Interpreting r and a line of best fit in a revision study
A maths department records, for 14 of its students, the hours of revision done in the week before a test (x) and the test score out of 100 (y). The values of x range from 1 hour to 9 hours. From these 14 pairs the department calculates the product moment correlation coefficient r = 0.82 and fits the line of best fit y = 31 + 7.5x. In a report to the governors, the head of maths writes: 'Each extra hour of revision adds 7.5 marks. If we required every student to revise 15 hours a week, average scores would rise to about 143.'
1. Interpret r = 0.82 and the value 7.5 in the context of this data, and state two things that r does not tell you.
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r = 0.82 means there is a strong positive linear correlation between hours of revision and test score for these 14 students: as revision hours increase, scores tend to increase, and the points lie fairly close to a straight line.
7.5 is the gradient of the line of best fit: for each additional hour of revision, the predicted score rises by 7.5 marks, on average, across this group.
What r does not tell you:
- It does not show that revision causes higher scores - a third variable such as prior attainment or motivation could drive both.
- It does not give the rate of change (that is the gradient), and it does not show that the relationship is linear rather than curved; r only measures how close the points are to a straight line.
- It says nothing about individual students: a particular student might gain far more or far less than 7.5 marks per hour. It is also sensitive to outliers, so the scatter diagram should be checked before r is trusted.
2. Give two separate reasons why the head's prediction for 15 hours a week is unreliable.
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- Extrapolation. 15 hours is far outside the observed range of 1 to 9 hours. Nothing in the data shows that the same straight-line pattern continues beyond 9 hours; the relationship may level off, and the estimate has no data to support it.
- An impossible answer. The predicted score is 31 + 7.5 × 15 = 31 + 112.5 = 143.5, but the test is marked out of 100. A model that produces a score of 143 is clearly invalid outside the data range.
- Correlation is not causation. Even inside the range, a positive correlation would not justify the claim that adding hours adds marks; students who revise more may already be higher attainers, so forcing everyone to revise 15 hours might produce no gain at all.
Any two, each explained, are enough. The strongest answers also note that an average gain of 7.5 marks per hour does not mean every student gains 7.5.
3. The department wants to claim that 'more revision causes higher test scores'. Describe what extra evidence would be needed and how the existing data could be checked first.
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Check the existing data first:
- Look at the scatter diagram for outliers or influential points - a single unusual student can inflate r and tilt the line of best fit.
- Check whether the relationship looks linear rather than curved.
- Recompute r with the outlier removed (if there is one) to see whether the conclusion survives.
Evidence needed to support a causal claim:
- Measure a likely confounding variable, such as prior test score or attendance, and check whether students who revise more were already scoring higher before the study. If the correlation disappears once prior attainment is taken into account, revision is not the driver.
- Better still, run an experiment: randomly allocate students to different set amounts of revision (for example 2 hours versus 6 hours) and compare test scores. Random allocation balances out other differences such as ability and motivation, which is what allows a causal conclusion.
- Repeat the study with a larger group and in a different class or term to check the result is not a one-off.