Mathematics • Year 10 • Unit 4 • Lesson 8

Range and IQR — Skill Drill

Build fluency with Lesson 8's two measures of spread. Practise the calculations: range = max − min; quartiles Q1, Q2 (median) and Q3 from ordered data; IQR = Q3 − Q1. Then apply the 1.5 × IQR outlier rule: any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is an outlier.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every step. Each one has a short reason on the right.

Problem. For the data 8, 12, 15, 18, 21, 24, 28, 32, 36, find the range, the IQR, and check for outliers using the 1.5×IQR rule.

Step 1 — Confirm ordered. Count n.

Already ordered. n = 9.

Step 2 — Range = max − min.

Range = 36 − 8 = 28.

Reason: Lesson 8 Key Terms — "Range: difference between max and min".

Step 3 — Median (Q2) splits data into halves.

Median position = (9+1)/2 = 5th value. Q2 = 21. Lower half: 8, 12, 15, 18. Upper half: 24, 28, 32, 36.

Step 4 — Q1 = median of lower half, Q3 = median of upper half.

Q1 = (12 + 15)/2 = 13.5. Q3 = (28 + 32)/2 = 30.

Reason: Lesson 8 misconception — "Q1 is a single number" (the value below which 25% sits), not a set.

Step 5 — IQR = Q3 − Q1.

IQR = 30 − 13.5 = 16.5.

Step 6 — Outlier rule.

Lower fence = Q1 − 1.5×IQR = 13.5 − 24.75 = −11.25.

Upper fence = Q3 + 1.5×IQR = 30 + 24.75 = 54.75.

All values lie between −11.25 and 54.75, so there are NO outliers.

Answer: range = 28, IQR = 16.5, no outliers.

Stuck? Revisit lesson § Key Terms — "IQR: spread of the middle 50% of data; Q3 − Q1".

2. We do — fill in the missing steps

Same structure as Section 1. Fill in each blank. 4 marks

Problem. For the data set 5, 7, 9, 11, 13, 15, 17, find the range, IQR, and check for outliers.

Step 1 — n = ____ (odd).

Step 2 — Range.

Range = ____ − ____ = ________

Step 3 — Median (Q2).

Median position = (7+1)/2 = ____. Q2 = ________

Step 4 — Lower half = ________; Upper half = ________.

Step 5 — Q1 = median of lower half = ________; Q3 = median of upper half = ________.

Step 6 — IQR = Q3 − Q1 = ________.

Step 7 — Outlier fences.

Lower = Q1 − 1.5×IQR = ________. Upper = Q3 + 1.5×IQR = ________. Outliers? ________

Stuck? With odd n, exclude the median from both halves when finding Q1 and Q3.

3. You do — independent practice

Eight graduated questions. Show ordering and Q1/Q3 working. Foundation (range only), Standard (IQR), Extension (1.5×IQR rule, why IQR is robust).

Foundation — range only

3.1 Find the range of 12, 5, 18, 9, 23, 7. 1 mark

3.2 Find the range of −3, 4, 2, −7, 5, 1, −1. 1 mark

3.3 A class's test marks were: 64, 71, 58, 82, 49, 73, 90, 67. Find the range. 1 mark

Standard — IQR

3.4 For the ordered data 2, 4, 6, 8, 10, 12, 14, 16, find Q1, Q3 and the IQR. (n = 8, so split into halves of 4 and 4.) 2 marks

3.5 Find Q1, Q3, range and IQR for 14, 16, 18, 20, 22, 24, 26, 28, 30. (n = 9, odd.) 2 marks

3.6 The daily rainfall (mm) at a school over 10 days was 0, 0, 2, 3, 5, 6, 8, 10, 12, 18. Calculate Q1, Q3 and the IQR. 2 marks

Extension — 1.5 × IQR outlier rule, robustness

3.7 For the data 5, 8, 9, 10, 11, 12, 13, 14, 15, 40: find Q1, Q3 and IQR. Then apply the 1.5×IQR rule: is 40 an outlier? Show the upper fence calculation. 3 marks

3.8 Compare these two data sets using the Lesson 8 misconception card.
Set A: 10, 20, 30, 40, 50. Set B: 10, 20, 30, 40, 200.
(a) Find the range of each.
(b) Find the IQR of each.
(c) In one sentence, explain why the range "exaggerates" the difference between A and B but the IQR does not. (This is exactly what the Lesson 8 misconception warns about.) 3 marks

Stuck on 3.8(c)? Lesson 8 misconception: "the range IS heavily affected by outliers — it uses max and min, so one extreme value changes it completely."

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (5, 7, 9, 11, 13, 15, 17)

Step 1: n = 7.
Step 2: range = 17 − 5 = 12.
Step 3: median position = 4. Q2 = 11.
Step 4: lower half = 5, 7, 9; upper half = 13, 15, 17.
Step 5: Q1 = 7; Q3 = 15.
Step 6: IQR = 15 − 7 = 8.
Step 7: lower fence = 7 − 12 = −5; upper fence = 15 + 12 = 27. All values lie between −5 and 27. No outliers.

3.1 — Range of 12, 5, 18, 9, 23, 7

Max = 23, min = 5. Range = 23 − 5 = 18.

3.2 — Range of −3, 4, 2, −7, 5, 1, −1

Max = 5, min = −7. Range = 5 − (−7) = 12.

3.3 — Test marks range

Max = 90, min = 49. Range = 41.

3.4 — Q1, Q3, IQR for 2,4,6,8,10,12,14,16

n = 8. Lower half: 2, 4, 6, 8. Upper half: 10, 12, 14, 16. Q1 = (4+6)/2 = 5. Q3 = (12+14)/2 = 13. IQR = 13 − 5 = 8.

3.5 — Quartiles for 14,16,...,30

n = 9, median (Q2) = 22 (5th value). Lower half (excluding median): 14, 16, 18, 20. Upper half: 24, 26, 28, 30. Q1 = (16+18)/2 = 17. Q3 = (26+28)/2 = 27. Range = 30 − 14 = 16. IQR = 27 − 17 = 10.

3.6 — Rainfall data

n = 10. Lower half: 0, 0, 2, 3, 5. Upper half: 6, 8, 10, 12, 18. Q1 = 2 (median of 5 values, 3rd). Q3 = 10. IQR = 10 − 2 = 8 mm.

3.7 — Outlier check for ...,15,40

n = 10. Lower half: 5, 8, 9, 10, 11. Upper half: 12, 13, 14, 15, 40. Q1 = 9, Q3 = 14, IQR = 5.
Upper fence = Q3 + 1.5×IQR = 14 + 7.5 = 21.5. Since 40 > 21.5, 40 IS an outlier.

3.8 — Range vs IQR sensitivity

(a) Range A = 40. Range B = 190.
(b) Set A: Q1 = 15, Q3 = 45, IQR = 30. Set B: Q1 = 15, Q3 = 40, IQR = 25.
(c) Replacing 50 with 200 in Set B blew up the range from 40 to 190 (using max and min only) but only nudged the IQR from 30 to 25 — the IQR ignores the extreme value because it only depends on the middle 50% of values. Lesson 8 misconception: the range IS heavily affected by outliers, which is exactly why IQR is preferred.