OCR A Level Mathematics A: Bivariate data: scatter diagrams, correlation and regression (H240) – Revision Notes

These notes condense the bivariate data strand of OCR A Level Mathematics A (H240), Version 3.1 (August 2026), for first assessment in 2018: references 2.02c, 2.02d and 2.02e, all Stage 1 content. For the full explanations and worked examples, see the study guide, then test yourself with the practice questions. Other links: course hub, printable checklist, OCR Statistics guide and the free diagnostics.

What is in scope

Ref	Skill
2.02c	Read scatter diagrams and regression lines, and spot separate groups of points. You may be asked to add to a diagram, not draw a whole one. Calculating regression equations is excluded
2.02d	Informal interpretation of correlation
2.02e	Correlation does not imply causation

Links to 2.05f and 2.05g (Pearson’s coefficient, A Level only) are covered in the hypothesis testing unit.

Describing a scatter diagram

Use this four-part checklist and write in context.

Part	Words to use
Direction	Positive, negative, none
Strength	Strong, weak (how tightly points cluster)
Shape	Linear, or curved and levelling off
Unusual features	Points far from the pattern, separate groups

Model sentence: “There is strong negative correlation between depth and temperature: deeper sites tend to be colder.”

Key definitions

Term	Meaning
Bivariate data	Two variables recorded for each item
Explanatory variable	Plotted on x. The variable thought to influence the other
Response variable	Plotted on y. The variable you predict
Regression line	A straight line used to predict y from x
Interpolation	Predicting for x inside the range of the data
Extrapolation	Predicting for x outside the range of the data
Confounding variable	A third variable that influences both of the others

Method boxes

Interpreting a regression line y = a + bx

1. Gradient b: when x goes up by 1, y is predicted to change
   by b units (rise if b > 0, fall if b < 0).
2. Intercept a: predicted y when x = 0. Meaningful only if x = 0
   is within or near the data. Otherwise say it has no real meaning.
3. Prediction: substitute x. Then state whether x is inside the
   data range (interpolation, fairly reliable) or outside it
   (extrapolation, unreliable).
4. Check the answer is possible in context (no negative counts).

Adding a line to a diagram

1. Choose two x values inside the plotted range.
2. Substitute to get two y values. Plot both points.
3. Join with a ruler and extend across the data range only.
4. If the means are given, plot (x̄, ȳ) as a check: the line of
   y on x passes through it.

Dealing with distinct sections

1. Notice two or more clusters or bands.
2. Say the data appear to come from different sections of the
   population.
3. Say one line through all the points would fit neither group well.
4. Suggest a separate line for each group, or investigate what
   separates them.

Worked reminder

The line y = 78 − 2.4x predicts resting heart rate (y beats per minute) from weekly training hours (x), for data between 2 and 10 hours.

• At 5 hours: 78 − 12 = 66. Interpolation.
• At 30 hours: 78 − 72 = 6. Extrapolation, and not credible.
• At the mean training time of 6 hours: 78 − 14.4 = 63.6, the mean heart rate.

Second worked reminder: two sections

Cycle couriers (distance 1 to 5 km) and van drivers (distance 4 to 14 km) are plotted together for journey time against distance. Both groups rise, so the overall correlation is positive. The cycle points sit higher and climb faster. One line through all of them would suggest about 17 minutes for 4 km, while the cyclist took 21.5 and the van driver 14. Say that the line suits neither group and that each group needs its own line.

Phrases that earn credit

Situation	Phrase
Describing	“Strong positive correlation between [x in context] and [y in context]”
Gradient	“Each extra [unit of x] is associated with about [b] more/fewer [units of y]”
Interpolation	“[x] is within the range of the data, so the estimate is reasonably reliable”
Extrapolation	“[x] is outside the range of the data, so the estimate may be unreliable”
Causation	“This shows an association, not that one causes the other. [Third variable] may affect both”
Sections	“The data appear to come from different groups, so one line is not appropriate”

Must-know distinctions

Pair	Difference
Correlation and causation	Correlation is an association between variables. Causation needs a mechanism or a controlled study
Interpolation and extrapolation	Inside the data range or outside it
Strong and linear	Points can follow a curve closely, so strong correlation does not make a straight line suitable
Gradient and intercept	Gradient is how much y changes when x rises by 1. Intercept is y at x = 0
y on x and x on y	A y-on-x line predicts y. Do not rearrange it to estimate x

Four explanations for a correlation

1. x causes y.
2. y causes x.
3. A third variable causes both.
4. Chance, particularly in a small sample.

Good wording: “This shows an association, not that one causes the other. A possible third factor is …, which affects both.”

Large data set reminders

• Use a spreadsheet scatter chart for two numerical variables.
• Filter or colour by a category to look for distinct sections.
• Note the range of each variable before predicting.
• Questions may give selected data or diagrams from the data set, so know the variables and units.

Quick self-test

1. A scatter diagram shows y falling steadily as x rises, with points tightly packed along a line. Describe the correlation.
2. A regression line is y = 4.2 + 0.75x with x from 10 to 40. Interpret the gradient 0.75 if y is a height in cm and x is age in days.
3. For the same line, find the prediction at x = 24 and say whether it is interpolation or extrapolation.
4. For the same line, find the prediction at x = 90 and comment.
5. Why might the intercept 4.2 be unreliable?
6. A scatter diagram has two separate bands. What should you say about a single line of best fit?
7. Towns with more fire stations have higher numbers of reported fires. Suggest a confounding variable.
8. A line of y on x has x̄ = 20 and ȳ = 50. Does the point (20, 50) lie on it?
9. List the four possible explanations for a correlation.
10. A newspaper writes “this proves that X causes Y” about a correlation. What is wrong?
11. Points follow a clear curve that rises then levels off, and a straight line has been drawn. What would you say?
12. For y = 100 − 3x, is a prediction at x = 5 reliable if the data run from x = 12 to x = 30?

Answers

1. Strong negative correlation (state the variables in context).
2. Each extra day of age is associated with about 0.75 cm more height on average.
3. y = 4.2 + 0.75 × 24 = 4.2 + 18 = 22.2. Interpolation, because 24 is between 10 and 40.
4. y = 4.2 + 0.75 × 90 = 4.2 + 67.5 = 71.7. Extrapolation, so unreliable, since growth may not stay linear.
5. It is the prediction at x = 0, outside the data range from 10 to 40, so it is extrapolation.
6. The data appear to be from distinct sections of the population, so one line fits neither group well. Use a line for each group.
7. Population (or town size), which increases both the number of stations and the number of fires.
8. Yes. The y-on-x regression line passes through (x̄, ȳ).
9. x causes y, y causes x, a third variable causes both, or coincidence.
10. Correlation shows association only. A third variable or coincidence could explain it, so causation is not proved.
11. A straight line is not appropriate: it fits the ends badly and extrapolating it would overestimate where the data level off. A curve is a better model.
12. No. x = 5 is below 12, so it is extrapolation (prediction 85, but not reliable).

Where marks are usually lost

• Describing correlation without naming the variables.
• Writing “positive correlation” when the fall in y shows it is negative.
• Calling a prediction reliable without checking the data range.
• Giving the intercept a meaning in context when x = 0 is outside the data.
• Failing to give a number and a unit in a gradient interpretation.
• Saying “causes” when the data only show association.
• Not offering a specific alternative explanation in a causation comment.
• Using one line for two groups and not noticing the separate sections.
• Leaving out the scale or units when adding points to a diagram.

Official syllabus

OCR, Advanced GCE in Mathematics A (H240), specification Version 3.1 (August 2026), for first assessment in 2018, references 2.02c to 2.02e and the large data set section.