IB Maths AI HLCorrelation & RegressionPaper 1 & 2~7 min read
Pearson’s Product-Moment Correlation Coefficient
Describing correlation as “strong negative” is fine, but the IB wants a number. Pearson’s product-moment correlation coefficient (PMCC), written r, puts a precise value on the strength and direction of linear correlation. It always lies between −1 and +1: the sign tells you the direction (positive or negative) and the size tells you the strength — the closer to ±1, the stronger. You’ll calculate r straight from your GDC’s statistics mode, then interpret it in words. One thing to keep front of mind: r only measures LINEAR correlation — a value near 0 doesn’t rule out a strong curved relationship.
📘 What you need to know
The PMCC, r, gives a numerical value to the linear relationship between two variables.
Range: −1 ≤ r ≤ 1. The sign = direction, the size = strength.
r > 0 → positive correlation; r < 0 → negative correlation; r = 0 → no linear correlation.
r = 1 → perfect positive linear; r = −1 → perfect negative linear (all points exactly on a line).
The closer to ±1, the stronger the linear correlation; the closer to 0, the weaker.
Calculate with your GDC (statistics mode) — you are not expected to use the formula by hand.
Critical values: if |r| > the given critical value, a linear model is appropriate. Critical values depend on sample size and are given in the exam.
r only measures LINEAR correlation — a low r can still hide a strong non-linear relationship.
Reading the value of r
Think of r as a dial from −1 to +1. Split it into two questions: which way (sign) and how strong (distance from 0).
the scale of r from −1 to +1
Sign = direction (− is negative, + is positive). Distance from 0 = strength (near ±1 is strong, near 0 is weak).
Value of r
Interpretation
r = 1
perfect positive linear correlation
r ≈ 0.8
strong positive linear correlation
r ≈ 0.4
weak positive linear correlation
r = 0
no linear correlation
r ≈ −0.4
weak negative linear correlation
r = −1
perfect negative linear correlation
🧠 Memory aid — “sign then size”
Read r in two steps. Sign first: minus = negative, plus = positive. Then size: ignore the sign and look at how close the number is to 1. Roughly, |r| above about 0.7 is “strong”, around 0.3–0.7 is “moderate/weak”, and near 0 is “no linear correlation”. Always include the word linear in your interpretation.
Calculating r on your GDC
You won’t compute the formula by hand — the IB expects the GDC. The process is the same on every calculator: enter the data, run a two-variable / linear regression analysis, read off r.
🧭 Recipe — finding the PMCC with a GDC
Enter the bivariate data into two lists (x in one, y in the other).
Run a linear regression (the “ax + b” model) in statistics mode.
Read off r from the output (turn on the diagnostics / correlation display if needed).
Round sensibly (usually 3 sf) and interpret in words: state direction, strength and the word “linear”.
The PMCC formula (for understanding only — use the GDC)r = SxySxSyin the formula booklet — but you’ll use the GDC, not this, in the exam
🤔 Why is a low r not the same as “no relationship”?
The PMCC measures linear correlation only — how well the points fit a straight line. Data that follows a strong curve (like a U-shape, or exponential growth that bends) can have a low or even zero r, because no single straight line fits it well. So r ≈ 0 means “no linear pattern”, not “no pattern at all”. Always plot the scatter diagram too — a curve in the points warns you that PMCC isn’t the right tool (that’s where Spearman’s rank comes in, next topic).
Critical values — is a linear model appropriate?
Some questions give you a critical value and ask whether a linear model is justified. The rule is a simple comparison.
Critical value test
if |r| > critical value → a linear model is appropriate
critical values depend on sample size and are GIVEN in the exam ✓
Use the absolute value. Compare |r| (ignore the sign) with the critical value. If |r| is bigger, the linear correlation is strong enough to justify a linear model.
Worked examples
WE 1
Calculate the PMCC
The table shows eight students’ maths (x) and English (y) test scores. Write down the value of the PMCC, r.
maths x
7
18
37
52
61
68
75
82
English y
5
3
9
12
17
41
49
97
enter the data into the GDC, run linear regressionr = 0.79433…r = 0.794 (3 sf)no hand calculation needed — read r straight off the GDC.
WE 2
Interpret the value
Comment on the value of the correlation coefficient from WE 1 (r = 0.794).
signpositive → positive correlationsize0.794 is close to 1 → strongstrong positive linear correlationalways include the word “linear” — r only measures linear correlation.
WE 3
Apply a critical value
For a sample of size 8, the critical value of r is 0.707. Using r = 0.794 from WE 1, decide whether a linear model is appropriate.
compare |r| with the critical value|0.794| = 0.7940.794 > 0.707 ✓yes — a linear model is appropriate|r| bigger than the critical value ⇒ the linear correlation is strong enough.
WE 4
Sign and strength from a value
A data set gives r = −0.92. Describe the linear correlation.
signnegative → as x increases, y decreasessize|−0.92| = 0.92, very close to 1 → strongstrong negative linear correlationthe minus sign survives in the description; the strength uses the size only.
WE 5
Low r — what it does (and doesn’t) mean
A scientist finds r = 0.05 between two variables, but the scatter diagram clearly shows a strong U-shaped curve. What can she conclude?
interpret rr ≈ 0 → no LINEAR correlationbut the scatter shows a curvethere IS a strong relationship — just not a linear one. PMCC can’t detect it.no linear correlation, but a clear non-linear relationshipr ≈ 0 never means “no relationship” — only “no straight-line relationship”.
💡 Top tips
Use the GDC — enter the data, run linear regression, read off r. No hand calculation expected.
Always say “linear” when interpreting r: “strong positive linear correlation”.
Sign then size: the sign gives direction, the distance from 0 gives strength.
Critical values: compare |r| with the given value — bigger means a linear model is appropriate.
Plot the scatter too — a curve warns you PMCC may understate a non-linear relationship.
Round to 3 sf unless told otherwise, and keep the full value stored on the GDC for any later steps.
⚠ Common mistakes
Forgetting the word “linear”.r measures linear correlation specifically — say so.
Reading r ≈ 0 as “no relationship”. It means no linear relationship; a curve may still fit.
Claiming causation from a high r. Correlation, however strong, never proves cause.
Comparing r (not |r|) with the critical value. Use the absolute value.
Calling a moderate r “strong”. Match the wording to how close |r| is to 1.
Mixing up sign and strength. A negative r can still be very strong (e.g. −0.95).
Next up — Spearman’s Rank Correlation Coefficient. PMCC only spots straight-line relationships. Spearman’s, rs, works on the ranks of the data instead of the raw values, so it can detect any monotonic (always-increasing or always-decreasing) relationship — even a curved one — and it’s far less affected by outliers.
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