Scatterplots, association and lines of best fit: QCE Essential Mathematics Unit 4
“Construct and interpret scatterplots of two numerical variables, identify the explanatory and response variables, describe the form, direction and strength of an association, draw a line of best fit by eye, and use it to make and judge predictions”
A scatterplot shows two numerical variables, with the explanatory variable on the horizontal axis. Describe association by direction, form and strength, draw a line of best fit by eye to predict and interpret the rate of change, trust interpolation more than extrapolation, and remember association does not prove causation.
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What this dot point is asking
In Unit 4 you work with bivariate data: two numerical variables recorded for each person or thing. You need to plot and interpret scatterplots, describe associations, draw and use a line of best fit, and judge how reliable predictions are.
The answer
Scatterplots
- The explanatory variable (the one that may explain or predict) goes on the horizontal axis.
- The response variable (the one being predicted) goes on the vertical axis.
- Each individual is one point.
Describing association
- Direction: positive (both increase together), negative (one increases as the other decreases) or no association.
- Form: linear (points follow a straight line) or non-linear (a curve).
- Strength: strong (points close to the line), moderate or weak (widely scattered).
Also mention any outliers.
Line of best fit
Draw a straight line by eye through the middle of the points, with roughly equal numbers of points above and below, following the trend. Use it to:
- predict a response value from an explanatory value (read up to the line and across);
- describe the rate of change: the slope tells how much the response changes for each one-unit increase in the explanatory variable.
Interpolation (predicting within the data range) is usually reliable. Extrapolation (beyond the data range) is risky because the trend may change.
Association is not causation
A strong association does not prove that one variable causes the other. There may be a third (lurking) variable, or it may be coincidence.
Data on hours of part-time work per week and hours of study per week for 12 Year 12 students shows a moderate, negative, linear association. A line of best fit passes through (5, 14) and (20, 5).
- Slope = (5 minus 14) divided by (20 minus 5) = negative 9 divided by 15 = negative 0.6. Each extra hour of work goes with about 0.6 fewer hours of study.
- Prediction for 12 hours of work: from (5, 14), increase by 7 hours, so study = 14 minus 0.6 × 7 = 9.8 hours (interpolation, reasonably reliable).
- Caution: other factors (subject load, motivation) also affect study time, so work hours alone do not cause the change.
- Swapping the axes
- The explanatory variable goes on the horizontal axis.
- Joining the dots
- A line of best fit is one straight line through the trend, not a zigzag through every point.
- Trusting extrapolation
- Predictions far outside the data can be impossible.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksAn ice-cream shop records the maximum temperature and number of ice-creams sold each day. (a) Which is the explanatory variable? (b) Describe the association you would expect. (c) Where should each variable be plotted?Show worked solution →
(a) Temperature is the explanatory variable (it helps explain sales).
(b) A positive association: as temperature increases, sales tend to increase.
(c) Temperature on the horizontal axis, ice-creams sold on the vertical axis.
Marking guide: 1 mark each.
core4 marksA line of best fit for a used car's value against its age is: value = 32 000 minus 2400 times age (in years), for cars aged 1 to 10 years. (a) Predict the value of a 6-year-old car. (b) Interpret the number 2400. (c) Is it sensible to use the line for a 15-year-old car? Explain.Show worked solution →
(a) Value = 32 000 minus 2400 × 6 = 32 000 minus 14 400 = $17 600.
(b) On average, the car's value decreases by $2400 for each extra year of age.
(c) No. 15 years is outside the data range (extrapolation). The line gives 32 000 minus 36 000 = negative $4000, which is impossible, showing the pattern does not continue.
Marking guide: 1 mark for (a), 1 mark for (b), 2 marks for (c).
exam5 marksA study of 30 suburbs finds a strong positive association between the number of coffee shops and the number of car thefts. A council member says coffee shops cause car theft. Evaluate this claim.Show worked solution →
- What the data shows
- There is a strong positive association: suburbs with more coffee shops tend to have more car thefts.
- Why it does not prove causation
- A third variable is likely to explain both. Busy suburbs with more people, shops and parked cars (for example near city centres) will have more coffee shops and more cars that can be stolen. Population or the number of parked cars is a possible confounding (lurking) variable.
- Better analysis
- Compare car thefts per 1000 parked cars or per 1000 residents, or look at other factors such as street lighting and public transport hubs.
- Judgement
- The claim is not supported. The association is real, but it does not show that coffee shops cause car theft.
Marking guide: 1 mark for describing the association, 2 marks for a plausible third variable, 1 mark for a better analysis, 1 mark for a judgement.