IB Maths AI HLCorrelation & RegressionPaper 1 & 2~6 min read
Comparison of Correlation Coefficients
You now have two correlation tools: Pearson’s r for linear relationships and Spearman’s rs for monotonic ones. This page is about choosing wisely and knowing each tool’s limits — exactly the kind of “comment” and “explain” marks examiners love. Three ideas do all the work: Pearson’s only sees straight lines while Spearman’s sees any always-increasing/decreasing trend; the two are linked but not equivalent (r = ±1 forces rs = ±1, but not the other way round); and outliers distort r but barely touch rs, because Spearman’s only uses ranks.
📘 What you need to know
Pearson’s r tests for a linear relationship. It won’t reveal a non-linear one (e.g. exponential growth).
Spearman’s rs tests for a monotonic relationship (always increasing or always decreasing) — it can detect non-linear monotonic trends.
Choose Pearson’s if you care about a linear model; choose Spearman’s if you suspect a non-linear monotonic relationship or have outliers.
Connection: r = 1 ⇒ rs = 1, and r = −1 ⇒ rs = −1 (perfect linear is also perfectly monotonic).
The converse is NOT true: rs = ±1 does not force r = ±1 (e.g. exponential data has rs = 1 but r < 1).
Pearson’s r is affected by outliers (it uses the actual numerical values).
Spearman’s rs is not usually affected by outliers (it only uses the ranks).
rs > r is a hint that the relationship is monotonic but curved, not a straight line.
Which coefficient should you use?
The choice comes down to what kind of relationship you’re testing for: a straight line, or just a consistent direction.
Pearson’s r
linear
Tests how well a STRAIGHT LINE fits. Use it when you want a linear model. Blind to curves.
Spearman’s rₛ
monotonic
Tests for an always-up or always-down trend, even if curved. Use it for non-linear monotonic data or when outliers are present.
Feature
Pearson’s r
Spearman’s rs
Tests for
linear relationship
monotonic relationship
Uses
actual values
ranks only
Detects curves?
No
Yes (if monotonic)
Affected by outliers?
Yes ✗
Not usually ✓
Tells you the model?
suggests linear
no — just “monotonic”
🧠 Memory aid — “Pearson is picky, Spearman is chill”
Pearson’s is picky: it only rewards a near-perfect straight line and gets thrown off by a single outlier. Spearman’s is relaxed: it just checks whether the order matches, so it forgives curves and shrugs off outliers. If a question mentions a curve, exponential growth, or an extreme value, that’s your cue to reach for Spearman’s.
How the two are connected
Perfect linear correlation is a special case of perfect monotonic correlation — so it forces Spearman’s to ±1 as well. But monotonic does not mean linear, so the connection only runs one way.
One-way implicationr = 1 ⇒ rs = 1 and r = −1 ⇒ rs = −1
the converse is FALSE: rₛ = ±1 does NOT force r = ±1 ✗
why exponential data splits the two coefficients
Exponential data is always increasing, so the ranks agree perfectly and rs = 1. But the points curve away from any straight line, so Pearson’s r < 1.
Outliers: the key difference
This is the most-asked comparison point. Because Pearson’s uses the raw numbers and Spearman’s uses only ranks, a single outlier can wreck r while leaving rs almost unchanged.
🤔 Why don’t outliers affect Spearman’s?
An outlier is extreme in value — say a point at 500 when everything else is under 50. Pearson’s r feels the full force of that 500, so it gets dragged. But when you rank the data, that 500 just becomes “the highest” — rank n — exactly as it would be if it were 51. The huge gap in value collapses to a single step in rank, so rs barely moves. That’s why Spearman’s is the safer choice when a data set contains outliers.
Reading the gap: if rs is much higher than r, the relationship is probably monotonic but non-linear (curved), or r has been pulled down by an outlier.
Worked examples
WE 1
Choose the right coefficient
A researcher believes the relationship between two variables follows an exponential (non-linear) curve, but is always increasing. Which correlation coefficient should she use, and why?
linear or monotonic?curve = non-linear, but always increasing = monotonicPearson’s r only detects straight-line relationships, so it would understate this.use Spearman’s rₛSpearman’s measures monotonic correlation, including curves.
WE 2
Interpret the connection
A data set gives rs = 1 but r = 0.91. What does this tell you about the relationship?
rₛ = 1the rankings agree perfectly → the relationship is perfectly monotonic (always increasing).r = 0.91 (not 1)the points don’t lie exactly on a straight line.monotonic but non-linear (e.g. a curve)perfect monotonic with imperfect linear ⇒ the data follows a curve, not a line.
WE 3
Is the converse true?
A student claims: “If r = 1 then rs = 1, so it must also be true that if rs = 1 then r = 1.” Is the student correct?
forward directionr = 1 ⇒ rₛ = 1 ✓ (perfect line is also perfectly monotonic)converserₛ = 1 does NOT ⇒ r = 1counterexample: exponential data has rₛ = 1 but r < 1.no — the converse is falsemonotonic doesn’t mean linear, so the implication only runs one way.
WE 4
Effect of an outlier
A data set has a strong increasing trend plus one extreme outlier. After including the outlier, r drops to 0.55 but rs stays at 0.94. Explain why.
Pearson’s uses actual valuesthe outlier’s extreme value pulls r down to 0.55.Spearman’s uses ranksthe outlier is just “the highest rank” — its extreme size doesn’t matter, so rₛ stays high at 0.94.outlier distorts r but not rₛranks compress the outlier to a single step, protecting Spearman’s.
WE 5
Compare two coefficients in context
For the maths/English test data, r = 0.794 and rs = 0.976. Comment on both values.
Pearson’s r = 0.794suggests strong positive LINEAR correlation.Spearman’s rₛ = 0.976suggests strong positive correlation that is not necessarily linear — the rankings agree almost perfectly.strong positive; likely monotonic & curvedrₛ noticeably higher than r ⇒ the trend is monotonic but probably bends rather than being a straight line.
💡 Top tips
Linear question → Pearson’s; monotonic/curved or outliers → Spearman’s.
The implication runs one way: r = ±1 ⇒ rs = ±1, but never assume the reverse.
Outliers hit r, not rs — quote this when a data set has an extreme value.
rs > r suggests a monotonic but non-linear (curved) relationship.
Spearman’s tells you “monotonic”, not the model — it won’t say which function to use.
Keep “linear” for r and “monotonic” for rs when you write up interpretations.
⚠ Common mistakes
Assuming the converse: thinking rs = 1 forces r = 1. It doesn’t.
Using Pearson’s for curved data. A non-linear monotonic trend needs Spearman’s.
Saying Spearman’s is affected by outliers. It uses ranks, so it usually isn’t.
Calling rs a test for linearity. It tests monotonic correlation.
Expecting r and rs to be equal. They differ for curves and for data with outliers — and the gap is informative.
Claiming Spearman’s identifies the model. It only confirms an increasing/decreasing trend.
Next up — Linear Regression. Once Pearson’s r (and a critical value) confirms a linear model is appropriate, you’ll find the actual line: the least-squares regression line of y on x, written y = ax + b. You’ll interpret the gradient, make predictions, and learn why interpolation is reliable but extrapolation isn’t.
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