- Put the independent (controlled) variable on the horizontal axis.
- Put the dependent (measured) variable on the vertical axis.
- Label both axes, including units.
- Plot one point for each pair of readings — with a small cross or dot.
- Do not join the points up.
| Correlation | Meaning | Example |
|---|---|---|
| Positive | As one increases, the other increases | Height and shoe size |
| Negative | As one increases, the other decreases | Age of a car and its value |
| None | No clear pattern | Shoe size and exam score |
| Strong | Points lie close to a straight line | — |
| Weak | Points are scattered but show a tendency | — |
e.g. "strong negative correlation", not just "negative"
A scatter graph of a car's age against its value shows points falling steeply and lying close to a straight line. Describe the correlation and what it means.
Strong negative correlation: as a car gets older, its value tends to decrease.
- Only draw one if the points show clear correlation.
- Draw a single straight line with a ruler.
- Follow the general direction of the points.
- Aim for roughly equal numbers of points above and below the line.
- Do not force it through the origin unless the data suggest it.
- Ignore any obvious outlier when positioning the line.
A line of best fit passes through $(2,\ 30)$ and $(8,\ 66)$ on a graph of revision hours against test score. Estimate the score for $5$ hours of revision.
Estimated score: $48$ marks.
Using $y = 6x + 18$ from above, interpret the gradient and the intercept in context.
Extrapolation — estimating outside the range. Unreliable.
A scatter graph shows revision hours from $1$ to $9$ against test scores. Comment on the reliability of estimates for (a) $6$ hours, (b) $20$ hours.
A line of best fit for children's heights against age is $h = 6a + 75$, fitted using ages $4$ to $11$. Use it to predict the height of a $40$-year-old, and comment.
The prediction is meaningless because $40$ is far outside the age range $4$–$11$ used to fit the line. People stop growing in their late teens, so the linear relationship simply does not hold at that age.
It does not show that one causes the other.
• $A$ causes $B$
• $B$ causes $A$
• A third factor causes both (a confounding variable)
• It is pure coincidence
| Correlation observed | The real explanation |
|---|---|
| Ice cream sales and drownings both rise | Hot weather causes both |
| Shoe size and reading ability in children | Age causes both |
| Number of firefighters and damage at a fire | The size of the fire causes both |
| Chocolate consumption and Nobel prizes by country | National wealth is linked to both |
A study finds a strong positive correlation between the number of ice creams sold and the number of people who get sunburnt. Does eating ice cream cause sunburn?
This third factor is called a confounding variable.
A scatter graph shows strong negative correlation between the amount of insulation in a house and its annual heating bill. Is it reasonable to suggest causation here?
Causation is plausible here — but the scatter graph alone still does not prove it. Other factors (house size, how warm people keep their homes) could also be involved. A controlled experiment would be needed to be sure.
On a graph of revision hours against test score, most points follow a rising pattern, but one student who revised for $2$ hours scored $95$. How should this be treated?
Do not simply delete the point from the graph — plot it, but do not let it pull the line.
Scatter graph
One point per item; never join the points.
Positive
Both variables increase together.
Negative
One increases as the other decreases.
Describe it fully
Strength and direction, then interpret in context.
Line of best fit
Straight, follows the trend, roughly equal points either side.
Gradient
The change in $y$ per unit increase in $x$ — give the units.
Interpolation
Inside the data range — reliable.
Extrapolation
Outside the data range — unreliable.
Correlation
Never proves causation; look for a third factor.
Outliers
Plot them, but ignore them when drawing the line.
Describe the correlation you would expect between: (a) a car's engine size and its fuel economy, (b) a person's height and their arm span, (c) a person's house number and their age.
▶ Show solution
(a) Negative — larger engines generally use more fuel, so economy falls.
(b) Strong positive — taller people generally have longer arm spans.
(c) No correlation — the two are unrelated.
Explain why you should never join the points on a scatter graph.
▶ Show solution
Each point represents a separate item (a different person, car or day), not consecutive stages of one thing changing.
Joining them would suggest a sequence or a path between the points, which has no meaning. If a line is needed, a single straight line of best fit is drawn instead.
A line of best fit passes through $(0,\ 12)$ and $(10,\ 52)$. Find its equation.
▶ Show solution
Gradient $= \dfrac{52 - 12}{10 - 0} = \dfrac{40}{10} = 4$
The line crosses the vertical axis at $12$, so $c = 12$.
$y = 4x + 12$
Using $y = 4x + 12$ from Question 3, where $x$ is hours worked and $y$ is pay in pounds, interpret the gradient and the intercept.
▶ Show solution
Gradient $4$: pay increases by £$4$ for every extra hour worked.
Intercept $12$: a fixed payment of £$12$ is received even for zero hours — perhaps a call-out fee or bonus.
Data were collected for ages $5$ to $16$. Explain whether an estimate for age $12$ and one for age $30$ would be reliable.
▶ Show solution
Age $12$: inside the range $5$–$16$, so this is interpolation — the estimate is reliable.
Age $30$: well outside the range, so this is extrapolation — unreliable, because there is no evidence that the pattern continues beyond age $16$.
A study finds a positive correlation between the number of storks nesting in a region and the number of babies born there. Does this mean storks deliver babies? Explain.
▶ Show solution
No. The correlation is real but there is no causal link.
A likely third factor is the size or type of the region: larger rural areas have both more buildings for storks to nest on and more people, and therefore more births.
This is a classic example of correlation without causation.
A line of best fit on a graph of a car's age (years) against value (£) is $V = 15\,000 - 1200a$. Interpret the gradient and use the equation to estimate the value of a $6$-year-old car.
▶ Show solution
Gradient $-1200$: the car loses about £$1200$ of value each year.
At $a = 6$: $V = 15\,000 - 1200(6) = 15\,000 - 7200 = £7800$
Using the equation from Question 7, find the value predicted for a $15$-year-old car, and comment.
▶ Show solution
$V = 15\,000 - 1200(15) = 15\,000 - 18\,000 = -£3000$
A negative value is impossible, so the prediction is meaningless.
This shows the danger of extrapolating too far: in reality a car's value falls steeply at first and then levels off towards a small positive amount, rather than continuing to drop by £$1200$ every year.
A scatter graph of temperature (°C) against ice cream sales shows a line of best fit through $(10,\ 40)$ and $(25,\ 130)$.
(a) Find the gradient and interpret it. (b) Find the equation. (c) Estimate sales at $18$°C. (d) Would you use the line to estimate sales at $35$°C?
▶ Show solution
(a) Gradient $= \dfrac{130 - 40}{25 - 10} = \dfrac{90}{15} = 6$
Sales increase by about $6$ ice creams for each $1$°C rise in temperature.
(b) $y = 6x + c$; using $(10, 40)$: $40 = 60 + c$, so $c = -20$.
$y = 6x - 20$
(c) At $x = 18$: $y = 108 - 20 = 88$ ice creams.
(d) $35$°C is outside the data range $10$–$25$°C, so this would be extrapolation and unreliable. The estimate of $190$ might also exceed what the shop can physically sell in a day.
The table shows the hours of sleep and reaction times (milliseconds) for eight people.
| Sleep (h) | $4$ | $5$ | $5.5$ | $6$ | $7$ | $7.5$ | $8$ | $9$ |
|---|---|---|---|---|---|---|---|---|
| Reaction (ms) | $420$ | $390$ | $385$ | $360$ | $330$ | $325$ | $300$ | $285$ |
(a) Describe the correlation. (b) A line of best fit passes through $(4,\ 420)$ and $(9,\ 285)$. Find its equation. (c) Estimate the reaction time for someone who sleeps $6.5$ hours. (d) A newspaper claims "sleeping more makes you react faster". Comment on this claim.
▶ Show solution
(a) As sleep increases, reaction time decreases, and the points follow the pattern closely.
This is strong negative correlation — people who sleep longer tend to have faster reaction times.
(b) Gradient $= \dfrac{285 - 420}{9 - 4} = \dfrac{-135}{5} = -27$
Using $(4, 420)$: $420 = -27(4) + c = -108 + c$, so $c = 528$.
$y = -27x + 528$
(c) At $x = 6.5$: $y = -27(6.5) + 528 = -175.5 + 528 = 352.5$ ms
(d) The data show a strong correlation, but that does not establish causation.
It is plausible that more sleep improves alertness, so the claim is not unreasonable — but a third factor could explain the pattern. For example, people who are ill, stressed or older may both sleep less and react more slowly. To support the claim properly you would need a controlled experiment in which sleep is varied deliberately.