Mathematics • Year 10 • Unit 4 • Lesson 9
Box Plots — Skill Drill
Build fluency with Lesson 9's five-number summary (min, Q1, median, Q3, max) and box-plot construction. Practise the two halves of the diagram: the BOX shows the IQR (middle 50%), the WHISKERS show the spread to min and max. Then read centre, spread and skewness directly off a box plot.
1. I do — fully worked example
Read every step. The reason on the right explains why.
Problem. Draw a box plot for the data: 12, 15, 18, 21, 24, 27, 30, 33, 36.
Step 1 — Order data and confirm n.
Already ordered. n = 9.
Step 2 — Find the five-number summary.
Min = 12. Max = 36. Median (Q2) = 5th value = 24.
Lower half (excluding median): 12, 15, 18, 21. Q1 = (15+18)/2 = 16.5.
Upper half: 27, 30, 33, 36. Q3 = (30+33)/2 = 31.5.
Reason: the box plot needs all five summary numbers — that is the only data it shows (Lesson 9 misconception: it does NOT show individual values).
Step 3 — Draw a labelled scale and the box plot.
Scale 10 → 40 in steps of 5 (label every tick). Draw box from Q1 (16.5) to Q3 (31.5).
Reason: Lesson 9 HSC Note (from Lesson 8 box-plot guidance) — examiners deduct marks for unlabelled axes.
Step 4 — Add the median line and whiskers.
Draw a vertical line inside the box at median = 24. Draw whiskers from the box ends out to min (12) and max (36).
Sketch of the result:
10 15 20 25 30 35 40
|____|____|____|____|____|____|
12 ┌──────│──────┐ 36
•───┤16.5 24 31.5├───•
└──────│──────┘
Answer: five-number summary 12, 16.5, 24, 31.5, 36; box plot as above.
2. We do — fill in the missing steps
Fill in the blanks and sketch the box plot on the scale provided. 4 marks
Problem. Draw a box plot for the test scores: 30, 35, 40, 45, 50, 55, 60, 65, 70, 75.
Step 1 — n = ____ (even).
Step 2 — Five-number summary.
Min = ____. Max = ____.
Median = average of the ____th and ____th values = ________.
Lower half (5 values): ____________ → Q1 = ________.
Upper half (5 values): ____________ → Q3 = ________.
Step 3 — Sketch on this scale (label the axis):
20 30 40 50 60 70 80
|____|____|____|____|____|____|
Box from Q1 to Q3, median line inside, whiskers to min and max.
3. You do — independent practice
Eight graduated questions. Use the box-plot construction process from Sections 1-2.
Foundation — read a box plot
3.1 A box plot shows: min = 5, Q1 = 10, median = 15, Q3 = 22, max = 30. State the range and the IQR. 1 mark
3.2 A box plot shows: min = 0, Q1 = 5, median = 10, Q3 = 15, max = 50. What fraction of the data lies between Q1 (5) and Q3 (15)? 1 mark
3.3 True or false: "The box in a box plot shows where ALL the data values are." Justify in one sentence using the Lesson 9 misconception card. 1 mark
Standard — build the five-number summary
3.4 Find the five-number summary for 4, 7, 9, 11, 13, 15, 18, 20. Then sketch a box plot. 2 marks
3.5 Find the five-number summary for 22, 25, 28, 30, 32, 35, 37, 40, 42. State the IQR. 2 marks
3.6 Find the five-number summary for the daily maximum temperatures (°C) over 11 days: 24, 26, 27, 28, 28, 29, 30, 31, 32, 33, 35. Sketch the box plot on a labelled scale. 2 marks
Extension — read shape from a box plot
3.7 A box plot has median = 60. The lower whisker stretches from 20 to 40 (length 20). The box covers 40 to 75. The upper whisker reaches max = 80. (a) Is the distribution symmetric, positively skewed or negatively skewed? Justify by comparing whisker lengths and box halves. (b) State the IQR. 3 marks
3.8 Two PARALLEL box plots are drawn for Class A (median 65, Q1 = 60, Q3 = 72) and Class B (median 65, Q1 = 50, Q3 = 80) on the same scale. Both classes have the same median, but the boxes differ. Using the Lesson 9 misconception ("a longer box means greater SPREAD in the middle 50%"), describe in one sentence what this comparison tells you about the two classes. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (30, 35, 40, 45, 50, 55, 60, 65, 70, 75)
Step 1: n = 10.
Step 2: min = 30, max = 75. Median = (5th + 6th)/2 = (50 + 55)/2 = 52.5. Lower half: 30, 35, 40, 45, 50 → Q1 = 40. Upper half: 55, 60, 65, 70, 75 → Q3 = 65.
Box from 40 to 65, median line at 52.5, whiskers to 30 and 75. Axis labelled 20-80 in steps of 10.
3.1 — Range and IQR from a box plot
Range = 30 − 5 = 25. IQR = Q3 − Q1 = 22 − 10 = 12.
3.2 — Fraction in the box
The box spans Q1 to Q3, which by definition contains the middle 50% of values (half of the data).
3.3 — True/false (box shows ALL values)
False. Lesson 9 misconception card: the box shows the IQR (middle 50%), NOT individual data values — those are hidden. A box plot does not let you read off any one student's score.
3.4 — Five-number summary 4–20
n = 8. Min = 4, max = 20. Median = (11 + 13)/2 = 12. Lower half: 4, 7, 9, 11 → Q1 = (7+9)/2 = 8. Upper half: 13, 15, 18, 20 → Q3 = (15+18)/2 = 16.5. Summary: 4, 8, 12, 16.5, 20.
3.5 — Five-number summary 22–42
n = 9, median = 32 (5th). Lower half: 22, 25, 28, 30 → Q1 = (25+28)/2 = 26.5. Upper half: 35, 37, 40, 42 → Q3 = (37+40)/2 = 38.5. Summary: 22, 26.5, 32, 38.5, 42. IQR = 12.
3.6 — Temperatures 24–35
n = 11, median = 29 (6th value). Lower half (5 values, excl. median): 24, 26, 27, 28, 28 → Q1 = 27. Upper half: 30, 31, 32, 33, 35 → Q3 = 32. Summary: 24, 27, 29, 32, 35. Box from 27 to 32, median line at 29, whiskers to 24 and 35; axis labelled 20-40 in steps of 5.
3.7 — Shape from box plot
(a) Lower whisker length 20 (from 20 to 40). Upper whisker length 5 (75 to 80). The lower whisker is much longer, AND the median (60) sits closer to Q3 (75) than to Q1 (40) — both signals point to negatively skewed data (tail to the left).
(b) IQR = 75 − 40 = 35.
3.8 — Parallel box plots
Both classes have the same median (65), so the "typical" student performed equally in each. However, Class B's box (IQR = 30) is much wider than Class A's (IQR = 12), so the middle 50% of Class B's students are far more spread out — Class A's middle students are clustered tightly around the centre, Class B's are scattered widely.