VCE Vault

VCE General Mathematics (Year 12) · Data Analysis · 14m read

Associations Between Two Variables

Do Year 12s play less sport than Year 11s? Does the median commute time change with the transport type? Questions like these ask whether two variables are associated. The trick is choosing the right tool for the pair of variables and then backing up your claim with percentages or medians — never just "it looks different".

Learning objective: Identify explanatory and response variables and investigate associations using two-way tables, segmented bar charts and parallel boxplots.

By the end of this lesson you can:

  • I identify the explanatory and response variables.
  • I calculate and compare column percentages in a two-way table.
  • I describe an association using parallel boxplots.

Associations are detective pairings. Are two clues linked? Choose the right tool — segmented bar charts for categorical, scatterplots for numerical.

Explanatory, response and the right tool

EXPLANATORY vs RESPONSE:
• EXPLANATORY variable — the one that might explain or predict changes (x).
• RESPONSE variable — the one that responds (y).
• Ask: "Does A affect B, or B affect A?" — the cause-like variable is explanatory.

TWO CATEGORICAL VARIABLES:
• Use a two-way frequency table.
• Convert to PERCENTAGES within each category of the EXPLANATORY variable.
• Display with a percentage segmented bar chart.
• If the percentages differ noticeably between categories → association.

ONE NUMERICAL, ONE CATEGORICAL:
• Categorical is explanatory; numerical is response.
• Use PARALLEL BOXPLOTS (or back-to-back stem plots for two groups).
• Compare MEDIANS (and spreads) across groups. Different medians → association.

TWO NUMERICAL VARIABLES:
• Use a scatterplot (next topic).

WRITING THE CONCLUSION:
• Name both variables.
• Quote a specific percentage or median for each group.
• E.g. "There is an association between year level and sport participation: 60% of Year 11s play sport compared with 35% of Year 12s."

Which display?

  • Categorical + categorical: Two-way table → column percentages → segmented bar chart.
  • Numerical + categorical: Parallel boxplots — compare medians.
  • Numerical + numerical: Scatterplot.

Quick reference

| Variables | Display |
|---|---|
| Two categorical | Two-way table, segmented bar chart |
| Numerical + categorical | Parallel boxplots |
| Two numerical | Scatterplot |
| Explanatory variable | Independent (x) |
| Response variable | Dependent (y) |

Worked example (Medium)

Survey: of 100 Year 11 students, 60 play organised sport; of 100 Year 12 students, 35 do. Is there an association between year level and playing sport?

  1. Variables. Explanatory: year level. Response: plays sport (yes/no).
  2. Percentages. Year 11: 60/100 = 60% play. Year 12: 35/100 = 35% play.
  3. Compare. The percentages differ substantially (60% vs 35%).

Answer: Yes. 60% of Year 11 students play organised sport compared with only 35% of Year 12 students, so participation is associated with year level.

Worked example (Easy)

Parallel boxplots of commute time show medians: car 25 min, bus 40 min, bike 20 min. Write a conclusion.

  1. Variables. Explanatory: transport type (categorical). Response: commute time (numerical).
  2. Compare medians. Medians vary from 20 to 40 minutes.

Answer: Commute time is associated with transport type: median bus commute (40 min) is longer than car (25 min) and bike (20 min).

Common mistakes

  • Calculating percentages the wrong way Percentages must be within each explanatory category, or the comparison is meaningless. Fix: Percentage across the response, within each explanatory group.
  • Saying "there is an association" with no evidence Marks require a quoted statistic. Fix: Quote at least two percentages or medians.

Correlation is not causation

Ice-cream sales and drownings are correlated — both rise in summer. Hot weather is a lurking variable causing both.

Only well-designed experiments can establish causation.

Practice questions

  1. A study investigates whether hours of sleep affect reaction time. The explanatory variable is:

    1. Reaction time
    2. Hours of sleep
    3. Both
    4. Neither

    (1 marks)

  2. Of 80 city residents, 20 own a car; of 50 rural residents, 40 own a car. Use percentages to describe the association.

    (2 marks)

Summary

  • Identify explanatory (x) and response (y) first.
  • Categorical pairs: compare percentages within explanatory groups.
  • Numerical vs categorical: compare medians with parallel boxplots.
  • Always back the claim with numbers.

More from VCE General Mathematics (Year 12)