M
hscscience Maths Std · Y12
0/100daily goal
0
0
L1 · 0 XP
KJ
Your weak spots
Insights load after your first practice round.
Module 8 · L8 of 12 ~30 min MST-12-S2-10 ⚡ +75 XP available

Comparing Distributions

Statistics is rarely about a single data set in isolation. The real power comes from comparison, does this year's class outperform last year's? Is the new teaching method producing more consistent results? This lesson teaches the systematic approach: examine centre, spread, shape and outliers side by side. Master this framework and you can draw meaningful conclusions from any pair of data sets.

Today's hook, Class A: most scores between 70–80, a few at 90. Class B: scores spread evenly from 50–100. Both have mean = 75. Which class would you rather be in, and why? Predict before reading on.
0/5QUESTS
1

Get oriented

Set up your goals, key ideas and key terms for comparing distributions.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Think First, gut answer before you read
+5 XP warm-up

Class A: most scores between 70–80, a few at 90. Class B: scores spread evenly from 50–100. Both have mean = 75. Which class would you rather be in, and why?

Before reading on write your gut feeling. We will revisit this at the end of the lesson.

auto-saved
02
Key ideas for this lesson
reference

Compare centre: Use medians or means, whichever group has a higher median/mean typically scores higher.

Compare spread: Use IQR or standard deviation, smaller spread means more consistent results.

Also compare shape (symmetric vs skewed, unimodal vs bimodal) and outliers (which group has more extreme values).

Always use the same measure for both groups, comparing median of one with mean of another invalidates the conclusion.
Consistent measures
If you use median for Group A, use median for Group B. Mixing median and mean is a common HSC error.
Consistency vs centre
A group with a lower mean but smaller IQR may be preferable in contexts where predictability matters (e.g. manufacturing).
Context always matters
State what the comparison means in context, "more consistent" is not enough; say consistent in what and for whom.
03
What you will master
Know

Key facts

  • The four dimensions: centre, spread, shape, outliers
  • Tools for each dimension
  • Why consistent measures matter
Understand

Concepts

  • Why context matters in comparison
  • How different spreads affect interpretation
  • When a lower mean might still be preferable
Can do

Skills

  • Compare two distributions using summary statistics
  • Read and compare parallel box plots
  • Write a full comparison paragraph in context
04
Key terms
CentreThe typical or middle value of a distribution, measured by mean or median.
SpreadHow varied or consistent the values are, measured by range, IQR or standard deviation.
ShapeThe overall pattern of the distribution, symmetric, positively skewed, negatively skewed, bimodal.
OutlierA value that lies far from the bulk of the data, identified using fences or z-scores.
IQRInterquartile range: $Q_3 - Q_1$, representing the spread of the middle 50% of data.
Parallel box plotsTwo box plots drawn on the same scale to visually compare two distributions.
2

Compare the centre

Decide which group is typically higher using mean and median.

05
Comparing centre, which group is typically higher?
core concept

The first step in comparing distributions is to compare the centre the typical or middle value.

  • Use medians when the data may be skewed or has outliers (median is resistant to extreme values).
  • Use means when the data is roughly symmetric and you need to consider all values.

Example:

School A: median = 78. School B: median = 72.

"School A typically achieves higher results than School B, with a median mark 6 points higher."

HSC tip: When comparing, explicitly state which group has the higher centre and by how much. Vague answers like "School A is better" lose marks. Always refer to the specific statistic.

To compare the centres of two distributions: compare their means (or medians for skewed data). The distribution with the higher mean has typically higher values. State the comparison in the language of the variable.

Pause, copy the centre comparison rule: the distribution with the higher mean has typically higher values, and the contextual language requirement: always say "the [group name] tends to have higher [variable name]" rather than "distribution A has a higher mean" into your book.

Quick check: Group A has median = 82 and Group B has median = 75. Which statement correctly compares centre?

3

Compare the spread

Decide which group is more consistent using standard deviation and range.

06
Comparing spread, which group is more consistent?
core concept

Comparing centres (means or medians) tells you which group is typically higher, the distribution with the larger mean has typically larger values. The second dimension is variability: compare standard deviations (or ranges/IQRs). The distribution with the smaller σ has values that cluster more tightly around the mean, meaning more consistent or predictable outcomes.

After comparing centre, compare the spread to assess consistency.

  • Smaller IQR or SD = more consistent (values cluster near the centre).
  • Larger IQR or SD = more variable (values are spread across a wider range).

Example:

Machine A: IQR = 2 mm. Machine B: IQR = 5 mm.

"Machine A produces more consistent parts than Machine B. Machine B's larger IQR indicates greater variability in screw length."

Context matters: A smaller spread is not always better. In education, wide spread might reflect a diverse classroom; in manufacturing, wide spread usually means poor quality control.

To compare spread: the distribution with larger standard deviation (or range/IQR) is more variable. Smaller σ = more consistent, more tightly clustered around the mean. Express spread comparisons in context (e.g., 'more consistent test scores').

Pause, copy the spread comparison rule: larger standard deviation = more variable/less consistent; smaller standard deviation = more consistent/tightly clustered, and write one contextual example (e.g., "Class B has more consistent test scores") into your book.

True or false: Two distributions with identical means must also have identical spreads.

4

Compare shape and outliers

Add shape and outliers to build the full comparison framework.

07
Shape, outliers and the full comparison framework
core concept

Comparing standard deviations (larger σ = more variable, smaller σ = more consistent) covers spread. A complete HSC comparison question requires four features: shape (normal/skewed/symmetric), centre (compare means or medians), spread (compare standard deviations or ranges), and any notable outliers. Always relate each feature to the real-world variable in a full sentence.

Shape:

  • Is one distribution symmetric while the other is skewed?
  • Does one have a single peak (unimodal) while the other is bimodal?
  • Right (positive) skew: tail extends to the right; median < mean.
  • Left (negative) skew: tail extends to the left; median > mean.

Outliers:

  • Does one group have more extreme values?
  • Are the outliers high or low? What effect do they have on the mean?

Full comparison framework:

"Group A has a higher median than Group B, indicating better typical performance. However, Group A also has a larger IQR, suggesting less consistency. Group B is more symmetric, while Group A is slightly right-skewed with one high outlier that inflates its mean."

Worked example, reading box plots: Class X: min=45, Q1=60, median=72, Q3=80, max=90. Class Y: min=50, Q1=65, median=68, Q3=78, max=95. Class X IQR = 20; Class Y IQR = 13. Class X has higher median (72 vs 68) but is less consistent. Class Y is more consistent with a slightly wider overall range.

A complete distribution comparison covers shape (normal, skewed, symmetric), centre (mean or median), spread (standard deviation or range), and outliers. In HSC responses, always relate each feature to the real-world variable.

Pause, copy the four-feature comparison framework for a full HSC response: (1) shape (normal, skewed, symmetric); (2) centre (higher/lower mean or median); (3) spread (larger/smaller standard deviation); (4) outliers (any unusual values), always relate each feature to the real-world variable into your book.

Fill the gap: A box plot has Q1 = 55 and Q3 = 75. The interquartile range is .

5

Work through examples

Follow two worked comparisons, including parallel box plots.

Worked examples · reveal each step

PROBLEM 1 · COMPARING WITH SUMMARY STATISTICS

Team A: mean = 80, SD = 5. Team B: mean = 75, SD = 12. Compare the two teams fully.

1
Centre: Team A mean (80) > Team B mean (75)
Team A typically scores higher by 5 points on average
1 of 4
PROBLEM 2 · READING PARALLEL BOX PLOTS

School X: min=45, Q1=60, median=72, Q3=80, max=90, two outliers at 95. School Y: min=50, Q1=65, median=68, Q3=78, max=95, no outliers. Compare the schools.

1
Centre: School X median (72) > School Y median (68)
School X has higher typical performance
1 of 4
6

Drill and consolidate

Work the quick-fire practice, then revisit your first answer.

Quick-fire practice · 2 activities

1

Group 1: mean = 65, SD = 8. Group 2: mean = 70, SD = 15. (a) Which group has higher typical scores? (b) Which group is more consistent? (c) If you wanted predictable results, which group is preferable?

2

Write a comparison paragraph for two sporting teams. Team Lions: median score = 88, IQR = 6, roughly symmetric. Team Tigers: median score = 82, IQR = 18, right-skewed with two outlier wins at 110.

Match each measure to what it tells you about a distribution:

  • Median
  • IQR
  • Standard deviation
  • Skewness
  • Average distance of values from the mean
  • Whether the distribution has a long tail
  • Typical middle value (resistant to outliers)
  • Spread of the middle 50% of data

Top 3 list: Name THREE real-world situations where comparing spread (consistency) is more important than comparing the average. Explain briefly for each.

09
Revisit your thinking

Class A is more predictable, most students score 70–80, so if you are an average student you know what to expect. Class B has wide variation: some score 100 while others score 50. For a risk-averse student, Class A is preferable because of its small spread. For a high achiever who believes they can outperform the group, Class B offers the possibility of a higher mark. The key insight is that identical means can hide very different classroom experiences, this is why comparing spread is just as important as comparing centre.

auto-saved
7

Practise comparing distributions

Answer the multiple-choice questions on centre, spread and shape.

01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next.

Q1. Class A has median = 75 and IQR = 8. Class B has median = 75 and IQR = 20. Which statement is correct?

Q2. When comparing two distributions with outliers, which measure of centre is most appropriate?

Q3. Factory A: mean = 50 mm, SD = 1 mm. Factory B: mean = 50.5 mm, SD = 3 mm. For precision engineering, which factory is preferable and why?

Q4. A distribution is right-skewed. Which relationship between mean and median is expected?

Q5. Two data sets have the same mean but different standard deviations. What does this tell you?

02
Short answer
ApplyBand 42 marks

SA 1. Factory A produces screws: mean = 50 mm, SD = 1 mm. Factory B: mean = 50.5 mm, SD = 3 mm. (a) Compare the centres of the two distributions. (b) Compare the spreads. (c) Which factory would you choose for precision engineering? Justify your answer using both measures. (2 marks)

auto-saved
ApplyBand 42 marks

SA 2. Two schools' HSC results: School X, median = 82, IQR = 8, two outliers at 95. School Y, median = 80, IQR = 15, no outliers. (a) Compare typical performance. (b) Compare consistency. (c) Which school has better top-end performance? (d) Write a balanced comparison paragraph. (2 marks)

auto-saved
AnalyseBand 53 marks

SA 3. A hospital compares two treatments for a condition. Treatment A: mean recovery = 10 days, SD = 2 days. Treatment B: mean recovery = 8 days, SD = 4 days. (a) A patient argues Treatment B is better because it has a shorter average recovery time. Evaluate this claim statistically. (b) A doctor prefers Treatment A. Explain why, using spread as the key reason. (c) Design a decision rule that helps patients choose between treatments based on their personal risk tolerance. (3 marks)

auto-saved
Comprehensive answers (click to reveal)

MC 1, B: Same median means same typical score; smaller IQR means Class A is more consistent.

MC 2, C: Median is resistant to outliers; mean is pulled toward extreme values.

MC 3, A: In precision engineering, consistency (small SD) matters more than the 0.5 mm difference in mean.

MC 4, D: Right skew means a long tail to the right, pulling the mean above the median.

MC 5, B: Same mean = same average level; different SD = different consistency.

SA 1 (2 marks): (a) Factory B mean 50.5 mm is slightly higher than Factory A mean 50 mm [0.5]. (b) Factory A SD = 1 mm is far smaller than Factory B SD = 3 mm, so Factory A is much more consistent [0.5]. (c) Factory A, in precision engineering, consistency (small SD) is far more important than the 0.5 mm difference in mean [1].

SA 2 (2 marks): (a) School X has higher typical performance, median 82 vs 80 [0.5]. (b) School X is more consistent, IQR 8 vs 15 [0.5]. (c) School X has outliers at 95, indicating exceptional high achievers; School Y has no outliers but a wider spread [0.5]. (d) School X generally achieves stronger and more consistent results with some exceptional high achievers. School Y has slightly lower typical performance but a wider range of outcomes [0.5].

SA 3 (3 marks): (a) Treatment B does have a shorter mean (8 vs 10 days), but its SD of 4 means some patients may take up to 16 days, the claim ignores spread [1]. (b) Treatment A is more predictable, doctors can plan confidently around a 8–12 day recovery window; Treatment B's high variability makes planning difficult [1]. (c) Risk-averse patients or those requiring predictable recovery (elderly, planned surgery follow-up): choose Treatment A. Risk-tolerant or young healthy patients willing to accept the chance of longer recovery for the possibility of a shorter one: may prefer Treatment B [1].