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

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions follow the bivariate data references 2.02c, 2.02d and 2.02e in the OCR A Level Mathematics A (H240) specification, Version 3.1 (August 2026), for first assessment in 2018. All of this content is Stage 1. Regression equations are always given, because calculating them is excluded. Read the study guide first, and use the revision notes for recall. Course pages: hub and checklist.

Questions

1. A letting agent records the distance from the city centre, x km, and the weekly rent, £y, for eight flats.

x (km)	1	2	3	4	5	6	7	8
y (£)	310	295	282	270	255	238	231	214

(a) Describe the correlation between distance and rent. [2] (b) State which variable is the explanatory variable. [1]

2. A gardener measures the height, h cm, of seedlings d days after planting, for d from 5 to 30. The regression line is h = 3.5 + 1.8d.

(a) Interpret the value 1.8 in context. [2] (b) Comment on the value 3.5 in context. [2]

3. Use the regression line in question 2.

(a) Estimate the height after 12 days. [1] (b) Estimate the height after 60 days, and comment on the reliability of your estimate. [3]

4. A report finds strong positive correlation between the number of computers per pupil and the average test score across a sample of schools. A headteacher concludes that buying more computers will raise test scores. Comment on this conclusion. [3]

5. A surveyor records the floor area, x m², and the annual heating bill, £y, for houses with loft insulation and houses without.

Area (m²)	60	80	100	120	140
Bill, insulated (£)	520	590	670	740	810
Bill, not insulated (£)	780	880	990	1090	1190

All ten points are plotted on one scatter diagram.

(a) Describe what the diagram shows. [2] (b) Explain why one line of best fit through all ten points would be unsuitable. [2] (c) State what should be done instead. [1]

6. A regression line y = 120 − 2.5x is to be added to a scatter diagram whose x values run from 10 to 40.

(a) Find the y values at x = 10 and x = 40, so the line can be drawn. [2] (b) The means are x̄ = 24 and ȳ = 60. Show that the line passes through the mean point. [1] (c) A further point (30, 52) is plotted. State whether it lies above or below the line and by how much. [2]

7. A researcher wants to know whether the hours of sunshine in a day affect the number of visitors to a beach.

(a) State which variable should go on the horizontal axis and give a reason. [2] (b) The regression line predicts a negative number of visitors for one value of sunshine. What does this tell you about that value? [1]

8. A warehouse records the number of items packed per hour, y, by new employees x weeks after they start.

x (weeks)	1	2	3	4	5	6	7	8
y	12	25	35	43	48	51	52	52.5

(a) Describe the pattern in the data. [2] (b) A straight line y = 14.6 + 5.6x is drawn. Show that it overestimates y at both x = 1 and x = 8. [2] (c) Use the line to estimate y at x = 12, and comment. [2]

9. A shop records weekly advertising spend, x (in £100s), and weekly sales, y (in £100s), for x from 12 to 45. The regression line is y = 15.2 + 0.46x.

(a) Interpret the gradient. [2] (b) Estimate the weekly sales when the advertising spend is £3000, and comment on reliability. [2] (c) The owner wants to spend £8000. Use the line to estimate sales and comment. [2] (d) The owner says the data prove advertising causes sales. Comment. [2]

10. An analyst plots daily mean temperature against hours of sunshine for two weather stations on one scatter diagram. The points from station A form a cluster at high sunshine and high temperature. The points from station B form a separate cluster at low sunshine and low temperature. Within each cluster there is no visible trend.

(a) Describe the overall correlation shown by the whole diagram. [2] (b) Explain why concluding that more sunshine raises temperature at either station would be unsafe. [2] (c) Describe how the analyst could use a spreadsheet to investigate further. [2] (d) State what you would expect to see in a single cluster if sunshine did affect temperature at that station. [1]

11. A survey of 200 children aged 4 to 16 finds strong positive correlation between daily hours of television watched and height.

(a) Suggest a third variable that could explain this and say how it would produce the correlation. [2] (b) Write a one-sentence conclusion suitable for a report. [2] (c) Give one other possible explanation of a correlation of this type. [1]

Answers

1. (a) Rent falls as distance from the centre rises, so the correlation is negative [1], and the points lie almost along a line, so it is strong [1]. (b) Distance from the city centre [1]. Examiner insight: name both variables in context.

2. (a) Each extra day after planting is associated with an increase in height of about 1.8 cm [1] on average [1]. (b) 3.5 cm is the predicted height at d = 0, when planted [1], but d = 0 is outside the data range of 5 to 30, so this is an extrapolation and not reliable [1]. Examiner insight: include the unit and the word “predicted” or “on average” in a gradient interpretation.

3. (a) h = 3.5 + 1.8 × 12 = 3.5 + 21.6 = 25.1 cm [1]. (b) h = 3.5 + 1.8 × 60 = 3.5 + 108 = 111.5 cm [1]. Day 60 is outside the range 5 to 30, so this is extrapolation [1] and unreliable, because seedlings may stop growing at a steady rate [1]. Examiner insight: give the reason the estimate is unreliable, not just the word “unreliable”.

4. The correlation shows an association between computers and scores [1], but it does not show that computers cause higher scores [1]. A third variable, such as school funding or the wealth of the area, could cause both [1]. Examiner insight: a causation comment normally needs a specific, plausible alternative explanation in the context. “Correlation is not causation” alone is thin.

5. (a) Both groups show positive correlation: larger houses have higher bills [1]. The points form two separate bands, with the uninsulated houses well above the insulated ones at every area [1]. (b) The data come from two distinct sections of the population [1]. A single line would pass between the bands and would predict a bill (about £826 at 100 m²) that fits neither group (£670 or £990) [1]. (c) Use a separate line of best fit for each group [1]. Examiner insight: quoting figures from the table to support “a single line is unsuitable” makes the explanation specific. A statement that the line is “inaccurate” without evidence is weaker.

6. (a) y = 120 − 2.5 × 10 = 95 [1]. y = 120 − 2.5 × 40 = 20 [1]. (b) At x = 24: y = 120 − 60 = 60, which is ȳ, so the line passes through (24, 60) [1]. (c) On the line at x = 30: y = 120 − 75 = 45 [1]. The point has y = 52, so it is above the line by 7 [1]. Examiner insight: when asked to add to a diagram, work out the values you will plot first and show the substitutions.

7. (a) Hours of sunshine on the horizontal axis [1], because it is the explanatory variable that may influence the number of visitors (the response) [1]. (b) That sunshine value is probably outside the range of the data, so the line should not be used there [1]. Examiner insight: explanatory goes on x and response on y.

8. (a) Positive correlation, since packing rate rises with weeks [1]. The rise is quick at first and then levels off near 52, so the pattern is curved, not linear [1]. (b) At x = 1: 14.6 + 5.6 = 20.2, above the observed 12 [1]. At x = 8: 14.6 + 44.8 = 59.4, above the observed 52.5 [1]. (c) y = 14.6 + 5.6 × 12 = 14.6 + 67.2 = 81.8 [1]. This is extrapolation and the data have levelled off, so the true value is likely to be far lower [1]. Examiner insight: a strong correlation can still go with a poor straight-line fit. Mention the curve when you reject the line, rather than just calling the correlation weak.

9. (a) Each extra £100 spent on advertising is associated with about £46 more weekly sales [1] (0.46 × £100, with units and “associated with”) [1]. (b) £3000 is x = 30: y = 15.2 + 0.46 × 30 = 15.2 + 13.8 = 29, so about £2900 [1]. 30 lies in the range 12 to 45, so this is interpolation and reasonably reliable [1]. (c) £8000 is x = 80: y = 15.2 + 36.8 = 52, so about £5200 [1]. 80 is outside the range 12 to 45, so this is extrapolation and may be unreliable [1]. (d) The data show an association only [1]. Other factors, such as the season, could affect both advertising and sales, so causation is not proved [1]. Examiner insight: convert units at the end. The line works in £100s, so reading y = 29 as £29 is an easy slip.

10. (a) Positive correlation overall [1]: warmer days go with more sunshine, because the whole cloud runs from the low cluster to the high cluster [1]. (b) The overall pattern is produced by the difference between the two stations [1]. Within each station there is no trend, so the data do not show that sunshine raises temperature at either [1]. (c) Add a column for station and colour the points by it [1]. Make a separate scatter chart for each station, and compare the sunshine and temperature ranges of each [1]. (d) A clear trend (points rising from left to right) inside that cluster [1]. Examiner insight: when a diagram has distinct sections, say explicitly that the groups differ.

11. (a) Age [1]: older children are taller and also watch more television, so both rise together [1]. (b) The data show an association between television watching and height [1], but watching television does not make children taller, and age is a likely explanation for both [1]. (c) Coincidence, or the relationship could run the other way [1]. Examiner insight: in a conclusion, report what the data show first, then qualify it.

Where marks are usually lost

• Describing correlation without using the names of the variables.
• Writing a gradient interpretation without units or without “on average” or “associated with”.
• Judging reliability without comparing x with the range of the data.
• Treating the intercept as meaningful when x = 0 is outside the data.
• Using the word “causes” when the data only show association.
• Making a causation comment with no specific alternative explanation.
• Missing separate groups on a scatter diagram.
• Forgetting to convert units, for example sales in £100s.

Next steps

• Review the revision notes and re-read the study guide.
• Tick off the topic on the printable checklist and see the course hub.
• Try all the free 10-minute diagnostics.
• Book a free trial class.

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.