IB Maths AI HLCorrelation & RegressionPaper 1 & 2~8 min read
Scatter Diagrams & Correlation
So far you’ve worked with one variable at a time. Now you’ll look at two variables together — like hours studied and exam score — to see how they move in relation to each other. This is bivariate data, and you plot it as a scatter diagram: one variable on each axis, one point per pair. From the picture you describe the correlation in two words — its type (positive, negative, or none) and its strength (strong or weak). The single biggest idea to take away: correlation does not imply causation — two things moving together doesn’t prove one causes the other.
📘 What you need to know
Bivariate data is collected on two variables, paired up — each x goes with a y.
A scatter diagram plots bivariate data: the independent (explanatory) variable on the x-axis, the dependent (response) variable on the y-axis.
Correlation = how the two variables change together. Describe it with type + strength.
Type: positive (both increase), negative (one up, one down), or no correlation.
Strength: strong (points close to a line) or weak (points scattered).
Perfect correlation: all points lie exactly on a straight line.
Line of best fit (by eye) passes through the mean point (x̄, ȳ) and follows the trend.
Correlation ≠ causation: a relationship in the data doesn’t prove one variable causes the other — there may be hidden factors.
Scatter diagrams can have outliers — points that don’t follow the trend.
Setting up a scatter diagram
The first decision is which variable goes on which axis. It comes down to which one you control and which one you measure.
x-axis
independent
The explanatory variable — the one you control or set. E.g. hours of practice, temperature.
y-axis
dependent
The response variable — the one you measure / discover. E.g. test score, ice creams sold.
🧠 Memory aid — “x explains, y responds”
The x-axis variable is the one doing the explaining (the cause you’re investigating); the y-axis variable responds to it. “Hours studied” (you choose how long) goes on x; “exam score” (the result) goes on y. If you’re unsure, ask: which one would I change to affect the other? That’s x.
Describing correlation: type + strength
Every “describe the correlation” question wants two things: the type and the strength. Miss one and you lose a mark.
the main types of correlation
Type tells you the direction (up = positive, down = negative); strength tells you how tightly the points hug a line. Tightly clustered = strong; widely scattered = weak.
If the points…
Type
Strength
rise tightly along a line
positive
strong
rise but scattered
positive
weak
fall tightly along a line
negative
strong
fall but scattered
negative
weak
show no clear trend
no (linear) correlation
The line of best fit
If the correlation is strong and linear, you can draw a line of best fit by eye. The one rule the IB checks: it must pass through the mean point.
The line of best fit passes through the mean point
mean point = (x̄, ȳ)
x̄ = mean of the x-values, ȳ = mean of the y-values
🧭 Recipe — drawing a line of best fit by eye
Find the mean point (x̄, ȳ) — the mean of the x-values and the mean of the y-values (use your GDC).
Plot the mean point on the scatter diagram.
Draw a straight line through it that follows the overall trend of the points.
Aim for roughly equal numbers of points above and below the line.
Correlation vs causation
This is the most-tested “explain” idea in the whole topic. Correlation just means two variables move together — it does not prove that one causes the other.
Likely causal
makes sense
Temperature ↑ and ice creams sold ↑ — there’s a believable mechanism linking them.
Correlation only
no mechanism
Global temperature and pet monkeys in the UK may correlate, but there’s no sensible cause — likely a hidden factor or coincidence.
🤔 Why doesn’t correlation prove causation?
Two variables can move together for reasons other than one causing the other. There may be a hidden third factor driving both (ice cream sales and drowning both rise in summer — because of the heat, not because ice cream causes drowning), or it could be pure coincidence. To argue a causal relationship, you need a sensible mechanism linking the two in context — not just a strong correlation. In the exam, always justify causation by reasoning about the actual variables.
Outliers in bivariate data: a point (x, y) can be an outlier on a scatter diagram (off the trend) even when x and y are not outliers in their own separate data sets — useful to know for your IA.
Worked examples
WE 1
Identify the variables & axes
A teacher investigates whether the time students spend on a phone per day affects the time they spend on a computer per day. Which variable goes on each axis?
independent (controlled / explanatory)phone time is the variable being investigated as the explanatory one → x-axis.dependent (measured / response)computer time is the response → y-axis.x = phone hours, y = computer hours“x explains, y responds.”
WE 2
Draw a scatter diagram
For a sample of 9 students, the hours on a phone (x) and on a computer (y) per day are recorded. Sketch the scatter diagram.
phone x
7.6
7.0
8.9
3.0
3.0
7.5
2.1
1.3
5.8
computer y
1.7
1.1
0.7
5.8
5.2
1.7
6.9
7.1
3.3
plot each (x, y) pairx-axis: phone hours 0–10; y-axis: computer hours 0–8. Mark a cross for each pair.see the diagram below
As phone hours increase, computer hours decrease — the points fall closely along a line.
WE 3
Describe the correlation
Describe the correlation shown in the scatter diagram from WE 2.
typeas phone hours ↑, computer hours ↓ → negativestrengthpoints lie close to a line → strongstrong negative (linear) correlationalways give BOTH type and strength.
WE 4
Find & use the mean point
For the WE 2 data, the means are x̄ = 5.13 and ȳ = 3.72 (3 sf). Explain how this helps you draw a line of best fit.
the line of best fit passes through (x̄, ȳ)mean point ≈ (5.13, 3.72)plot this point, then draw a straight line through it following the downward trend.anchor the line at (5.13, 3.72)the mean point is the one fixed point every line of best fit must pass through.
WE 5
Correlation vs causation
A study finds a strong positive correlation between the number of ice creams sold and the number of people who get sunburnt at a beach. Does eating ice cream cause sunburn? Explain.
is there a sensible mechanism?no — eating ice cream cannot cause sunburn.what’s really going on?a hidden factor — hot, sunny weather — increases BOTH ice cream sales and sunburn.correlation, not causationstrong correlation alone never proves causation — look for a hidden variable.
💡 Top tips
Independent on x, dependent on y — the variable you control explains the one you measure.
Describe correlation with TWO words: type (positive/negative/none) and strength (strong/weak).
Line of best fit must pass through (x̄, ȳ) — plot the mean point first.
Correlation ≠ causation. To claim cause, give a sensible mechanism in context; otherwise mention a possible hidden factor.
Use your GDC to find the means (and to plot the scatter diagram to visualise the data).
“No correlation” means no linear trend — the data could still follow a curve.
⚠ Common mistakes
Giving only type or only strength. “Negative correlation” alone misses the strength mark — say “strong negative”.
Swapping the axes. The independent/explanatory variable goes on x, not y.
Claiming causation from correlation. A strong r doesn’t prove one variable causes the other.
Line of best fit not through the mean point. It must pass through (x̄, ȳ).
Saying “no correlation” for curved data. There may be a strong non-linear relationship.
Forcing a line of best fit on weak/no correlation. Only draw one when the linear correlation is strong.
Next up — Pearson’s Product-Moment Correlation Coefficient (PMCC). Describing correlation in words is useful, but the IB wants a number. The PMCC, r, measures the strength and direction of linear correlation on a scale from −1 to +1 — and your GDC calculates it in seconds.
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