Mathematics · AQA 8300 §S5
Correlation and Scatter Graphs
- Plot and interpret scatter graphs from bivariate data
- Identify and describe the type and strength of correlation
- Draw a line of best fit through the mean point $(\bar{x}, \bar{y})$
- Use the line of best fit for interpolation and extrapolation, evaluating reliability
- Distinguish clearly between correlation and causation
🔑 Core Concepts
2.1 Scatter Graphs and Bivariate Data
A scatter graph (or scatter diagram) is used to investigate the relationship between two numerical variables. Each data point is plotted as a coordinate $(x, y)$ where $x$ is the independent variable (cause/explanatory) and $y$ is the dependent variable (effect/response). Because we are looking at two variables for each observation, this is called bivariate data.
DEFINITION — Bivariate Data
Data that records two measurements for each individual or item, allowing us to investigate whether there is a relationship (association) between the two variables.EXAM TIP — Plotting Scatter Graphs
Label both axes with the variable name and its units. Plot each point with a small cross (×). Do not join the points with a line — they are not a line graph.COMMON MISTAKE — Joining Points
Students sometimes join scatter graph points with straight-line segments. This is wrong. A scatter graph shows individual data points only; any line drawn should be the line of best fit.2.2 Types of Correlation
Correlation describes the direction and strength of the linear relationship between two variables. There are three types of correlation:
Positive Correlation
r > 0
As $x$ increases, $y$ tends to increase. Points cluster around a line sloping upward left-to-right.
Example: Hours studied vs. exam score.
Negative Correlation
r < 0
As $x$ increases, $y$ tends to decrease. Points cluster around a line sloping downward left-to-right.
Example: Temperature vs. number of coats sold.
No Correlation
r ≈ 0
No clear linear pattern. Points are scattered randomly across the diagram.
Example: Shoe size vs. intelligence.
DEFINITION — Strength of Correlation
Strong: Points lie close to a straight line.Moderate: Points show a clear trend but with noticeable scatter.
Weak: There is a slight trend but points are widely scattered.
EXAM TIP — Describing Correlation
Always use two adjectives: the strength (strong / moderate / weak) and the direction (positive / negative). For example: "strong positive correlation" or "weak negative correlation." A single-word answer will not earn full marks.2.3 Line of Best Fit
The line of best fit is a straight line drawn through the scatter graph to best represent the overall linear trend of the data. It is drawn by eye such that:
- Roughly equal numbers of points lie above and below it.
- It passes through (or very near) the mean point $(\bar{x}, \bar{y})$.
- It represents the general direction of the data.
Mean Point
$$(\bar{x},\,\bar{y}) = \left(\frac{\sum x}{n},\,\frac{\sum y}{n}\right)$$
$\bar{x}$ = mean of all $x$-values
$\bar{y}$ = mean of all $y$-values
$n$ = number of data points
$\sum$ = sum of all values
IMPORTANT — The Mean Point Must Lie on the Line
The line of best fit must pass through the mean point $(\bar{x}, \bar{y})$. In an exam, if you are asked to draw a line of best fit, calculate the mean point first, plot it on the graph, and draw your line through it. This is the single most important property of the line of best fit.EXAM TIP — Drawing the Line
Mark the mean point $(\bar{x}, \bar{y})$ with a special symbol (e.g., a circle around a cross) so the examiner can see you have calculated it. Extend the line to cover the full range of $x$-values shown.COMMON MISTAKE — Forcing the Line Through the Origin
The line of best fit does not have to pass through the origin $(0, 0)$ unless the context makes this certain. Always let the data guide you.2.4 Interpolation and Extrapolation
Once a line of best fit is drawn, it can be used to predict values of one variable given the other. The reliability of predictions depends on whether you are working within or outside the range of the data.
Given an $x$-value for prediction
→
Is it within the data range?
→
YES → Interpolation (reliable)
→
NO → Extrapolation (unreliable)
DEFINITION — Interpolation
Reading off a predicted value from the line of best fit for an $x$-value that lies within the observed data range. This is generally reliable because the observed data supports the trend in that region.DEFINITION — Extrapolation
Reading off a predicted value from the line of best fit for an $x$-value that lies outside the observed data range. This is generally unreliable because we are assuming the linear trend continues beyond the region we have evidence for, which may not be true.COMMON MISTAKE — Treating All Predictions as Equally Reliable
Students often make a prediction by extrapolation without commenting on its unreliability. In an exam, if the question asks "how reliable is this prediction?" and the value is outside the data range, you must say it is unreliable because it involves extrapolation.EXAM TIP — Evaluating Reliability
When evaluating reliability, consider: (1) Is it interpolation or extrapolation? (2) How strong is the correlation? A prediction from a strong correlation is more reliable than from a weak one, even for interpolation. (3) Is the sample size large enough? (4) Is the sample representative?2.5 Correlation vs Causation
This is one of the most important conceptual distinctions in statistics and is heavily tested at Grade 9. Correlation does not imply causation.
DEFINITION — Causation
A causal relationship exists when a change in one variable directly causes a change in the other. For example, applying more fertiliser directly causes plants to grow taller (within limits).DEFINITION — Spurious Correlation
A correlation between two variables that arises not because one causes the other, but because both are influenced by a third (confounding) variable, or purely by coincidence. For example, ice cream sales and drowning rates both correlate positively — but ice cream does not cause drowning; both are caused by hot weather.MEMORY TRICK — Correlation ≠ Causation
"Just because two things happen together does not mean one makes the other happen."Classic exam example: "A study found that countries with more TVs per household have higher life expectancies. Does this mean watching TV makes you live longer?" Answer: No — both are caused by a higher standard of living (confounding variable).
EXAM TIP — Causation Questions
If the exam asks "Does this scatter graph prove that X causes Y?", the answer is always "No." You must state: (1) there is a correlation between X and Y, (2) this does not prove causation, (3) there may be a third variable (confounding factor) responsible for both, or it could be coincidence.2.6 Pearson's Product Moment Correlation Coefficient (PMCC)
At GCSE, you are not required to calculate the PMCC, but you must be able to interpret its value. The PMCC, denoted $r$, measures the strength and direction of the linear correlation between two variables on a scale from $-1$ to $+1$.
PMCC Range
$$-1 \leq r \leq 1$$
$r = +1$: perfect positive linear correlation
$r = -1$: perfect negative linear correlation
$r = 0$: no linear correlation
$|r|$ close to 1 indicates strong correlation
$r = -1$
Perfect negative $r = 0$
No linear correlation $r = +1$
Perfect positive
Perfect negative $r = 0$
No linear correlation $r = +1$
Perfect positive
| Value of $r$ | Interpretation |
|---|---|
| $r = 1$ | Perfect positive linear correlation — all points lie exactly on an upward-sloping line |
| $0.7 \leq r < 1$ | Strong positive correlation |
| $0.4 \leq r < 0.7$ | Moderate positive correlation |
| $0 < r < 0.4$ | Weak positive correlation |
| $r = 0$ | No linear correlation |
| $-0.4 < r < 0$ | Weak negative correlation |
| $-0.7 < r \leq -0.4$ | Moderate negative correlation |
| $-1 < r \leq -0.7$ | Strong negative correlation |
| $r = -1$ | Perfect negative linear correlation — all points lie exactly on a downward-sloping line |
COMMON MISTAKE — PMCC and Non-Linear Relationships
The PMCC measures linear correlation only. Two variables can have a strong non-linear (e.g., quadratic or exponential) relationship and yet have an $r$ value close to 0. A value of $r \approx 0$ means no linear correlation, not necessarily no relationship at all.EXAM TIP — Interpreting $r$
When given a PMCC value, write a full sentence: "A value of $r = -0.85$ indicates a strong negative linear correlation between the two variables." Always state both the strength (strong/moderate/weak) and direction (positive/negative) in context.🗺️ Visual Notes
Correlation & Scatter Graphs
📊 Scatter Graphs
- Plot bivariate data as $(x, y)$ points
- Each axis: variable name + units
- Use × symbols for each point
- Do NOT join the points
↗ Types of Correlation
- Positive: both increase together
- Negative: one increases as other falls
- No correlation: no linear pattern
- Strength: strong / moderate / weak
📏 Line of Best Fit
- Drawn by eye through mean point $(\bar{x}, \bar{y})$
- Equal points above and below
- Extends across full data range
- Only draw if correlation exists
🔮 Predictions
- Interpolation: within range (reliable)
- Extrapolation: outside range (unreliable)
- Stronger correlation → more reliable
- Read off line, not nearest point
⚖️ Causation vs Correlation
- Correlation ≠ causation
- Confounding variables may exist
- Need controlled experiment to prove cause
- Can be coincidence (spurious correlation)
📐 PMCC ($r$)
- Range: $-1 \leq r \leq 1$
- $r = \pm 1$: perfect linear correlation
- $r = 0$: no linear correlation
- Measures linear relationships only
Interpolation vs Extrapolation — Comparison
| Feature | Interpolation | Extrapolation |
|---|---|---|
| Where on graph? | Within the plotted data range | Outside the plotted data range |
| Reliability | Generally reliable | Generally unreliable |
| Assumption required? | Minimal — trend is supported by data | Assumes trend continues beyond data |
| Risk | Low (data supports the trend) | High (trend may change outside range) |
| Exam wording | "Estimate the value of $y$ when $x = 5$" (within range) | "Predict the value of $y$ when $x = 20$" (beyond data) |
Correlation vs Causation — Comparison
| Aspect | Correlation | Causation |
|---|---|---|
| Definition | Association/relationship between two variables | One variable directly makes the other change |
| Evidence needed | Scatter graph / statistical analysis | Controlled experiment |
| Proven by scatter graph? | Yes | No |
| May have third variable? | Yes (confounding variable) | No — direct link established |
| Example | Ice cream sales and drowning rates | Smoking and lung cancer |
Decision Tree — Using a Scatter Graph for Prediction
Is there a clear correlation?
→
NO → Do not draw line of best fit. Cannot make reliable predictions.
→
YES → Calculate $(\bar{x}, \bar{y})$ and draw line of best fit
→
Is the $x$-value within the data range?
→
YES → Interpolate (reliable estimate)
→
NO → Extrapolate (unreliable — state this in answer)
Scatter Graph Illustrations
✏️ Worked Examples
Grade 4–5 | Basic Interpretation
The scatter graph shows the number of hours 8 students revised and their test scores (out of 50). Describe the correlation and use the line of best fit to estimate the score for a student who revised for 5 hours.
1
Identify the correlation type
Look at the overall pattern: as revision hours increase, test scores also increase. The points cluster reasonably close to a straight line sloping upward. This is a strong positive correlation.
2
Use the line of best fit
Locate $x = 5$ on the horizontal axis. Draw a vertical line up to the line of best fit, then read across horizontally to the vertical axis. The line of best fit gives approximately $y = 38$.
3
Comment on reliability
The data range was 1 to 9 hours, so $x = 5$ is within the range. This is interpolation — the estimate is reliable. Also, the correlation is strong, which increases confidence in the estimate.
The correlation is strong and positive. The estimated score for 5 hours revision is approximately 38 marks. This is a reliable estimate as it uses interpolation within the data range.
Grade 6–7 | Mean Point and Line of Best Fit
Eight data points show the age of a car (years) and its value (£000s):
Ages: 1, 2, 3, 4, 5, 6, 7, 8 | Values: 14, 12, 10, 8.5, 7, 5.5, 4, 3
Calculate the mean point and state that the line of best fit must pass through it.
Ages: 1, 2, 3, 4, 5, 6, 7, 8 | Values: 14, 12, 10, 8.5, 7, 5.5, 4, 3
Calculate the mean point and state that the line of best fit must pass through it.
1
Calculate $\bar{x}$ (mean age)
$$\bar{x} = \frac{1+2+3+4+5+6+7+8}{8} = \frac{36}{8} = 4.5 \text{ years}$$
2
Calculate $\bar{y}$ (mean value)
$$\bar{y} = \frac{14+12+10+8.5+7+5.5+4+3}{8} = \frac{64}{8} = £8000$$
3
State the mean point and its significance
The mean point is $(4.5, 8)$. Plot this point on the scatter graph and draw the line of best fit through it with approximately equal numbers of points on each side. The line should slope downward (negative correlation) since car values decrease with age.
4
Describe the correlation
As the age of the car increases, the value decreases. This is a strong negative correlation.
Mean point: $(4.5, 8)$ — i.e., mean age = 4.5 years, mean value = £8000. The line of best fit must pass through this point. The scatter graph shows strong negative correlation: as car age increases, its value decreases.
Grade 9 | Evaluating Claims About Causation
A researcher collects data from 50 countries. She finds a PMCC of $r = 0.89$ between the number of hospitals per 100,000 people and the average life expectancy. She concludes: "Building more hospitals causes people to live longer."
Evaluate this conclusion fully. [4 marks]
Evaluate this conclusion fully. [4 marks]
1
Interpret the PMCC value
$r = 0.89$ indicates a strong positive linear correlation between the number of hospitals per 100,000 people and average life expectancy across the 50 countries. As the number of hospitals increases, life expectancy tends to increase.
2
Challenge the causal claim
The researcher's conclusion that "building more hospitals causes people to live longer" is not justified by the correlation alone. Correlation does not imply causation. To establish a causal relationship, a controlled experiment would be needed.
3
Identify the confounding variable
A third variable — such as national wealth (GDP per capita) — is likely responsible for both. Wealthier countries can afford both to build more hospitals AND to provide better nutrition, clean water, and education, all of which increase life expectancy. The correlation between hospitals and life expectancy may be entirely due to this confounding variable.
4
Comment on additional limitations
The sample is 50 countries — it is worth asking whether this is a representative sample. Also, aggregating data by country (ecological fallacy) may mask individual-level patterns. The $r$ value of 0.89 is very strong, but this could partly reflect the confounding effect of wealth rather than a genuine causal mechanism of hospitals themselves.
Full Grade 9 answer:
The PMCC of $r = 0.89$ shows a strong positive linear correlation (B1). However, the conclusion that hospitals cause longer life expectancy is not valid — correlation does not prove causation (B1). A confounding variable such as national wealth (GDP) is likely responsible for both the number of hospitals and life expectancy, as wealthier nations can invest in both healthcare infrastructure and other life-extending factors (M1, A1). A controlled experiment or further analysis controlling for GDP would be needed to establish causation.
The PMCC of $r = 0.89$ shows a strong positive linear correlation (B1). However, the conclusion that hospitals cause longer life expectancy is not valid — correlation does not prove causation (B1). A confounding variable such as national wealth (GDP) is likely responsible for both the number of hospitals and life expectancy, as wealthier nations can invest in both healthcare infrastructure and other life-extending factors (M1, A1). A controlled experiment or further analysis controlling for GDP would be needed to establish causation.
❓ Exam Questions
Q11 mark
A scatter graph shows that as temperature increases, the number of hot drinks sold decreases. What type of correlation does this show?
Answer: Negative correlation. (1 mark)
Mark scheme: Accept "negative correlation". Do not accept "inverse" alone without "correlation".
Mark scheme: Accept "negative correlation". Do not accept "inverse" alone without "correlation".
Q22 marks
The data below shows the height (cm) and shoe size of 6 students.
Heights: 155, 160, 165, 170, 175, 180 | Shoe sizes: 4, 5, 6, 7, 8, 9
Calculate the mean point that the line of best fit must pass through.
Answer:
$\bar{x} = \frac{155+160+165+170+175+180}{6} = \frac{1005}{6} = 167.5$ cm (1 mark)
$\bar{y} = \frac{4+5+6+7+8+9}{6} = \frac{39}{6} = 6.5$ (1 mark)
Mean point: $(167.5, 6.5)$
$\bar{x} = \frac{155+160+165+170+175+180}{6} = \frac{1005}{6} = 167.5$ cm (1 mark)
$\bar{y} = \frac{4+5+6+7+8+9}{6} = \frac{39}{6} = 6.5$ (1 mark)
Mean point: $(167.5, 6.5)$
Q33 marks
A student draws a line of best fit on a scatter graph showing hours of sunshine ($x$) and ice cream sales in £ ($y$). The data range is $x = 2$ to $x = 10$ hours.
(a) The student uses the line to estimate sales when $x = 7$ hours. Is this interpolation or extrapolation? Is the estimate reliable? (2 marks)
(b) The student uses the line to estimate sales when $x = 15$ hours. Comment on the reliability. (1 mark)
Answer:
(a) This is interpolation because $x = 7$ is within the data range of 2 to 10 hours. The estimate is reliable. (2 marks: 1 for interpolation, 1 for reliable)
(b) $x = 15$ is outside the data range, so this is extrapolation. The estimate is unreliable because we are assuming the linear trend continues beyond the observed data, which may not be the case. (1 mark)
(a) This is interpolation because $x = 7$ is within the data range of 2 to 10 hours. The estimate is reliable. (2 marks: 1 for interpolation, 1 for reliable)
(b) $x = 15$ is outside the data range, so this is extrapolation. The estimate is unreliable because we are assuming the linear trend continues beyond the observed data, which may not be the case. (1 mark)
Q44 marks
A journalist reports: "A study of 200 primary school children found a strong positive correlation ($r = 0.82$) between the number of books in the home and reading age. This proves that buying more books makes children better readers."
Evaluate the journalist's conclusion. [4 marks]
Mark scheme (4 marks):
• B1: States that $r = 0.82$ indicates a strong positive linear correlation between books in the home and reading age.
• B1: States that the conclusion is not valid / correlation does not prove causation.
• M1: Identifies a plausible confounding variable, e.g., parental education level, socioeconomic background, or parents reading to children — all of which would increase both the number of books and reading age.
• A1: Concludes that a controlled study (experiment) would be needed to establish a causal relationship, or gives a further developed argument about the confounding variable.
• B1: States that $r = 0.82$ indicates a strong positive linear correlation between books in the home and reading age.
• B1: States that the conclusion is not valid / correlation does not prove causation.
• M1: Identifies a plausible confounding variable, e.g., parental education level, socioeconomic background, or parents reading to children — all of which would increase both the number of books and reading age.
• A1: Concludes that a controlled study (experiment) would be needed to establish a causal relationship, or gives a further developed argument about the confounding variable.
Q54 marks
Two scatter graphs are drawn for the same 10 data points:
Graph A: PMCC $r_A = 0.76$
Graph B: PMCC $r_B = -0.91$
(a) Which graph shows stronger correlation? Justify your answer. (2 marks)
(b) For which graph would predictions using the line of best fit be more reliable, assuming both use interpolation? Justify your answer. (2 marks)
Answer:
(a) Graph B shows stronger correlation because $|r_B| = 0.91 > |r_A| = 0.76$. The strength of correlation depends on the absolute value of $r$, not the sign. (2 marks: 1 for Graph B, 1 for comparing absolute values)
(b) Predictions from Graph B would be more reliable because it has a stronger correlation (points lie closer to the line of best fit). The closer the points are to the line, the more accurate predictions made from the line will be. (2 marks: 1 for Graph B, 1 for valid justification about strength of correlation)
(a) Graph B shows stronger correlation because $|r_B| = 0.91 > |r_A| = 0.76$. The strength of correlation depends on the absolute value of $r$, not the sign. (2 marks: 1 for Graph B, 1 for comparing absolute values)
(b) Predictions from Graph B would be more reliable because it has a stronger correlation (points lie closer to the line of best fit). The closer the points are to the line, the more accurate predictions made from the line will be. (2 marks: 1 for Graph B, 1 for valid justification about strength of correlation)
Q66 marks
The table shows data on advertising spend (£000s) and monthly sales (£000s) for a company over 8 months.
| Advertising (£000s), $x$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| Sales (£000s), $y$ | 18 | 22 | 24 | 27 | 30 | 33 | 35 | 38 |
(a) Calculate the mean point $(\bar{x}, \bar{y})$. (2 marks)
(b) Describe the correlation and explain what it means in context. (2 marks)
(c) A manager says "If we spend £15,000 on advertising next month, we can expect sales of about £55,000." Evaluate this prediction. (2 marks)
Answer:
(a) $\bar{x} = \frac{2+3+4+5+6+7+8+9}{8} = \frac{44}{8} = 5.5$ (1 mark)
$\bar{y} = \frac{18+22+24+27+30+33+35+38}{8} = \frac{227}{8} = 28.375 \approx 28.4$ (1 mark)
Mean point: $(5.5, 28.4)$
(b) Strong positive correlation. (1 mark) As advertising spend increases, monthly sales also increase. (1 mark — must be stated in context)
(c) $x = 15$ is outside the data range of 2 to 9, so this is extrapolation and is therefore unreliable. (1 mark) We are assuming the linear trend continues beyond the observed data, which may not be true — for example, there may be a ceiling on how much sales can increase regardless of advertising. (1 mark)
(a) $\bar{x} = \frac{2+3+4+5+6+7+8+9}{8} = \frac{44}{8} = 5.5$ (1 mark)
$\bar{y} = \frac{18+22+24+27+30+33+35+38}{8} = \frac{227}{8} = 28.375 \approx 28.4$ (1 mark)
Mean point: $(5.5, 28.4)$
(b) Strong positive correlation. (1 mark) As advertising spend increases, monthly sales also increase. (1 mark — must be stated in context)
(c) $x = 15$ is outside the data range of 2 to 9, so this is extrapolation and is therefore unreliable. (1 mark) We are assuming the linear trend continues beyond the observed data, which may not be true — for example, there may be a ceiling on how much sales can increase regardless of advertising. (1 mark)
⭐ Grade 9 Model Answers
Below is a full annotated answer to Q4 above (the journalist / causation question), broken down to show exactly which marks each part earns and the examiners' expectations at Grade 9.
Grade 9 | Evaluating Correlation and Causation Claims
Q4: A journalist reports: "A study of 200 primary school children found a strong positive correlation ($r = 0.82$) between the number of books in the home and reading age. This proves that buying more books makes children better readers." — Evaluate the journalist's conclusion. [4 marks]
1
Interpret the PMCC correctly (B1)
A value of $r = 0.82$ indicates a strong positive linear correlation between the number of books in a home and a child's reading age. As the number of books increases, reading age tends to increase.
WHY THIS EARNS A MARK
You must explicitly state: (1) the word "strong", (2) the word "positive", (3) "correlation", and (4) name both variables. Saying merely "there is a relationship" is not enough.2
Refute the causal claim (B1)
The journalist's conclusion is incorrect. The scatter graph (or PMCC) demonstrates a correlation, but this does not prove that buying books causes children to become better readers. Correlation does not imply causation.
WHY THIS EARNS A MARK
You must explicitly state that correlation does not prove causation. This is a standalone mark — don't bury it inside a longer sentence where the examiner might miss it.3
Identify a confounding variable (M1)
A more likely explanation is that a confounding variable — such as parental education level or socioeconomic status — is responsible for both variables. Educated or more affluent parents are likely to: (a) own more books, and (b) read to their children more often, support their learning, and have access to better educational resources. Both the number of books and the reading age are therefore caused by this third variable, rather than one causing the other.
WHY THIS EARNS A MARK
Name a specific, credible confounding variable and explain how it affects both variables. A vague "there might be another reason" will not earn this mark.4
Conclude with what would be needed (A1)
To determine whether buying more books genuinely causes an improvement in reading age, a controlled experiment would be needed — for example, randomly assigning some families books and comparing outcomes with a control group, while controlling for parental education and income. Until such a study is conducted, it is impossible to conclude causation from this correlational data alone.
WHY THIS EARNS THE FINAL MARK
Stating what type of evidence would be needed to prove causation shows you understand the distinction at the highest level. This is what separates Grade 8 from Grade 9 responses.
Full Grade 9 Model Answer:
The PMCC of $r = 0.82$ shows a strong positive linear correlation between the number of books in the home and reading age [B1]. However, the journalist's conclusion is not valid — correlation does not prove causation [B1]. A confounding variable such as parental education level is likely responsible for both, as educated parents tend to own more books and also engage more with their children's reading development [M1]. To establish that buying books causes improved reading, a controlled experiment randomly assigning books to families while controlling for socioeconomic factors would be needed [A1].
The PMCC of $r = 0.82$ shows a strong positive linear correlation between the number of books in the home and reading age [B1]. However, the journalist's conclusion is not valid — correlation does not prove causation [B1]. A confounding variable such as parental education level is likely responsible for both, as educated parents tend to own more books and also engage more with their children's reading development [M1]. To establish that buying books causes improved reading, a controlled experiment randomly assigning books to families while controlling for socioeconomic factors would be needed [A1].
📋 Revision Sheet
Key Definitions
| Term | Meaning |
|---|---|
| Scatter graph | A graph plotting bivariate data as $(x,y)$ points |
| Bivariate data | Two measurements recorded for each individual |
| Positive correlation | Both variables increase together |
| Negative correlation | One increases as the other decreases |
| Mean point | $(\bar{x}, \bar{y})$ — must lie on line of best fit |
| Interpolation | Prediction within the data range (reliable) |
| Extrapolation | Prediction outside the data range (unreliable) |
| Causation | One variable directly causes the other to change |
| PMCC ($r$) | Measures linear correlation: $-1 \leq r \leq 1$ |
Essential Formulae
Mean point (must lie on line of best fit):
$$\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}$$PMCC range:
$$-1 \leq r \leq 1$$Strength interpretation:
- $|r| \geq 0.7$: strong
- $0.4 \leq |r| < 0.7$: moderate
- $|r| < 0.4$: weak
Memory Hooks
- "MEANBEST" — the Mean point is the BESt point for the line
- "IN=Inside=reIN-forceable" — interpolation (inside) is reliable
- "EX=EXtreme=EXtra risky" — extrapolation is unreliable
- "Ice cream drownings" — remember, correlation ≠ causation
- PMCC closer to ±1 = points closer to line
- Two adjectives always: strong/moderate/weak + positive/negative
- r is always between -1 and +1 — if you get outside this, recheck
Exam Tips
- Always calculate and mark the mean point before drawing the line of best fit
- Describe correlation with TWO words: strength + direction
- State whether predictions use interpolation or extrapolation — and comment on reliability
- Correlation ≠ causation — say it explicitly even if it feels obvious
- Name the confounding variable specifically in causation questions
- When comparing PMCC values, compare absolute values $|r|$ for strength
- A strong correlation makes predictions more reliable, even within range
- Lines of best fit should not be forced through the origin
🔄 Flashcards
Click each card to reveal the answer.
✗ Common Mistakes
MISTAKE 1 — Describing Correlation with Only One Word
What students do: Write "positive correlation" without stating the strength, or say "strong" without giving the direction.Why marks are lost: A full description requires both strength (strong/moderate/weak) and direction (positive/negative). One word answers earn at most 1 out of 2 marks.
How to avoid: Always write two adjectives: e.g., "strong positive correlation". Practise until it is automatic.
MISTAKE 2 — Not Passing the Line Through the Mean Point
What students do: Draw a line of best fit by eye without calculating $(\bar{x}, \bar{y})$ first, then fail to pass the line through it.Why marks are lost: Questions specifically asking you to draw the line of best fit may award a mark for passing through $(\bar{x}, \bar{y})$. Missing this loses a guaranteed mark.
How to avoid: Always calculate $\bar{x}$ and $\bar{y}$ first, plot the mean point clearly, then draw the line through it.
MISTAKE 3 — Claiming Correlation Proves Causation
What students do: Agree with a statement like "the data shows that X causes Y" when given a scatter graph or PMCC value.Why marks are lost: This is the most conceptually tested aspect of this topic at higher grades. Simply agreeing loses all the evaluation marks.
How to avoid: Remember: a scatter graph (or PMCC) can only show correlation. To establish causation you need a controlled experiment. Always ask "what else could explain this?"
MISTAKE 4 — Not Commenting on Reliability of Predictions
What students do: Make a prediction from the line of best fit but do not say whether it is interpolation or extrapolation, or whether it is reliable.Why marks are lost: Questions often have a mark specifically for stating whether the prediction is reliable and why. Not including this loses a free mark.
How to avoid: After every prediction, ask yourself: "Is this $x$-value within the data range?" If yes, say "This is interpolation and the estimate is reliable." If no, say "This is extrapolation and the estimate may be unreliable."
MISTAKE 5 — Confusing the Sign of $r$ with the Strength
What students do: Say that $r = 0.8$ shows stronger correlation than $r = -0.9$ because "$0.8$ is bigger."Why marks are lost: The sign only tells you the direction. The strength of correlation depends on the absolute value $|r|$. Here, $|-0.9| = 0.9 > |0.8| = 0.8$, so $r = -0.9$ is actually the stronger correlation.
How to avoid: When comparing PMCC values, always compare $|r_A|$ vs $|r_B|$.
MISTAKE 6 — Drawing a Line of Best Fit When There Is No Correlation
What students do: Draw a line of best fit on a scatter graph that clearly shows no correlation (random scatter).Why marks are lost: A line of best fit is only meaningful when there is a correlation. Drawing one on a no-correlation graph is incorrect and may lose the mark for the line, as it misrepresents the data.
How to avoid: Before drawing any line, first check whether there is a clear trend. If the points are randomly scattered, state "no correlation" and do not draw a line.
✅ Final Checklist
Tick each item as you master it. Your progress is saved automatically.
- I can plot bivariate data as $(x, y)$ points on a scatter graph
- I can identify positive, negative, and no correlation from a scatter graph
- I can describe correlation using both strength (strong/moderate/weak) and direction
- I can calculate the mean point $(\bar{x}, \bar{y})$ correctly from a data set
- I always draw the line of best fit through the mean point $(\bar{x}, \bar{y})$
- I can use the line of best fit to make predictions by reading off the graph
- I know interpolation is within the data range and is generally reliable
- I know extrapolation is outside the data range and is generally unreliable
- I can explain why correlation does not prove causation
- I can identify and explain confounding variables in context
- I can interpret a PMCC value: state its strength and direction
- I know $r = \pm 1$ means perfect linear correlation and $r = 0$ means no linear correlation
- I compare PMCC strength using absolute values $|r|$, not signed values
- I can evaluate research conclusions involving correlation data at Grade 9 level
- I have practised at least 3 full exam questions on this topic