Mathematics • Year 10 • Unit 4 • Lesson 11
Comparing Data Sets — Mixed Challenge
Pull every Lesson 11 idea together: centre, spread, shape, parallel box plots, back-to-back stem-and-leaf, and the misconception that "higher mean = better". Spot a Year 10 mistake and design your own comparison.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 11. Decide whether the question is about centre, spread, shape or a misconception before you start writing. 3 marks each
1.1 Two Year 10 classes' assessment marks have summary statistics:
Class A: mean 70, range 20, IQR 8.
Class B: mean 70, range 60, IQR 30.
Why is the warning "they have the same average" misleading here? Quote both spread statistics.
1.2 Decide whether each statement is True or False, and give a one-sentence reason: (a) "If two distributions have the same median, they have the same shape." (b) "If two distributions have different IQRs, one must have a higher mean." (c) "A parallel box plot lets you compare centre, spread AND skew visually."
1.3 A back-to-back stem-and-leaf plot shows: stem 5 leaves Boys 1 2 4 vs Girls 0 3 5 7; stem 6 Boys 0 1 5 vs Girls 1 2 8; stem 7 Boys 2 vs Girls 0 3. Count totals, then state which group has the higher median and explain how the back-to-back display made the comparison easy.
1.4 Two parallel box plots for Year 10 swim times (s): Squad P has min 30, Q1 33, median 35, Q3 37, max 40. Squad Q has min 30, Q1 31, median 35, Q3 39, max 40. Both have the same median (35) and the same range (10). Compare them using IQR and shape.
1.5 "Higher mean is always better" — give two contexts from Australian life where the OPPOSITE is true (lower mean is better). Briefly explain why in each case.
1.6 A class of 25 students has mean test score 70 and IQR 10. One new student with a score of 90 joins. Without recalculating exactly, predict: (a) what happens to the mean (up/down/same)? (b) what happens to the median (likely larger/smaller change than the mean)? (c) what likely happens to the IQR? Justify each in one sentence.
2. Find the mistake
Another Year 10 student has tried to compare two cricket bowlers' wicket counts per match. Their reasoning is shown below. Exactly one claim is wrong (a Lesson 11 misconception). Spot it, explain why it's wrong, and re-do the comparison correctly. 3 marks
Student's reasoning — Bowler X vs Bowler Y, last 10 matches:
Line 1: Bowler X mean wickets = 2.5, Bowler Y mean wickets = 2.5.
Line 2: Bowler X range = 5, Bowler Y range = 1.
Line 3: Conclusion: "Because the means are equal, the two bowlers are equally good."
(a) Which line contains the misconception?
(b) Explain in one or two sentences why that claim is wrong, quoting the lesson misconceptions card.
(c) Write a corrected conclusion that uses both a centre AND a spread statistic.
Stuck? Lesson 11 misconception: "You can fully compare two data sets using only one statistic" — wrong. You need both centre AND spread.3. Open-ended challenge — design two data sets
This question has many valid answers. Be creative but follow every rule. 4 marks
3.1 Design two small data sets (n = 7 each) about a real-world topic of your choice (e.g. café wait times, basketball shots made, study minutes per day). Your two sets must satisfy ALL of the following:
- they have the same median,
- they have different ranges (one at least double the other),
- one is right-skewed and the other is roughly symmetric,
- your worked comparative statement uses centre, spread AND shape with numbers.
For each set:
(i) list the seven values,
(ii) state the median, range and IQR,
(iii) state the shape (use the lesson's definitions).
Then write your one-sentence comparative statement and (bonus) sketch the two parallel box plots.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Same mean, different spread
The means are equal (70) but Class B has triple the IQR (30 vs 8) and triple the range (60 vs 20). Lesson 11: a complete comparison needs BOTH centre and spread; quoting the mean alone hides the very different consistency of the two classes.
1.2 — True/False
(a) FALSE — same median says nothing about IQR or skew.
(b) FALSE — IQR is a spread statistic and is independent of mean (the mean could be higher, lower or equal).
(c) TRUE — parallel box plots show median (centre), box width (IQR/spread) and whisker asymmetry (skew/shape).
1.3 — Back-to-back stem-and-leaf
Boys n = 3+3+1 = 7, values: 51, 52, 54, 60, 61, 65, 72. Median = 60.
Girls n = 4+3+2 = 9, values: 50, 53, 55, 57, 61, 62, 68, 70, 73. Median = 61.
Girls have a slightly higher median. The back-to-back display puts the two groups on the same stem, so the comparison is visual — you can see both centres at a glance.
1.4 — Same median, same range — IQR + shape
Squad P: IQR = 37 − 33 = 4. Squad Q: IQR = 39 − 31 = 8. Q's middle 50% is twice as spread out as P's. Shape: P is symmetric around the median (33→35→37 is even). Q is also symmetric around 35 but with wider whiskers and a wider box, so Q is more variable in the middle as well as overall.
1.5 — "Lower is better" contexts
Examples (any two): (i) Injury rates per 1000 workers — lower is safer. (ii) Hospital wait times (minutes) — shorter is better. (iii) Sprint times (seconds) — faster runners have lower times. (iv) CO₂ emissions per car (g/km) — lower is greener. (v) Number of bugs in a software release — fewer is better.
1.6 — One new outlier joins
(a) Mean: goes up — adding 90 to a class with mean 70 pulls the average upward.
(b) Median: smaller change than the mean. With 26 values, the median is the average of the 13th and 14th value; one new value at the top barely shifts these positions.
(c) IQR: almost no change. IQR uses the middle 50% (Q1 to Q3); one new extreme value above Q3 does not move Q1 or Q3 significantly.
2 — Find the mistake
(a) The mistake is on Line 3 (the conclusion).
(b) The Lesson 11 misconceptions card warns: "You can fully compare two data sets using only one statistic" is wrong. Equal means do not mean equal performance — you also need to look at spread.
(c) Corrected: Both bowlers average 2.5 wickets per match, but Bowler X is far more variable (range 5 vs 1). Bowler Y is the more consistent bowler; Bowler X has higher highs but also lower lows.
3 — Open-ended challenge (sample solution)
Topic: Minutes spent on homework per night for two students over a week.
Student A (symmetric): 25, 28, 30, 30, 30, 32, 35.
Median = 30; range = 35 − 25 = 10; Q1 = 28, Q3 = 32, IQR = 4. Roughly symmetric (mean ≈ 30, balanced around the median).
Student B (right-skewed): 10, 15, 25, 30, 40, 60, 90.
Median = 30; range = 90 − 10 = 80; Q1 = 15, Q3 = 60, IQR = 45. Right-skewed (long upper tail, mean ≈ 38.6 > median).
Comparative statement: "Although both students have the same median homework time (30 min), Student B's homework time is far more variable (range 80 vs 10, IQR 45 vs 4) and is right-skewed, with occasional long nights pulling the distribution up; Student A is roughly symmetric and very consistent."
Marking: 1 mark for same median, 1 for range ratio ≥ 2, 1 for one right-skew + one symmetric set, 1 for a comparative statement quoting centre + spread + shape with numbers.