S6: Scatter Graphs
Use and interpret scatter graphs; correlation; lines of best fit; interpolation and extrapolation
📋 Key Concepts
Definition: A scatter graph plots two variables against each other to show the relationship between them. Each point represents one data item.
Key Terms
| Term | Definition |
|---|---|
| Correlation | The relationship between two variables |
| Line of Best Fit | A straight line through the data showing the trend |
| Interpolation | Predicting within the range of data |
| Extrapolation | Predicting beyond the range of data |
| Outlier | A point that doesn't fit the general pattern |
📝 Types of Correlation
Positive Correlation: As one variable increases, the other also increases.
Negative Correlation: As one variable increases, the other decreases.
No Correlation: No clear relationship between the variables.
Negative Correlation: As one variable increases, the other decreases.
No Correlation: No clear relationship between the variables.
Strength of Correlation:
- Strong: Points closely follow a line
- Weak: Points are more scattered but still show a trend
- None: Points are randomly scattered
Example 1
Describe the correlation you would expect between:
a) Height and weight
b) Price and demand
c) Shoe size and IQ
Solution:
a) Positive correlation - taller people tend to weigh more
b) Negative correlation - higher prices usually mean lower demand
c) No correlation - these are unrelated
📝 Drawing a Line of Best Fit
Rules for Line of Best Fit:
- Use a ruler to draw a straight line
- The line should follow the general trend
- Approximately equal number of points above and below the line
- The line does NOT have to go through the origin
- Ignore any outliers when drawing the line
Example 2
A scatter graph shows temperature (x-axis) and ice cream sales (y-axis). The data shows a positive correlation. How would you draw a line of best fit?
Solution:
Draw a straight line that:
- Has a positive gradient (slopes up from left to right)
- Has roughly equal points above and below
- Passes through the "middle" of the data cloud
📝 Using Lines of Best Fit
Interpolation: Using the line to predict a value WITHIN the range of data.
Extrapolation: Using the line to predict a value OUTSIDE the range of data (less reliable).
Extrapolation: Using the line to predict a value OUTSIDE the range of data (less reliable).
Example 3
A scatter graph shows revision hours (0-20) and test scores (30-95). The line of best fit passes through (5, 45) and (15, 80).
a) Estimate the score for 10 hours revision.
b) Estimate the score for 25 hours revision.
Solution:
a) 10 hours is within the range (interpolation).
Read up from 10 on x-axis to the line, then across to y-axis.
Estimated score ≈ 62.5
b) 25 hours is outside the range (extrapolation).
Extend the line and read off: ≈ 97.5
Less reliable - we don't know if the pattern continues.
Warning about Extrapolation: Predictions outside the data range may be unreliable because the relationship might not continue in the same way.
📝 Identifying Outliers
Outlier: A data point that is significantly different from the general pattern. It may be an error or a genuine unusual case.
Example 4
A scatter graph shows age and salary. Most points follow a trend, but one point shows a 22-year-old earning £200,000. Is this an outlier?
Solution:
Yes, this point is an outlier. It is far from the general pattern where salary increases with age. This could be:
- A data entry error (perhaps £20,000?)
- A genuine case (e.g. a successful entrepreneur or athlete)
Outliers should be investigated - don't just delete them without checking.
📝 Causation vs Correlation
Important: Correlation does NOT prove causation!
Just because two variables are correlated, it doesn't mean one causes the other.
Just because two variables are correlated, it doesn't mean one causes the other.
Example 5
Studies show positive correlation between ice cream sales and drowning deaths. Does ice cream cause drowning?
Solution:
No! Both are caused by a third variable: hot weather.
Hot weather → More people buy ice cream
Hot weather → More people go swimming → More drowning
This is called a "spurious correlation" or "confounding variable".
Possible explanations for correlation:
- Variable A causes B
- Variable B causes A
- A third variable causes both A and B
- It could be a coincidence
📝 Equation of Line of Best Fit
Form: y = mx + c
where m = gradient (change in y ÷ change in x)
and c = y-intercept (where line crosses y-axis)
where m = gradient (change in y ÷ change in x)
and c = y-intercept (where line crosses y-axis)
Example 6
A line of best fit passes through (2, 10) and (8, 40). Find the equation.
Solution:
Gradient = (40 - 10) ÷ (8 - 2) = 30 ÷ 6 = 5
y = 5x + c
Substitute (2, 10): 10 = 5(2) + c, so c = 0
Equation: y = 5x
Example 7
Using the line y = 5x, predict y when x = 6.
Solution:
y = 5 × 6 = 30
🧮 Worked Examples — Exam Style
Scatter graphs: correlation direction/strength, line of best fit by eye, predictions INSIDE the data range (interpolation — reliable) vs outside (extrapolation — risky).
Describe the correlation between hours of practice and errors made, and what it does NOT prove.
Negative correlation (more practice → fewer errors). It does NOT prove practice CAUSES fewer errors — other factors may explain the pattern (correlation ≠ causation).
B1 correlation, B1 the causation caveat. The 'does not prove' clause is the examined AO3 line.
A best-fit line has equation y = 4x + 1 (x = hours, y = score). Predict the score for 6 hours, and comment on predicting for 50 hours.
y = 25 → 25 (interpolation, reliable). For 50 hours: y = 201 — extrapolation, unreliable (the trend need not continue that far).
M1 for the substitution, A1 for 25, B1 for the extrapolation caveat.
Exam wording to expect: "describe the correlation", "draw the line of best fit", "use it to estimate…", "comment on the reliability".
❓ Practice Questions
Q1: What type of correlation would you expect between hours of sleep and tiredness?
Q2: A line of best fit has equation y = 3x + 5. Predict y when x = 7.
Q3: What is the difference between interpolation and extrapolation?
Q4: A scatter graph shows test scores between 40% and 90%. Is predicting a score of 95% interpolation or extrapolation?
Q5: State two rules for drawing a line of best fit.
Q6: A best-fit line passes through (2, 10) and (6, 30). Write its equation and predict y at x = 10
Q7: A scatter graph shows an outlier far from the trend. What should you do before drawing the line of best fit?
✅ Answers
- Negative correlation - more sleep means less tiredness.
- y = 3(7) + 5 = 21 + 5 = 26
- Interpolation is predicting within the data range; extrapolation is predicting outside the range.
- Extrapolation - 95% is outside the range of 40-90%.
- Any two: use a ruler; follow the trend; equal points above and below; don't force through origin; ignore outliers.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Negative correlation — more sleep, less tiredness
Q2. y = 3(7) + 5 = 26
Q3. Interpolation: predicting within the data range (reliable); extrapolation: outside it (unreliable)
Q4. Extrapolation — 95% exceeds the 40–90% data range
Q5. Any two: ruler; follow the trend; balance points above/below; pass through the mean point if drawn; ignore outliers
Q6. Gradient = (30 − 10)/4 = 5 → y = 5x (through origin? check intercept: 10 = 5×2 ✓) → y = 5x; at x = 10: 50
Q7. Identify and check the outlier — if it is a data error, exclude it; otherwise keep it but note it — then draw the best fit ignoring it if spurious
📌 Key Facts to Memorise
- Correlation: positive/negative × strong/weak/none — describe DIRECTION and STRENGTH
- Best fit: single straight line, balanced points, drawn with a ruler
- Interpolation (inside range) reliable; extrapolation (outside) not
- Correlation ≠ causation — always the AO3 caveat
- Outliers: investigate before excluding; they drag the best-fit line
🗺️ In the Exam
Scatter graphs: 3–5 marks on Papers 2/3 — describing, fitting, estimating. The reliability commentary (interpolation/extrapolation, causation) is where top marks live.
Links to: describing populations (S5), graphs (A9), proportion (R7).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always label axes when drawing scatter graphs
- Use "positive" or "negative" - not "good" or "bad"
- Interpolation is more reliable than extrapolation
- Correlation does NOT mean causation
- When drawing a line of best fit, balance points above and below
- Check for outliers and consider if they are errors
- Use the line of best fit to make predictions, not individual points
🧠 Problem-Solving Strategies
Problem-Solving
For scatter graph problems: (1) Describe correlation using "positive", "negative", or "none" — never "good" or "bad", (2) Draw lines of best fit with roughly equal points above and below, (3) For predictions, use the line of best fit — not individual points, (4) Interpolation (within data range) is more reliable than extrapolation (outside), (5) Correlation does NOT prove causation — always consider confounding variables, (6) Find the equation of the line of best fit using y = mx + c.
Multi-Step Problem
A scatter graph shows revision hours (x) and test scores (y). The line of best fit passes through (5, 40) and (20, 85). (a) Find the equation of the line of best fit. (b) Predict the score for 12 hours revision. (c) Explain why predicting the score for 30 hours may be unreliable.
Solution: (a) Gradient = (85−40)/(20−5) = 45/15 = 3. y − 40 = 3(x − 5), so y = 3x + 25. (b) y = 3(12) + 25 = 61. (c) 30 hours is outside the data range (0–20 hours), so this is extrapolation. The linear relationship may not continue — students might reach a maximum score or tire out, so the prediction may be too high.
⚠️ Common Errors
Watch Out!
1. Wrong: Saying "strong positive correlation" means one variable causes the other Correct: Correlation does NOT imply causation — a third variable could cause both, or it could be coincidence
2. Wrong: Drawing a line of best fit that passes through (0,0) because "it should start at the origin" Correct: The line of best fit does NOT have to go through the origin — it should follow the trend of the data with roughly equal points above and below
3. Wrong: Describing correlation as "good" or "bad" instead of positive/negative/none Correct: Use "positive correlation" (both increase), "negative correlation" (one increases, other decreases), or "no correlation" — avoid value judgements
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Data is collected on ice cream sales and temperature for 12 days:
Temperature (°C): 15, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 35
Sales (£hundreds): 5, 8, 10, 14, 16, 18, 19, 22, 25, 28, 30, 35
(a) Describe the correlation. (b) The line of best fit has equation y = 1.5x − 18. Use this to predict sales at 21°C and 40°C. (c) Which prediction is more reliable? Explain why.
(a) Strong positive correlation — as temperature increases, ice cream sales also increase. The points follow a clear upward trend.
(b) At 21°C: y = 1.5(21) − 18 = 31.5 − 18 = 13.5, so £1,350. At 40°C: y = 1.5(40) − 18 = 60 − 18 = 42, so £4,200.
(c) The prediction at 21°C is more reliable because it is interpolation — 21°C is within the data range (15–35°C). The prediction at 40°C is extrapolation — it's outside the range, and the relationship might not continue linearly. For example, at very high temperatures, people might stay indoors and sales could drop.
Mark scheme: M1 for identifying positive correlation, A1 for "strong positive", M1 for correct substitution, A1 for both predictions, M1 for identifying interpolation vs extrapolation, A1 for explaining why 21°C is more reliable
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A study finds a strong positive correlation between the number of books in a home and children's test scores.
(a) Describe the correlation in context.
(b) A politician says "If we give every family more books, test scores will improve." Evaluate this claim.
(c) Suggest a confounding variable that could explain the correlation.
Answers: (a) Strong positive correlation — homes with more books tend to have children with higher test scores. (b) The claim is not well supported — correlation does not prove causation. Having more books doesn't necessarily cause higher scores; the relationship might be due to other factors. (c) Income/wealth — wealthier families can afford more books AND better education/tutoring. Parental education level is another confounding variable — more educated parents may have more books and also provide more academic support.
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