Scatter Graphs, Correlation, Lines of Best Fit, Interpolation & Sampling

SPEC S1 + S5 + S6 • STATISTICS

Scatter Graphs & Sampling

Correlation, lines of best fit, interpolation vs extrapolation, and the basics of sampling. High-priority on every paper.

Scatter Graphs

A scatter graph shows a relationship between two variables by plotting pairs of values as points. Unlike a line graph, the points are not joined — the whole picture comes from the cloud.

Bivariate data

Data that pairs two measurements from the same subject — e.g. a person's height and weight, or a car's age and its price.

Scatter graph conventions

One variable on each axis — usually the independent variable (input) on the x-axis and the dependent variable (output) on the y-axis.

Plot each pair as a single point (usually a cross ×).

Do not join the points — leave them as a cloud.

Label both axes with the variable names and units.

MCE approach

Method — identify the type and strength of correlation; if appropriate, draw a line of best fit. Calculate — use the line to make predictions (reading off both axes). Evaluate — check whether the prediction is inside the data range (reliable) or outside it (unreliable).

Types of Correlation

The shape of the cloud tells you the relationship between the two variables.

Correlation must be specified by type

Writing "correlation" alone doesn't score marks. The type must be stated: positive, negative, or no correlation. AQA mark schemes are explicit about this.

Correlation Strength

The tightness of the cloud around a straight-line trend tells you how strong the correlation is. Strong, moderate, or weak.

How to describe

Strong: points lie close to an obvious straight-line trend.

Moderate: trend is visible but with noticeable scatter around it.

Weak: trend is just barely visible — points are very spread out but still tend in one direction.

Correlation ≠ Causation

The key principle

A correlation between two variables does not prove that one causes the other. There may be a third variable (a confounder) influencing both, or the link may be pure coincidence.

Classic example

Data shows a strong positive correlation between ice cream sales and shark attacks. Does ice cream cause shark attacks? No — the confounding variable is hot weather, which independently boosts both ice cream sales and the number of people swimming at beaches.

What to write for the mark

AQA examiners want a named alternative explanation, not a vague statement. "Correlation doesn't mean causation" alone is weak. Better: "There may be a third factor — e.g. hot weather increases both ice cream sales and swimming, which is why they are correlated."

Other classic confounders

Shoe size and reading ability in children — both grow with age.

Number of firefighters at a fire and damage caused — both grow with the fire's size.

Chocolate consumption and Nobel Prize winners — both tend to correlate with national wealth.

When a Line of Best Fit is Appropriate

A straight line of best fit is only useful when the data follows a roughly linear trend. Not every scatter graph supports one.

Ask yourself before drawing

Do the points follow a roughly straight-line trend? If yes → draw a straight LOBF.

Do the points follow a curve? → a straight LOBF is inappropriate.

Are the points randomly scattered? → no correlation, so don't draw a LOBF.

Outliers

A point that stands out from the general trend of the rest of the data. Outliers are ignored when drawing a LOBF — the line follows the main cloud, not the odd point.

Outlier recognition on AQA

A common 1-mark question shows four scatter graphs and asks "which has one outlier?" The outlier is the single point that clearly doesn't fit the pattern of the others — typically well above or below the trend of the rest of the cloud.

Drawing a Line of Best Fit

Four rules

Use a ruler — freehand lines are not accepted.

Place the line so roughly equal numbers of points sit above and below it.

Try to balance how far points are from the line — not just their count.

Ignore outliers when placing the line.

Common myths

❌ "The line must pass through the origin." Not true — only if the data supports it.

❌ "The line must pass through specific data points." Not true — it doesn't have to pass through any point.

❌ "The line must pass through the first and last points." Not true — often misleading.

Good LOBF habit

Mark the mean of x and the mean of y — the best-fit line should pass roughly through this "mean point" (x̄, ȳ). This is a useful sanity check even if not required in the mark scheme.

Using the LOBF for Predictions

Once drawn, the LOBF lets you estimate the value of one variable given a value of the other.

Three-step routine

Find the given value on its axis.

Draw a line to the LOBF, then a perpendicular to the other axis.

Read the estimated value off that axis.

Show your read-off lines

Always draw the two straight lines (horizontal + vertical) on the graph paper to show which value you're reading. AQA awards method marks for visible construction even if the final number is slightly off.

Worked Example — Scatter Graph & LOBF01

The scatter graph shows the number of hours 9 students spent studying and their exam scores out of 100. (a) Describe the correlation. (b) Draw a line of best fit. (c) Estimate the score of a student who studied for 5.5 hours. [4 marks]

MMethod. (a) Describe both the direction and the strength of the cloud. (b) Place ruler so roughly equal points sit above/below; ignore outliers. (c) Read 5.5 up to the line, across to the y-axis.

Calculate.

(a) As hours studied increases, exam score increases. The cloud clusters closely around a straight-line trend. Strong positive correlation.

(b) LOBF drawn as shown (roughly passing through (0, 30) and (10, 90)).

(c) At h = 5.5, reading off gives approximately 63 marks.

EEvaluate. 5.5 hours is inside the data range (2 to 10), so this is interpolation — the prediction is reasonably reliable. 63 is between the observed values at 5 hours (55) and 6 hours (70), which makes sense. AQA accepts readings within ±2 marks of the expected answer.

Interpolation vs Extrapolation

Interpolation

Predicting a value inside the range of the observed data. Generally reliable because the relationship has been demonstrated in that range.

Extrapolation

Predicting a value outside the range of the observed data. Generally unreliable — there's no evidence the pattern continues beyond what was measured.

WE1 example

Data range: 2 to 10 hours studied. Predicting a score at 5.5 hours is interpolation → reliable.

Predicting a score at 15 hours is extrapolation — the formula gives 6(15) + 30 = 120, but exam scores can't exceed 100. Unreliable.

AQA examiner phrase

If a prediction is outside the data range, state explicitly: "This prediction is an extrapolation, so it may be unreliable as we have no evidence the pattern continues outside the observed data range." This is worth a mark on "explain" questions.

Sampling — Why We Use Samples

A population is the whole group being studied. A sample is a subset of that population. Asking everyone is usually impossible, so we use a sample to infer properties of the whole.

Key sampling terms

Population: the complete group of interest.

Sample: a smaller group taken from the population.

Random sample: every member of the population has an equal chance of being picked.

Stratified sample: the population is split into groups (strata), and a proportional number from each is sampled.

Biased sample: doesn't fairly represent the population — skewed in some way.

Why bigger samples are better

Larger samples give more reliable estimates. A sample of 1000 gives a better picture of a population than a sample of 10. But larger samples are also more expensive and time-consuming — there's a trade-off.

Stratified Sampling

When a population has distinct groups (strata) — boys vs girls, different year groups, different regions — a stratified sample draws from each group proportionally. This keeps the sample representative of the population's composition.

The stratified sampling formula

sample from stratum = (stratum size ÷ population) × sample size

Four-step routine

Find the total population (sum the strata).

For each stratum, compute (stratum size ÷ population).

Multiply each fraction by the sample size.

Check that your stratum samples add up to the sample size — they must.

Round to whole numbers

People can't be partial — your stratum sample sizes must be whole numbers. AQA questions are usually designed so the numbers divide cleanly, but if rounding is needed, round each stratum's figure and check the sizes still total the intended sample size.

Worked Example — Stratified Sample02

A youth club has 350 members under the age of 16, and 150 members aged 16 or over. A stratified sample of 40 members is to be taken. How many from each age group should be in the sample? [3 marks]

MMethod. Total population; for each stratum, compute (stratum ÷ total) × sample size.

Calculate.

Total: 350 + 150 = 500 members.

•

Under 16 sample: (350 ÷ 500) × 40 = 0.7 × 40 = 28 members .

•

16 or over sample: (150 ÷ 500) × 40 = 0.3 × 40 = 12 members .

Check: 28 + 12 = 40 ✓

EEvaluate. The under-16s form 70% of the population (350/500), so 70% of the sample (28 of 40) comes from that group. The proportions match — sample is representative.

Bias & Limitations of Sampling

Even a correctly-sized sample can give misleading conclusions if it's biased — not representative of the population.

Common sources of bias

Self-selection: only motivated people respond (e.g. online surveys reach tech-savvy users).

Convenience sampling: asking only people nearby (e.g. outside one supermarket).

Time of day: surveying at 10 am misses people who work nine-to-five.

Location: surveying one street doesn't represent a whole town.

Fundamental limitation

Even the best sample is an estimate of the population — never a guarantee. The larger and more representative the sample, the more reliable the conclusion, but there is always some uncertainty.

"Give a reason" AQA answers

If asked to criticise a sampling method, name the bias and explain how it skews the result: "Asking only people leaving a gym will overestimate fitness levels in the whole town because gym-users are more active than average."

Examiner Alerts

Alert 1 — "Correlation" without specifying type

"There is correlation" alone is not enough. Mark schemes require positive, negative, or no correlation to be stated.

Alert 2 — Extrapolating without a caveat

Predicting a value outside the data range is extrapolation and may be unreliable. If you use a LOBF to predict outside the data, always explicitly state this in the answer.

Alert 3 — Correlation = causation

"A causes B because they are correlated" scores zero. Say the correlation doesn't imply causation and name a plausible third variable.

Alert 4 — Freehand line of best fit

AQA mark schemes require LOBFs to be drawn with a ruler. Freehand lines are penalised.

Alert 5 — Equal-size stratified samples

A stratified sample must be proportional to each stratum's size. Splitting the sample equally between strata ignores their relative sizes — this is not stratified sampling.

Practice Question

The scatter graph shows daily temperatures (°C) and ice cream sales at a kiosk on nine days. A line of best fit has been drawn.

(a) Describe the type and strength of correlation. [1 mark]

(b) Estimate the ice cream sales at a temperature of 27 °C. [2 marks]

(c) The owner estimates sales at 45 °C using this line. Explain why this estimate may be unreliable. [2 marks]

(d) Explain why a positive correlation here does not prove that hotter weather directly causes more ice creams to be sold. [1 mark]

MMethod. (a) Describe direction and tightness. (b) Interpolation using LOBF. (c) Extrapolation argument. (d) State correlation ≠ causation with a named alternative.

Calculate.

(a) Strong positive correlation — as temperature increases, ice cream sales increase, and the points lie close to a straight line.

(b) At 27 °C, read up to the LOBF and across to the y-axis. Sales ≈ 74 ice creams (AQA accepts approximately 70–78).

(c) 45 °C is outside the data range (15–35 °C). This is extrapolation — there is no evidence the linear trend continues beyond 35 °C, so the prediction may be unreliable.

(d) Correlation does not prove causation. A third variable (a confounder) could influence both temperature and ice cream sales — for example, time of year: in summer, both temperatures and factors unrelated to heat (school holidays, tourists, longer opening hours) drive up sales. Correlation alone cannot rule these out.

EEvaluate. All four answers are statistical — correlation described precisely, interpolation within range gives reliable value, extrapolation caveat is clear, and the causation answer includes a named alternative explanation.

Summary — MCE Checkpoint

Scatter graphs & correlation

Positive : as x increases, y increases. Negative : as x increases, y decreases. None : random scatter.

Strong (tight cluster) vs weak (loose cluster).

Correlation ≠ causation — always consider a confounding variable.

Line of best fit

Draw with a ruler , balanced above/below the points, ignoring outliers.

Does not have to pass through origin or any data point.

Should pass roughly through the mean point (x̄, ȳ).

Use it for predictions: find value on one axis, read across/up to the line, then off the other axis.

Interpolation vs extrapolation

Interpolation (inside data range): reliable.

Extrapolation (outside data range): unreliable — always state the caveat.

Sampling

sample from stratum = (stratum size ÷ population) × sample size

Stratified sample = proportional to stratum size.

Check stratum samples add to the total sample.

Samples give estimates , never guarantees — bigger, more representative samples are more reliable.

Examiner traps

State the type of correlation (positive/negative/none), not just "correlation".

Extrapolation → add a reliability caveat.

Causation → name a third variable .

LOBF with a ruler , not freehand.

Stratified samples are proportional , not equal.

✓ You should now be able to

Construct and interpret scatter graphs with appropriate axes.

Identify correlation type (positive, negative, none) and strength (strong, moderate, weak).

Identify outliers and know when a straight LOBF is appropriate.

Draw a line of best fit with a ruler, respecting the data trend.

Make predictions using a LOBF (interpolation) and recognise the risk of extrapolation.

Explain why correlation does not imply causation, naming an alternative explanation.

Compute a stratified sample proportional to stratum sizes.

Identify sources of bias and describe limitations of sampling.