A visible calculation trail
Expose the means and sums so a learner can locate an error instead of receiving an unexplained number.
Learning support · Demonstration study
Understand the calculation. Then challenge the conclusion.
A worked statistical example that moves from arithmetic to interpretation, feedback and an independent transfer task.
The brief
What needs to change?
Teach correlation using four invented observations, then distinguish a correct calculation from an unjustified causal claim. This is an original teaching example, not an answer to a real assessment.
The challenge
A learner can reproduce a formula without understanding what its result means. The demonstration makes the calculation inspectable and requires the learner to reason beyond it.
Working excerpt · Illustrative material
Worked example and annotated reasoning
| x | y | x − 2.5 | y − 60 | Product |
|---|---|---|---|---|
| 1 | 50 | −1.5 | −10 | 15 |
| 2 | 60 | −0.5 | 0 | 0 |
| 3 | 60 | 0.5 | 0 | 0 |
| 4 | 70 | 1.5 | 10 | 15 |
| Sum | — | 0 | 0 | 30 |
Σ(x − x̄)² = 5 · Σ(y − ȳ)² = 200
r = 30 / √(5 × 200) = 0.948683…
A strong positive linear association in these four invented observations. This calculation does not establish why the variables are associated.
Transfer exercise: check your reasoning
For (1, 72), (2, 68), (3, 64), (4, 60), every one-unit increase in x corresponds to a four-unit decrease in y. The observations lie on a straight line: r = −1. The relationship is negative and perfectly linear in this small constructed dataset; it is not proof of causation.
01 / Decode the task
Demonstration prompt: “Describe the relationship between study time and a quiz score, and explain what the data cannot tell you.” Separate three tasks: calculate the descriptive association, interpret the pattern in context, and examine alternative explanations. A numerical answer alone does not complete the reasoning.
02 / Show the working
Entirely synthetic teaching data—not student records: (hours, score) pairs are (1, 50), (2, 60), (3, 60), (4, 70). Means: 2.5 hours and 60 score points. Centred cross-products sum to 30; squared hour deviations sum to 5; squared score deviations sum to 200. Pearson's r = 30 / sqrt(5 × 200) ≈ 0.949. This describes only these four invented pairs.
03 / Explain the result
Reasoning model: “The example has a strong positive linear association: higher study-time values tend to appear with higher quiz scores. It does not establish that an extra hour causes a particular score increase. Prior knowledge, task difficulty and other unrecorded factors could affect both variables.” No significance test or population claim is attached to the invented observations.
04 / Give actionable draft feedback
Fictional learner sentence: “Because the correlation is almost one, studying causes better results.” Feedback: “Your calculation and causal conclusion are different claims. Keep the descriptive association; explain what a comparison capable of addressing causation would need to rule out. Could prior knowledge also influence study time?” This supports the learner's own revision.
05 / Test transfer, not imitation
Transfer task using a second invented set: (1, 72), (2, 68), (3, 64), (4, 60). Predict the direction of association before calculating. Explain why a negative relationship would still not show that study time harms performance. Self-check: Can I describe the sample, pattern and one alternative explanation without relying on the formula?
Read alongside the reasoning and limitations below. These excerpts are not a complete commissioned deliverable.
Editorial & methodological judgement
Why these decisions?
The value is in the choices that connect the brief to the result.
Interpretation is a separate skill
Describe the association in the units and context of the example, then explain what the design cannot establish.
The learner does the next step
A changed example tests whether the reasoning transfers. Feedback guides the learner's own decisions and submission in line with course requirements.
Review checklist
What to check.
- Means, cross-products and correlation reconcile.
- Hours and score units remain distinct.
- Association strength is not equated with causality.
- The transfer exercise requires independent reasoning and retains its synthetic label.
A possible commissioned handover
What to specify.
- Worked example and calculation trail
- Interpretive paragraph
- Annotated draft feedback
- Independent transfer exercise
The actual deliverable list, format and review stages are agreed in your quote.
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