Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 4 - Continuous distributions and boxplots

Continuous distributions and boxplots

Represent and compare continuous data using histograms, boxplots, quartiles and distribution features.

Updated 2026-07-26 - 11 min read

Continuous distributions and boxplots 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.

Construct and read histograms

Histograms represent continuous intervals with adjacent bars; frequency density is required when interval widths differ. 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 construct and read histograms 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: Continuous intervals meet at boundaries, so adjacent bars communicate continuity.

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

Why do histogram bars usually touch?

Step 1 - identify the governing idea: Histograms represent continuous intervals with adjacent bars; frequency density is required when interval widths differ.

Step 2 - apply it to this evidence: There are no categorical gaps between successive measurement intervals.

Result: The variable is continuous across adjacent intervals

The relationship is visible in the working: There are no categorical gaps between successive measurement intervals. 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:

  • Every frequency is equal — It does not agree with the required relationship: There are no categorical gaps between successive measurement intervals.
  • The graph has no axes — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Bars represent means — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Intervals have unequal widths. What should bar area represent?

Step 1 - identify the governing idea: Histograms represent continuous intervals with adjacent bars; frequency density is required when interval widths differ.

Step 2 - apply it to this evidence: Using frequency density as height makes area proportional to frequency.

Result: Frequency

The relationship is visible in the working: Using frequency density as height makes area proportional to frequency. 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:

  • Interval midpoint — It does not agree with the required relationship: Using frequency density as height makes area proportional to frequency.
  • Maximum value — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Median only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A class 10≤x<20 has frequency 30 and width 10. Find frequency density.

Step 1 - identify the governing idea: Histograms represent continuous intervals with adjacent bars; frequency density is required when interval widths differ.

Step 2 - apply it to this evidence: Density = frequency/width = 30/10.

Result: 3

The relationship is visible in the working: Density = frequency/width = 30/10. 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:

  • 300 — It does not agree with the required relationship: Density = frequency/width = 30/10.
  • 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.3 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Interpret boxplots and quartiles

A boxplot shows minimum, lower quartile, median, upper quartile and maximum; the box length is the interquartile range. 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 interpret boxplots and quartiles 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: Box length shows spread of the middle 50%, not sample size or frequency.

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

Q1=12 and Q3=20. Find IQR.

Step 1 - identify the governing idea: A boxplot shows minimum, lower quartile, median, upper quartile and maximum; the box length is the interquartile range.

Step 2 - apply it to this evidence: IQR = Q3 − Q1 = 8.

Result: 8

The relationship is visible in the working: IQR = Q3 − Q1 = 8. 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:

  • 32 — It does not agree with the required relationship: IQR = Q3 − Q1 = 8.
  • 16 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

What proportion of observations lie between Q1 and Q3?

Step 1 - identify the governing idea: A boxplot shows minimum, lower quartile, median, upper quartile and maximum; the box length is the interquartile range.

Step 2 - apply it to this evidence: The interquartile interval contains the middle half.

Result: About 50%

The relationship is visible in the working: The interquartile interval contains the middle half. 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:

  • 25% — It does not agree with the required relationship: The interquartile interval contains the middle half.
  • 75% — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 100% — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Two boxplots have equal medians but one larger IQR. What differs?

Step 1 - identify the governing idea: A boxplot shows minimum, lower quartile, median, upper quartile and maximum; the box length is the interquartile range.

Step 2 - apply it to this evidence: Their typical centre matches while central variability differs.

Result: The spread of the middle 50%

The relationship is visible in the working: Their typical centre matches while central variability differs. 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:

  • Their sample sizes for certain — It does not agree with the required relationship: Their typical centre matches while central variability differs.
  • Their means for certain — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Their units — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare continuous distributions

Compare centre, spread, shape, overlap and unusual values using consistent scales and qualified contextual language. 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 compare continuous distributions 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: One extreme does not establish typical performance; compare medians and distribution overlap.

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 has median 72 and IQR 8; B has median 68 and IQR 16. Which is supported?

Step 1 - identify the governing idea: Compare centre, spread, shape, overlap and unusual values using consistent scales and qualified contextual language.

Step 2 - apply it to this evidence: A has higher median and smaller IQR.

Result: A is typically higher and less variable

The relationship is visible in the working: A has higher median and smaller IQR. 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:

  • B is higher and less variable — It does not agree with the required relationship: A has higher median and smaller IQR.
  • A has more observations — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The groups do not overlap — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A histogram is strongly right-skewed. Which centre is usually more resistant?

Step 1 - identify the governing idea: Compare centre, spread, shape, overlap and unusual values using consistent scales and qualified contextual language.

Step 2 - apply it to this evidence: The long right tail can pull the mean upward.

Result: Median

The relationship is visible in the working: The long right tail can pull the mean upward. 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:

  • Mean — It does not agree with the required relationship: The long right tail can pull the mean upward.
  • Maximum — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Range — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

What does substantial boxplot overlap imply?

Step 1 - identify the governing idea: Compare centre, spread, shape, overlap and unusual values using consistent scales and qualified contextual language.

Step 2 - apply it to this evidence: Overlap tempers claims of complete group separation.

Result: Many values may be in a similar range

The relationship is visible in the working: Overlap tempers claims of complete group separation. 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 medians are equal — It does not agree with the required relationship: Overlap tempers claims of complete group separation.
  • The groups are identical — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No comparison is possible — 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. Intervals have unequal widths. What should bar area represent?
  2. What proportion of observations lie between Q1 and Q3?
  3. A histogram is strongly right-skewed. Which centre is usually more resistant?

Answers

  1. Frequency — Using frequency density as height makes area proportional to frequency.
  2. About 50% — The interquartile interval contains the middle half.
  3. Median — The long right tail can pull the mean upward.

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