Mathematics Advanced • Year 12 • Module 5 • Lesson 9
Bivariate Data Analysis
Practise HSC-style writing on scatter plots, Pearson's r and the correlation-vs-causation distinction, with a structured extended response on critiquing a causal claim.
1. Short-answer questions
1.1 For the data (1, 5), (2, 7), (3, 11), (4, 13), (5, 17), compute Pearson's r using the computational formula. 3 marks Band 3-4
1.2 A study reports r = −0.85 between hours of social media use per day and self-reported sleep hours. (a) State r² and interpret it in context. (b) Explain in one sentence what the negative sign of r means in this context. 3 marks Band 3
1.3 A researcher sees a strong correlation (r = 0.9) in a sample but, when plotting the scatter, notices a perfect U-shape. (a) Explain in 1-2 sentences why Pearson's r is misleading here. (b) State two of the lesson's five "limitations of r" that would lead you to mistrust this number. 3 marks Band 4
Stuck on 1.3(b)? Revisit lesson § Limitations of r.2. Extended response
2.1 A wellness-influencer's blog post claims:
"My new study of 500 Australians shows a correlation of r = 0.72 between daily green-tea consumption (cups) and self-reported happiness score (out of 10). Drinking more green tea makes you happier, clearly the only sensible recommendation is two cups a day."
Write an HSC-style critique of this claim. Your response must:
(a) State and interpret r = 0.72 (strength, direction, r²), being precise about what "explained" means.
(b) Identify and explain at least three alternative explanations for the correlation, covering at least one each of (i) confounding variable, (ii) reverse causation, and (iii) limitations of r (selection bias / measurement / restricted range).
(c) Conclude with a one-paragraph recommendation about what kind of study would support the blogger's causal claim, explicitly referencing the principle "correlation does not imply causation".
8 marks Band 5-6
Explicit marking criteria
Part (a), 2 marks
• 1 mark strength (strong) and direction (positive) named, with r² = 0.5184 ≈ 0.52 calculated.
• 1 mark interprets r² in context: about 52% of the variation in happiness score is "explained" by the linear relationship with cups of green tea, and flags that "explained" does not mean "caused".
Part (b), 4 marks
• 1 mark, confounder: identifies a plausible third variable (e.g. income / education / overall healthy-lifestyle bundle) that drives both green-tea drinking and self-reported happiness.
• 1 mark, reverse causation: argues that happier people may simply be more likely to take up green-tea rituals, not the other way around.
• 1 mark, limitation of r / selection bias: notes the sample is self-selected (blog readers), self-reported and possibly restricted in range (only people who drink > 0 cups).
• 1 mark, explanation quality: each alternative is described in enough detail that the marker can see how it produces the observed r without requiring a green-tea → happiness causal arrow.
Part (c), 2 marks
• 1 mark recommends a randomised controlled trial (or equivalent) with placebo / control group, blinding, and pre-specified outcome measure, in enough detail to count as a real proposal.
• 1 mark closes by explicitly invoking the lesson principle "correlation does not imply causation" (or equivalent) and linking it to why a controlled experiment, not an observational correlation, is needed to justify the recommendation "two cups a day".
Your response:
Stuck on (b)? Pick three distinct mechanisms, don't lump "confounder" and "selection bias" together.How did this worksheet feel?
What I'll revisit before next class:
1.1, r for (1, 5), (2, 7), (3, 11), (4, 13), (5, 17) (3 marks)
Sample response. Σx = 15, Σy = 53, Σx² = 55, Σy² = 653, Σxy = 1(5)+2(7)+3(11)+4(13)+5(17) = 5+14+33+52+85 = 189, n = 5.
Numerator = 5(189) − 15(53) = 945 − 795 = 150.
n·Σx² − (Σx)² = 275 − 225 = 50.
n·Σy² − (Σy)² = 3265 − 2809 = 456.
r = 150 / √(50 × 456) = 150 / √22,800 = 150 / 151.0 ≈ 0.993.
Marking notes. 1 mark, all sums correct. 1 mark, numerator and denominator computed correctly. 1 mark, final r ≈ 0.99 (accept 0.99 – 1.00). Stopping with the right formula but no arithmetic scores 1; getting r > 1 or r < −1 is a hard zero on the last mark (impossible value).
1.2, Social media use and sleep (3 marks)
Sample response. (a) r² = 0.7225 ≈ 0.72: about 72% of the variation in self-reported sleep hours is explained by the linear relationship with daily social-media use. (b) The negative sign means more social-media use is associated with fewer sleep hours (and vice versa), when one goes up, the other tends to go down.
Marking notes. (a) 1 mark, r² value correct; 1 mark, "% of variation in y explained by x" phrasing in context. (b) 1 mark, clear "as x increases, y decreases" interpretation. Students who write "r negative means there is no relationship" score 0/3 on (b).
1.3, Misleading r & limitations (3 marks)
Sample response. (a) Pearson's r measures only the linear component of association. A perfect U-shape (e.g. y = x² − 1) has equal-and-opposite linear trends on the two arms that partially cancel, so r can be small or misleading, even though the relationship is strong and entirely deterministic. Reporting r = 0.9 without showing the scatter plot would mask the curved shape.
(b) Two limitations from the lesson: (1) Non-linear relationships r is blind to curved patterns; (2) Outliers a single extreme point can dramatically inflate or deflate r.
Marking notes. (a) 1 mark, explicit statement that r measures linear association only; 1 mark, names how a curved relationship can produce a misleading r. (b) 1 mark, two distinct limitations correctly named (accept any two of: non-linear, outliers, heteroscedasticity, restricted range, ecological fallacy).
2.1, Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Interpretation of r = 0.72. r = 0.72 indicates a strong positive linear association between daily green-tea consumption and happiness score. r² = 0.5184, so about 52% of the variation in happiness score in this sample is "explained" by the linear relationship with cups of green tea. The word "explained" is statistical jargon, not a causal claim: it tells us how tightly the points cluster around a linear trend, not whether tea causes happiness. [1 mark, strength + direction + r²; 1 mark, context interpretation that flags "explained" ≠ "caused".]
(b) Three alternative explanations.
(i) Confounding variable, income / overall healthy-lifestyle bundle. People with higher disposable income, more time for self-care, regular exercise, and a balanced diet are more likely to both drink green tea (a discretionary purchase) and report higher happiness scores. The third variable "healthy-lifestyle bundle" plausibly drives both x and y, generating the correlation without green tea itself causing happiness. [1 mark, confounder.]
(ii) Reverse causation. Happier people may be more likely to take up wellness rituals (including green-tea drinking) because they have more energy, optimism and openness to new habits, not the other way around. The blogger has assumed an x → y arrow; the data are equally consistent with a y → x arrow. [1 mark, reverse causation.]
(iii) Selection bias / measurement issues (limitation of r). The 500 respondents are drawn from blog readers (a self-selected sample of people already interested in wellness), happiness is self-reported (subjective and prone to social-desirability bias), and the sample may be range-restricted to people who drink some green tea, none of these support a generalisable causal claim. [1 mark, selection bias / limitation of r.] Each of these alternatives can produce r = 0.72 without any causal arrow from green tea to happiness, that is, the data are consistent with all three at once, and Pearson's r alone cannot distinguish them. [1 mark, explanation quality: ties the three together by showing each alone can produce the observed r.]
(c) What would actually support a causal claim?
The blogger's data are observational. To support a causal "two cups a day will make you happier" claim, we would need a randomised controlled trial: recruit a representative sample (not blog readers), randomly assign participants to a "two cups of green tea per day" arm and a matched placebo arm (e.g. caffeine-matched herbal tea), blind both participants and assessors to which beverage they receive, then measure happiness with a validated scale over a pre-specified period. Randomisation balances confounders (income, lifestyle) across groups, the placebo controls for the act of "drinking a special tea" and for any caffeine effect, and blinding controls for reporting bias. Until such an experiment is run, the data show only association: this is precisely the lesson principle correlation does not imply causation, and it is why no health recommendation should be issued from r = 0.72 alone. [1 mark, concrete RCT proposal; 1 mark, explicit invocation of the lesson principle and link to why observational correlation cannot justify the recommendation.]
Total: 8/8.
Band descriptors for marker.
Band 3: States "strong positive correlation" and computes r²; lists one alternative explanation but not three distinct categories; no controlled-trial proposal. ≈ 3-4 marks.
Band 4: (a) complete; (b) names two of three categories (confounder + reverse causation, say) but omits limitations of r; (c) vague proposal ("do a better study"). ≈ 5-6 marks.
Band 5: All three alternative-explanation categories present and clearly distinguished; (c) names an RCT but with weak justification. ≈ 7 marks.
Band 6: Full marks across (a)-(c); explicit "explained ≠ caused" flag in (a); three categorically distinct alternatives in (b), each plausibly producing the observed r; (c) concrete RCT with randomisation + placebo + blinding, and an explicit invocation of "correlation does not imply causation" linking the lesson principle to the practical recommendation. 8/8.