Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 4 - Bivariate data, association and investigation

Bivariate data, association and investigation

Conduct bivariate investigations, describe association and use fitted models while distinguishing association from causation.

Updated 2026-07-26 - 12 min read

Bivariate data, association and investigation is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Design bivariate investigations

Measure two variables on the same observational units, define units and collect a sample suited to the population question. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by design bivariate investigations and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Bivariate analysis requires paired measurements so each point describes one unit.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Which data are bivariate?

Step 1 - identify the governing idea: Measure two variables on the same observational units, define units and collect a sample suited to the population question.

Step 2 - apply it to this evidence: Each student contributes a paired numerical measurement.

Result: Height and arm span for each student

The relationship is visible in the working: Each student contributes a paired numerical measurement. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • A list of heights only — It does not agree with the required relationship: Each student contributes a paired numerical measurement.
  • Heights from one school and arm spans from another — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Favourite colour counts — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

What graph is suitable for two numerical variables?

Step 1 - identify the governing idea: Measure two variables on the same observational units, define units and collect a sample suited to the population question.

Step 2 - apply it to this evidence: Each ordered pair is plotted as one point.

Result: Scatterplot

The relationship is visible in the working: Each ordered pair is plotted as one point. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Pie chart — It does not agree with the required relationship: Each ordered pair is plotted as one point.
  • Single boxplot only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Pictograph — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Why record units for both variables?

Step 1 - identify the governing idea: Measure two variables on the same observational units, define units and collect a sample suited to the population question.

Step 2 - apply it to this evidence: Gradient and practical conclusions require known measurement units.

Result: Scale and interpretation depend on them

The relationship is visible in the working: Gradient and practical conclusions require known measurement units. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Units force causation — It does not agree with the required relationship: Gradient and practical conclusions require known measurement units.
  • Units remove outliers — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Units determine sample size — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Describe association in scatterplots

Describe direction, form, strength and unusual points, supported by the plotted evidence. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by describe association in scatterplots and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Association describes an overall tendency and can include scatter or exceptions.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Points cluster closely around a downward line. How is association described?

Step 1 - identify the governing idea: Describe direction, form, strength and unusual points, supported by the plotted evidence.

Step 2 - apply it to this evidence: As x increases, y tends to decrease with little scatter.

Result: Strong negative linear association

The relationship is visible in the working: As x increases, y tends to decrease with little scatter. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Strong positive — It does not agree with the required relationship: As x increases, y tends to decrease with little scatter.
  • No association — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Nonlinear growth — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A plot forms a clear U-shape. What type of association?

Step 1 - identify the governing idea: Describe direction, form, strength and unusual points, supported by the plotted evidence.

Step 2 - apply it to this evidence: The relationship is systematic but not well described by one straight line.

Result: Nonlinear association

The relationship is visible in the working: The relationship is systematic but not well described by one straight line. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • No association — It does not agree with the required relationship: The relationship is systematic but not well described by one straight line.
  • Perfect positive linear — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Perfect negative linear — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

One point lies far from the overall pattern. What should be done?

Step 1 - identify the governing idea: Describe direction, form, strength and unusual points, supported by the plotted evidence.

Step 2 - apply it to this evidence: It may be error, a valid unusual case or important subgroup evidence.

Result: Investigate and report it, not delete automatically

The relationship is visible in the working: It may be error, a valid unusual case or important subgroup evidence. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Always remove it — It does not agree with the required relationship: It may be error, a valid unusual case or important subgroup evidence.
  • Move it onto the line — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Ignore the entire data set — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Fit and evaluate bivariate models

A fitted line should follow the centre of a roughly linear cloud; use interpolation cautiously and avoid causal or distant extrapolation claims. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by fit and evaluate bivariate models and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The line summarises the overall trend, so residuals occur above and below it.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

A fitted line is y=2.5x+10. Predict y at x=8.

Step 1 - identify the governing idea: A fitted line should follow the centre of a roughly linear cloud; use interpolation cautiously and avoid causal or distant extrapolation claims.

Step 2 - apply it to this evidence: 2.5(8)+10=30.

Result: 30

The relationship is visible in the working: 2.5(8)+10=30. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 28 — It does not agree with the required relationship: 2.5(8)+10=30.
  • 90 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 20 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Why is predicting far beyond observed x-values risky?

Step 1 - identify the governing idea: A fitted line should follow the centre of a roughly linear cloud; use interpolation cautiously and avoid causal or distant extrapolation claims.

Step 2 - apply it to this evidence: Extrapolation assumes the fitted pattern continues under unobserved conditions.

Result: The relationship may change outside the data range

The relationship is visible in the working: Extrapolation assumes the fitted pattern continues under unobserved conditions. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The equation stops working algebraically — It does not agree with the required relationship: Extrapolation assumes the fitted pattern continues under unobserved conditions.
  • Scatterplots cannot show lines — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • All future values are zero — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A strong association appears in observational data. What can be concluded?

Step 1 - identify the governing idea: A fitted line should follow the centre of a roughly linear cloud; use interpolation cautiously and avoid causal or distant extrapolation claims.

Step 2 - apply it to this evidence: Confounding and reverse causation remain possible.

Result: The variables are related, but causation needs further evidence

The relationship is visible in the working: Confounding and reverse causation remain possible. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x definitely causes y — It does not agree with the required relationship: Confounding and reverse causation remain possible.
  • y definitely causes x — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The sample is unbiased — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. What graph is suitable for two numerical variables?
  2. A plot forms a clear U-shape. What type of association?
  3. Why is predicting far beyond observed x-values risky?

Answers

  1. Scatterplot — Each ordered pair is plotted as one point.
  2. Nonlinear association — The relationship is systematic but not well described by one straight line.
  3. The relationship may change outside the data range — Extrapolation assumes the fitted pattern continues under unobserved conditions.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources