A scatter diagram plots two measurements for each item to show whether they are related. Describe the correlation, draw a line of best fit by eye, and use it to estimate, but only inside the range of the data.
Positive correlation: as one variable increases, the other tends to increase. Negative: as one increases, the other tends to decrease. A line of best fit is ONE ruled line across the whole data set with roughly equal numbers of points on each side; use it to estimate only within the data.
Fig. 1: temperature and hot drinks sold on 10 days, drawn to scale. The points show negative correlation. The line of best fit passes close to the mean point (13.7, 67.1) and gives an estimate of 61 drinks at 16°C.
Section 01
Plotting and correlation.
In the lab, pick a data set and drag any cross. Watch the description change from positive to negative or to no correlation as the pattern changes. Try 'Ice creams sold and sunburn cases' and ask yourself whether one really causes the other.
Plot each pair of values as a small cross (\(\times\)), with the two variables on the two axes. Do not join the crosses.
Positive correlation: the points rise from left to right (height and arm span).
Negative correlation: the points fall from left to right (temperature and hot drinks sold).
Zero (no) correlation: no pattern (shoe size and test mark). Points close to a line show strong correlation; a loose cloud shows weak correlation.
An outlier is a point that does not fit the pattern of the others.
Practise reading the scale: the graph scales trainer has you work out one small square, plot values and read points back, on printed scales like the exam grids.
Worked example Describe and interpret
The table shows the temperature and the number of hot drinks sold at a cafe on 10 days. Temperature (°C): 4, 6, 8, 10, 12, 14, 17, 19, 22, 25 Hot drinks: 92, 88, 80, 76, 70, 66, 58, 55, 46, 40 Describe the correlation and explain what it means.
Plot the 10 points (Fig. 1). They fall from left to right and lie close to a straight line.
So there is strong negative correlation.
In context: as the temperature increases, the number of hot drinks sold decreases.
Answer: Strong negative correlation: the warmer the day, the fewer hot drinks sold.
Worked example Spot the outlier
Hours of revision and test marks for 6 students: (1, 30), (2, 37), (3, 44), (4, 15), (5, 58), (6, 63). Which point is an outlier, and give a possible reason.
The other five points rise steadily, by about 6 or 7 marks per hour.
(4, 15) is far below this pattern: about 50 would be expected.
Answer: (4, 15) is the outlier; perhaps that student was ill on the day of the test.
Examiner trap
Correlation does not prove cause. Ice cream sales and sunburn cases have positive correlation, but ice cream does not cause sunburn: both rise in hot, sunny weather.
Section 02
The line of best fit.
In the line of best fit tab, drag the two blue ends to draw your own line by eye. Aim for one straight line across the whole data set with about as many crosses above it as below. Tick 'show the mean point' and make your line pass through it, then drag the green marker to read an estimate.
A single ruled straight line, drawn by inspection (not a curve, not dot-to-dot).
It should extend across the full data set, from the first points to the last.
It does not need to pass through any of the points, but there should be a roughly even spread of points on each side along its whole length.
A good check: it should pass close to the mean point \((\bar{x}, \bar{y})\).
Worked example Use the line to estimate
Using the line of best fit in Fig. 1, estimate (a) the number of hot drinks sold when the temperature is 16°C (b) the temperature on a day when 71 hot drinks are sold.
(a) Go up from 16 on the temperature axis to the line, then across to the other axis: about 61 drinks.
(b) Go across from 71 on the drinks axis to the line, then down: about 12°C.
Both values are inside the range of the data (4°C to 25°C), so the estimates are reliable.
Answer: (a) about 61 drinks (b) about 12°C
Worked example Is the estimate reliable?
The cafe owner uses the same line to estimate the hot drinks sold at 45°C. Comment.
45°C is far outside the data, which go up to 25°C. Extending the line is called extrapolation.
The line in Fig. 1 is \(y = 101 - 2.5x\), so at 45°C it gives \(101 - 2.5 \times 45 = -11.5\) drinks, which is impossible.
Answer: Unreliable: the line should not be used outside the range of the data.
Your turn
A line of best fit passes through (2, 38) and (8, 68). Estimate \(y\) when \(x = 5\).
From \(x = 2\) to \(x = 8\), \(y\) rises 30, so 5 per unit. At \(x = 5\): \(38 + 3 \times 5 = 53\).
Examiner trap
Do not join the points dot-to-dot, and do not force the line through the origin. Examiners also reject a line that stops short of the data or has most points on one side.
Stage 1 · Learn
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The points on a scatter diagram fall from left to right. What type of correlation is this?
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Which of these is a rule for a line of best fit?
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Which pair of variables would you expect to show zero correlation?
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How should points be plotted on a scatter diagram?
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A scatter diagram shows the age of 12 cars and their value. Older cars tend to be worth less. Which statement is correct?
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A line of best fit passes through the points (10, 30) and (50, 90). Use it to estimate \(y\) when \(x = 35\).
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A scatter diagram shows hours of practice (from 2 to 12 hours) against a typing speed. The line of best fit gives 95 words per minute at 30 hours. Which comment is correct?
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Ice cream sales and sunburn cases in a town show strong positive correlation. Which conclusion is correct?
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