Scatter Graphs and Correlation

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1. 1

   Plot the pairs as points

   each student → (hours revised, exam mark)

   Each point carries two pieces of information about one individual. This is what makes the graph able to show a relationship at all — the pairing is the data, and plotting the two variables separately would destroy it.

2. 2

   Describe the correlation in words

   points rising left to right → positive correlation

   Say both the direction and the strength: strong positive, weak negative, or no correlation. Questions ask for a description in words, and 'positive' alone often scores less than 'strong positive correlation'.

3. 3

   Draw a line of best fit through the trend

   a straight line following the points, roughly equal numbers each side

   The line should pass through the middle of the cloud of points with a similar number above and below. It does not need to touch any point and should not be forced through the origin.

4. 4

   Use the line to estimate within the data

   8 hours revised → read up to the line, across to ≈ 62 marks

   Reading an unknown value from within the range of the data is interpolation, and it is reliable because the pattern is evidenced there. This is what the line of best fit is for.

5. 5

   Do not extend the line beyond the data

   predicting the mark for 40 hours from data covering 0–12 is extrapolation

   Outside the range of the data there is no evidence the pattern continues, and it usually does not. Questions frequently ask why a prediction is unreliable, and extrapolation is the expected answer.

Describing what you see

Positive correlation means both variables increase together — more revision, higher marks. Negative means one rises as the other falls — more absences, lower marks. No correlation means the points are scattered with no trend, which is a legitimate and examinable answer.

Strength is about how closely the points hug a line. A tight band is strong correlation; a loose cloud with a discernible slope is weak. Both direction and strength are usually needed for full marks.

• Positive — both rise together
• Negative — one rises as the other falls
• None — no discernible trend
• Strong — points close to a line
• Weak — points loosely scattered around it

Correlation is not causation

Two variables moving together does not mean one causes the other. Ice cream sales and drowning incidents correlate strongly, and neither causes the other — hot weather causes both. A third factor explaining both is the usual reason for a surprising correlation.

This is examined regularly, usually by presenting a correlation and asking whether it proves a causal claim. The answer is no, and a good response names a plausible third variable.

Outliers and what to do with them

A point far from the trend is an outlier. It might be a recording error, or it might be a genuine and interesting case, and the graph alone cannot tell you which. Questions often ask students to identify one and comment.

The line of best fit should not be dragged towards an outlier. Drawing the line for the bulk of the data and noting the outlier separately is the correct treatment, and saying so earns the interpretation mark.

How we teach scatter graphs

We drill the interpolation and extrapolation distinction explicitly, because 'why might this estimate be unreliable?' is a guaranteed question and the answer is a single sentence students either have or do not.

We also use a genuinely absurd correlation — ice cream and drownings — early in the topic. It makes the causation point memorable in a way that a careful explanation does not, and students reach for it correctly in exams months later.