Mathematics Standard • Year 12 • Module 8 • Lesson 3
Measures of Spread — Skill Drill
Build fluency in calculating range, quartiles, IQR and standard deviation (calculator-mode), and judging which best describes a data set's variability.
1. Quick recall
Answer each in the space provided. 1 mark each
Q1.1 Formulas:
Range = ____________ − ____________. IQR = ____________ − ____________.
Q1.2 The sample standard deviation symbol is ____ ; the population standard deviation symbol is ____. On HSC questions, use the value labelled by your calculator as ____ unless told otherwise (Standard 2 uses the population SD by default).
Q1.3 Which measure of spread is most affected by an outlier: range, IQR or standard deviation? ____________. Which is least affected? ____________.
2. Worked example — range, quartiles, IQR, SD
Weekly sales ($): 1,200, 1,350, 1,400, 1,450, 1,500, 1,550, 1,600, 2,500. n = 8.
Step 1 — Order the data.
1200, 1350, 1400, 1450, 1500, 1550, 1600, 2500 (already ordered)
Step 2 — Range.
R = max − min = 2,500 − 1,200 = $1,300.
Step 3 — Quartiles. Lower half (4 values): 1200, 1350, 1400, 1450 → Q1 = (1350 + 1400)/2 = 1,375. Upper half: 1500, 1550, 1600, 2500 → Q3 = (1550 + 1600)/2 = 1,575.
Step 4 — IQR.
IQR = Q3 − Q1 = 1,575 − 1,375 = $200.
Step 5 — Mean and SD (population, σ from calculator).
Mean = 12,550 / 8 = $1,568.75. σ ≈ $370.8 (1 d.p.).
Conclusion. The large gap between range ($1,300) and IQR ($200) shows that the $2,500 day is an outlier. SD ($370.8) is between the two and is also affected by the outlier.
3. Faded example — fill in the missing spread calculations
Class A test marks (n = 9): 42, 55, 60, 65, 68, 72, 75, 80, 88. 4 marks
Step 1 — Range: R = ________ − ________ = ____________.
Step 2 — Median position: n = 9 → median is the ________th value = ____________.
Step 3 — Q1 and Q3: Lower 4 values (exclude median) → Q1 = avg of 2nd and 3rd = ____________. Upper 4 values → Q3 = avg of 7th and 8th = ____________.
Step 4 — IQR: IQR = ________ − ________ = ____________.
Step 5 — SD (calculator, σ): σ ≈ ____________ marks (1 d.p.).
4. Graduated practice — spread calculations
Foundation — one-step (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | Find the range of: 12, 18, 25, 28, 34, 41. | |
| 4.2 1 | If Q1 = 15 and Q3 = 38, find the IQR. | |
| 4.3 1 | A data set has min = 4 and max = 67. Find the range. | |
| 4.4 1 | For Q1 = 6.2 and Q3 = 9.5, find the IQR (1 d.p.). |
Standard — typical HSC difficulty (6 questions)
4.5 Find Q1, Q3 and IQR for: 4, 6, 7, 9, 10, 12, 14, 16, 18, 20 (n = 10). 2 marks
4.6 Find Q1, median, Q3 and IQR for: 14, 18, 22, 25, 28, 30, 33, 38, 45 (n = 9). 2 marks
4.7 Use a calculator to find σ (population SD) for: 50, 52, 54, 56, 58, 60, 62 (n = 7). Give your answer to 1 d.p. 2 marks
4.8 For the values 8, 12, 15, 20, 22, 28, find the mean and σ (1 d.p.). 2 marks
4.9 Two data sets both have mean 50. Set A: 48, 49, 50, 51, 52. Set B: 30, 40, 50, 60, 70. Without a calculator, which set has the larger SD and why? 2 marks
4.10 A small data set is 5, 7, 9, 11, 13. Show that the range is 8 and use a calculator to find σ (1 d.p.). 2 marks
Extension — interpret the spread (2 questions)
4.11 Two soccer strikers have the following goals-per-game records over 10 games. Striker P: 1, 1, 2, 2, 2, 2, 2, 3, 3, 3 (mean 2.1). Striker Q: 0, 0, 1, 1, 2, 2, 3, 4, 4, 4 (mean 2.1). For each, calculate the range and use a calculator to find σ (1 d.p.). Which striker is more consistent? 3 marks
4.12 A data set of daily temperatures has IQR = 4 and range = 28. State what this combination suggests about the distribution, and explain in one sentence why IQR is normally preferred over range when one or two unusual values are present. 3 marks
5. Self-check the easy 3
Tick once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Formulas
Range = max − min. IQR = Q3 − Q1.
Q1.2 — SD symbols
Sample SD = s. Population SD = σ. Standard 2 uses σ.
Q1.3 — Spread sensitivity
Most affected: range (depends only on the extremes). Least affected: IQR (ignores the outer 25% at each end).
Q3 — Faded example (Class A)
Step 1: Range = 88 − 42 = 46. Step 2: median = 5th value = 68. Step 3: lower half (4 values): 42, 55, 60, 65 → Q1 = (55 + 60)/2 = 57.5. Upper half: 72, 75, 80, 88 → Q3 = (75 + 80)/2 = 77.5. Step 4: IQR = 77.5 − 57.5 = 20. Step 5: σ ≈ 13.5 marks (population SD from calculator).
Q4.1 – Q4.4 — Foundation
4.1: range = 41 − 12 = 29. 4.2: IQR = 38 − 15 = 23. 4.3: range = 67 − 4 = 63. 4.4: IQR = 9.5 − 6.2 = 3.3.
Q4.5 — n = 10
Lower half (5 values): 4, 6, 7, 9, 10 → Q1 = 7. Upper half: 12, 14, 16, 18, 20 → Q3 = 16. IQR = 16 − 7 = 9.
Q4.6 — n = 9
Median = 5th value = 28. Lower half (exclude median): 14, 18, 22, 25 → Q1 = (18 + 22)/2 = 20. Upper half: 30, 33, 38, 45 → Q3 = (33 + 38)/2 = 35.5. IQR = 35.5 − 20 = 15.5.
Q4.7 — σ for symmetric set
Mean = 56. Deviations squared: 36 + 16 + 4 + 0 + 4 + 16 + 36 = 112. σ = √(112/7) = √16 = 4.0.
Q4.8 — Mean and σ
Mean = (8+12+15+20+22+28)/6 = 105/6 = 17.5. σ ≈ 6.7 (from calculator, 1 d.p.).
Q4.9 — Compare without calculator
Set B has the larger SD. Set A's values are all within ±2 of the mean (small deviations); Set B's values are up to ±20 from the mean (large deviations). Larger deviations from the mean → larger SD.
Q4.10 — 5, 7, 9, 11, 13
Range = 13 − 5 = 8. Mean = 9. Deviations: −4, −2, 0, 2, 4 → squared: 16, 4, 0, 4, 16 → sum = 40. σ = √(40/5) = √8 ≈ 2.8.
Q4.11 — Striker P vs Q
P: range = 3 − 1 = 2; σ ≈ 0.7. Q: range = 4 − 0 = 4; σ ≈ 1.5. Both average 2.1 goals but P's SD is half Q's → Striker P is more consistent (P scores 1-3 every week; Q has games with 0 and games with 4).
Q4.12 — IQR 4 vs range 28
The huge gap (range 28, IQR 4) suggests the middle 50% of days cluster tightly (within 4°), but at least one day is far from the centre — likely an outlier (e.g. a heatwave). IQR is preferred when outliers are present because it only depends on Q1 and Q3, so one extreme value cannot distort it.