	continuous or ordinal
Effect of outliers	strongly affected	resistant
Assumes normality	yes (for inference)	no (non-parametric)
In formula booklet	regression is; r itself is not	no (GDC / technique)

Common Mistake

Reporting a low Pearson's $r$ as no relationship when the scatter is clearly a smooth curve: the relationship is monotonic, so $r_s$ is the right measure.
Pearson's $r$ answers how linear, not whether a trend exists.

The Effect Of Outliers

Pearson's $r$ multiplies the actual sizes of the values, so a single extreme point can pull $r$ a long way toward $0$ or $\pm 1$.
Spearman's $r_s$ replaces each value by its rank, so the largest value is only rank $1$ no matter how extreme it is.
This bounds an outlier's influence: pushing the biggest value even higher does not change any rank, so $r_s$ is unchanged.
So when a scatter plot shows one clear outlier, $r_s$ usually gives the more honest summary of the trend.

Example question

Solution

As the dose rises the response always rises, so the relationship is strictly monotonic, but the increases get bigger each step, so it is curved, not linear.
Pearson's $r = 0.936$ is held below $1$ only because the trend is not a straight line, so it understates the true relationship.
Both variables have the same ranking order, so ranking each ($1$ = highest) gives two identical rank lists.
Pearson's $r$ on those ranks gives $r_s = 1.00$ ($3$ s.f.), a perfect positive monotonic relationship.
Spearman's $r_s$ is the appropriate measure here, because the relationship is monotonic but non-linear.

Active recall

What does $r_s$ measure, and how does a monotonic relationship differ from a linear one?
Describe how to rank paired data and how to assign ranks when two values are tied.
In AI, how do you compute $r_s$ on the GDC, and why is entering the ranks (not the raw values) the key step?
Give two situations where Spearman's $r_s$ is more appropriate than Pearson's $r$, and say why.
Why is $r_s$ far less affected by a single extreme outlier than Pearson's $r$ is?

IB Mathematics Applications & Interpretation SL 4.10—spearman’s Rank Correlation Coefficient Notes

IB Mathematics Applications & Interpretation SL 4.10—spearman’s Rank Correlation Coefficient Notes

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

What $r_s$ Measures: Monotonic, Not Just Linear

How To Rank Paired Data

Handling Ties With Average (Mid) Ranks

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

The $d^2$ Formula: A No-Ties Aside

Interpreting $r_s$: Strength And Direction

Spearman's Vs Pearson's: Which To Use

The Effect Of Outliers

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

What $r_s$ Measures: Monotonic, Not Just Linear

How To Rank Paired Data

Handling Ties With Average (Mid) Ranks

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

The $d^2$ Formula: A No-Ties Aside

Interpreting $r_s$: Strength And Direction

Spearman's Vs Pearson's: Which To Use

The Effect Of Outliers

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 4.10—spearman’s Rank Correlation Coefficient Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 4.10—spearman’s Rank Correlation Coefficient Notes

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

What $r_s$ Measures: Monotonic, Not Just Linear

How To Rank Paired Data

Handling Ties With Average (Mid) Ranks

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

The $d^2$ Formula: A No-Ties Aside

Interpreting $r_s$: Strength And Direction

Spearman's Vs Pearson's: Which To Use

The Effect Of Outliers

Spearman's Rank Correlation: Do Two Rankings Rise And Fall Together?

What $r_s$ Measures: Monotonic, Not Just Linear

How To Rank Paired Data

Handling Ties With Average (Mid) Ranks

Computing $r_s$ In AI: Pearson's $r$ On The Ranks

The $d^2$ Formula: A No-Ties Aside

Interpreting $r_s$: Strength And Direction

Spearman's Vs Pearson's: Which To Use

The Effect Of Outliers