Mathematics • Year 10 • Unit 4 • Lesson 8
Range and IQR — Mixed Challenge
Pull together every idea from Lesson 8: range, quartiles, IQR, the 1.5 × IQR outlier rule, and the misconception that "the range is unaffected by outliers". Then spot a Year 10 mistake and design a data set to a strict spread brief.
1. Mixed problems
Each question uses a different idea from Lesson 8. 3 marks each
1.1 Find the range, Q1, Q3 and IQR of 4, 6, 8, 10, 12, 14, 16, 18, 20, 22.
1.2 For the ordered set 3, 5, 7, 9, 11, 13, 15, find Q1, Q3 and IQR. (n = 7, odd — exclude the median from each half.)
1.3 A data set has Q1 = 50 and Q3 = 80. Use the 1.5×IQR rule to find the upper and lower outlier fences. Then state whether each value 12, 60, 130 is an outlier.
1.4 Two classes' test marks are summarised below. Compare their spreads.
Class A: range 30, IQR 10. Class B: range 30, IQR 22.
Explain in two sentences what this comparison tells you about the shape of each distribution (where most of the data sits).
1.5 Given the data 11, 13, 15, 17, 19, 21, 23, 25, 27, 29: if we ADD 5 to every value, what happens to (i) the range and (ii) the IQR? Show why with a one-line calculation each. (Hint: shift both Q1 and Q3 by the same amount.)
1.6 Two data sets have the same minimum (5), the same maximum (95), the same median (50), but Set A has IQR 10 while Set B has IQR 40. Sketch a quick rough description (in words, no graph required) of how each set's data is distributed.
2. Find the mistake
A Year 10 student is finding the IQR of 6, 9, 12, 15, 18, 21, 24, 27. Their working is below. Exactly one line contains the error. Spot it, explain, re-do. 3 marks
Student's working:
Line 1: n = 8 (even).
Line 2: Lower half = {6, 9, 12}. Upper half = {21, 24, 27}.
Line 3: Q1 = median of lower half = 9. Q3 = median of upper half = 24.
Line 4: IQR = 24 − 9 = 15.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Re-do the calculation correctly, showing the full lower and upper halves.
Stuck? With even n, BOTH halves include 4 values each. The student dropped 15 and 18 from the halves.3. Open-ended challenge — design a data set
This question has many valid answers. Follow every rule. 4 marks
3.1 Design a data set of exactly 10 positive integers with ALL of these properties:
- range = 50,
- IQR = 10,
- exactly ONE outlier (by the 1.5×IQR rule),
- median = 20.
Show:
(i) the 10 values in order,
(ii) the calculation confirming range = 50, median = 20, IQR = 10,
(iii) the 1.5×IQR fences and the value(s) outside them,
(iv) one sentence linking your design to the Lesson 8 misconception that "range alone tells you about typical spread".
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Quartiles of 4–22 (steps of 2)
n = 10. Lower half: 4, 6, 8, 10, 12. Upper half: 14, 16, 18, 20, 22. Range = 22 − 4 = 18. Q1 = 8 (3rd of lower half). Q3 = 18. IQR = 18 − 8 = 10.
1.2 — Quartiles of 3,5,...,15
n = 7, median = 9 (4th). Lower half (excluding median): 3, 5, 7. Upper half: 11, 13, 15. Q1 = 5, Q3 = 13. IQR = 8.
1.3 — Outlier fences for Q1=50, Q3=80
IQR = 30. Lower fence = 50 − 45 = 5. Upper fence = 80 + 45 = 125. Outliers: 12 is NOT an outlier (12 > 5); 60 is NOT an outlier (within fences); 130 IS an outlier (130 > 125).
1.4 — Class spread comparison
Class A has a much smaller IQR (10) despite the same range (30) — most of Class A's data is clustered tightly in the middle, with one or two outliers stretching the range. Class B (IQR 22) has data spread more evenly across the full 30-mark range, with no tight central cluster.
1.5 — Adding 5 to every value
(i) Range = max − min. New max = 29 + 5 = 34, new min = 11 + 5 = 16. New range = 34 − 16 = 18 (unchanged).
(ii) Q1 and Q3 both shift up by 5. New IQR = (Q3 + 5) − (Q1 + 5) = Q3 − Q1 = unchanged. Both range and IQR are invariant under shifts.
1.6 — Two distributions, same range, different IQR
Set A (IQR 10) has data heavily concentrated around the median 50 (middle 50% lies between roughly 45 and 55), with thin tails reaching out to 5 and 95. Set B (IQR 40) has data much more spread across the full range — the middle 50% spans roughly 30 to 70, so even the "middle" group is widely scattered.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) With even n = 8, BOTH halves contain 4 values (positions 1-4 and 5-8). The student incorrectly left out 15 (4th value) and 18 (5th value) from each half.
(c) Lower half: {6, 9, 12, 15}. Upper half: {18, 21, 24, 27}. Q1 = (9+12)/2 = 10.5. Q3 = (21+24)/2 = 22.5. IQR = 22.5 − 10.5 = 12.
3 — Open-ended challenge (sample solution)
Data set (ordered): 15, 16, 18, 19, 20, 20, 22, 24, 25, 65.
(ii) Range = 65 − 15 = 50 ✓. n = 10, median = (20 + 20)/2 = 20 ✓. Lower half: 15, 16, 18, 19, 20. Upper half: 20, 22, 24, 25, 65. Q1 = 18 (median of lower half). Q3 = 24. IQR = 24 − 18 = 6 — wait, this doesn't quite give IQR = 10.
Alternative set: 15, 15, 17, 17, 20, 20, 25, 27, 27, 65. Lower half: 15, 15, 17, 17, 20. Upper half: 20, 25, 27, 27, 65. Q1 = 17, Q3 = 27, IQR = 10 ✓. Range = 65 − 15 = 50 ✓. Median = (20 + 20)/2 = 20 ✓.
(iii) Upper fence = 27 + 1.5×10 = 42. 65 > 42 → outlier. Lower fence = 17 − 15 = 2. All other values > 2, so only 65 is the outlier ✓.
(iv) The range (50) is dragged up by the single outlier 65, while the IQR (10) shows the middle 50% is actually tightly clustered around 20 — Lesson 8 misconception: range alone overstates "typical" spread.
Marking: 1 mark for 10 positive integers with median 20, 1 for range = 50, 1 for IQR = 10, 1 for exactly one outlier by the 1.5×IQR rule plus a sentence linking design to the Lesson 8 misconception.