Mathematics • Year 10 • Unit 4 • Lesson 9
Box Plots — Mixed Challenge
Pull together every idea from Lesson 9: the five-number summary, drawing and reading a box plot, parallel box plots for comparing data sets, and identifying skewness from whiskers and box halves. Then spot a Year 10 mistake and design a five-number summary to a strict brief.
1. Mixed problems
Each question uses a different idea from Lesson 9. 3 marks each
1.1 Find the five-number summary for 18, 22, 25, 27, 30, 33, 35, 38, 40, 44, 50.
1.2 Sketch the box plot for the summary in 1.1 on a labelled axis 15-55.
1.3 From the summary in 1.1, what percentage of the data lies BETWEEN the median and Q3?
1.4 Two box plots on the same scale show: Plot A median 50, IQR 20; Plot B median 70, IQR 20. Same spread, different centres. Describe in two sentences what conclusion you can draw about the two data sets.
1.5 A box plot has a very short LEFT whisker, a long RIGHT whisker, and a median that sits close to Q1 inside the box. Classify the skewness (positive, negative, or symmetric) and explain in one sentence.
1.6 A box plot summary is min = 10, Q1 = 15, median = 18, Q3 = 22, max = 60. Apply the 1.5×IQR rule: which value(s), if any, are outliers? Show the fence calculations.
2. Find the mistake
A Year 10 student is drawing a box plot from the summary min = 20, Q1 = 25, median = 30, Q3 = 38, max = 60. Their description of the diagram is below. Exactly one statement is wrong. Spot it, explain, re-do. 3 marks
Student's working:
Line 1: Draw a horizontal scale from 15 to 65 (labelled every 5).
Line 2: Draw the box from 20 (min) to 60 (max).
Line 3: Draw the median line inside the box at 30.
Line 4: Add whiskers from the box ends out to min and max.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong (use the Lesson 9 misconception that the BOX shows the IQR, not the range).
(c) Re-do the description correctly so the box plot matches the summary.
Stuck? The box spans Q1 to Q3 (25 to 38), NOT min to max. Whiskers go from Q1/Q3 to min/max.3. Open-ended challenge — design a five-number summary
This question has many valid answers. Follow every rule. 4 marks
3.1 Design a five-number summary for a data set of test marks (out of 100) that satisfies ALL of the following Lesson 9 properties:
- median = 60,
- the data is positively skewed (right tail longer than left),
- IQR = 20,
- range = 70,
- no value is above 100 or below 0.
Show:
(i) all five summary numbers (min, Q1, median, Q3, max),
(ii) a sketch of the box plot on a labelled 0-100 axis,
(iii) verification that the upper whisker is LONGER than the lower whisker, and the median is closer to Q1 than to Q3 (the two signals of positive skew),
(iv) one sentence using the Lesson 9 phrase "positively skewed".
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Five-number summary for 18–50
n = 11, median (6th value) = 33. Lower half (5 values, excl. median): 18, 22, 25, 27, 30 → Q1 = 25 (median of 5). Upper half: 35, 38, 40, 44, 50 → Q3 = 40. Summary: 18, 25, 33, 40, 50.
1.2 — Box plot sketch
On axis labelled 15-55 in steps of 5: box from 25 to 40, median line at 33, whiskers to 18 and 50.
1.3 — Percentage between median and Q3
By definition, the median splits the data in half, and Q3 marks the 75th percentile. So the proportion between the median and Q3 is 25%.
1.4 — Two box plots, same spread different centres
Both data sets have the same middle-50% width (IQR 20), so they are equally spread. However, Plot B sits 20 units higher than Plot A (medians 70 vs 50), meaning every typical value in Plot B is shifted upward — a clear translation of the entire distribution, not a change in shape.
1.5 — Short left whisker, long right whisker, median near Q1
Positively skewed (right-skewed). The long right whisker and the median sitting close to Q1 inside the box both signal a tail of high values stretching to the right.
1.6 — Outliers from summary
IQR = 22 − 15 = 7. Upper fence = 22 + 1.5×7 = 32.5. Lower fence = 15 − 10.5 = 4.5. Max = 60, which is greater than 32.5 → 60 IS an outlier. Min = 10, between 4.5 and 32.5 → not an outlier.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The Lesson 9 misconception card says the BOX shows the IQR (middle 50%), not the full range. The student drew the box from min (20) to max (60), which is the whisker span. Two distinct parts of the plot got conflated.
(c) Corrected: Draw the box from Q1 = 25 to Q3 = 38; median line inside at 30; whiskers from the box ends out to min (20) and max (60).
3 — Open-ended challenge (sample solution)
Summary: min = 30, Q1 = 58, median = 60, Q3 = 78, max = 100.
(ii) Box from 58 to 78, median line at 60, whiskers from 30 to 58 (left) and 78 to 100 (right) on a labelled 0-100 axis.
(iii) Check: IQR = 78 − 58 = 20 ✓. Range = 100 − 30 = 70 ✓. Lower whisker length = 58 − 30 = 28. Upper whisker length = 100 − 78 = 22. (Or adjust min to 25 to make the upper whisker longer.) Median to Q1 = 60 − 58 = 2 (close). Median to Q3 = 78 − 60 = 18 (far). Median sits much closer to Q1 → positive skew ✓.
Alternative cleaner summary: min = 35, Q1 = 55, median = 60, Q3 = 75, max = 100. Lower whisker = 20, upper whisker = 25 (upper longer ✓); median 60 is 5 above Q1 (55) and 15 below Q3 (75) → closer to Q1 ✓.
(iv) The data is positively skewed — a longer right tail of high marks pulls the distribution to the right.
Marking: 1 mark for valid summary in 0-100, 1 for median 60 + IQR 20 + range 70, 1 for verifying upper whisker > lower whisker AND median closer to Q1, 1 for sentence using "positively skewed".