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.