Mathematics • Year 10 • Unit 4 • Lesson 7
Median and Mode — Mixed Challenge
Pull together every idea from Lesson 7: median for odd/even n, mode (including bimodal/no-mode), median from frequency tables, and "choosing the right measure" for different data shapes. Then spot a Year 10 mistake and design a data set to a strict brief.
1. Mixed problems
Each question uses a different idea from Lesson 7. 3 marks each
1.1 Find the median and mode of: 6, 9, 4, 6, 12, 4, 8, 6, 10.
1.2 Find the median of 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 (even n).
1.3 A frequency table shows the number of after-school activities done per week by 25 students.
x: 0 1 2 3 4 5
f: 3 6 8 5 2 1
Find the median and the modal number of activities. (Tip: think where the 13th student sits in the cumulative count.)
1.4 Compare the data sets A: {10, 12, 14, 16, 18} and B: {10, 12, 14, 16, 100}. Find each set's mean and median. Which measure changes more when 18 is replaced by 100, and why? (Reference Lesson 7's "outlier-resistant" key term.)
1.5 A school records the most-popular sport played by 100 students: soccer 38, netball 22, basketball 18, rugby 12, other 10. Mean, median or mode — which is the only sensible measure of centre here, and why?
1.6 Design two short data sets, each with 5 numbers, so that:
(a) Set X has mean = median (symmetric).
(b) Set Y has mean > median (positively skewed by a large value).
Show both data sets and the calculations.
2. Find the mistake
A Year 10 student is finding the median of the data set 15, 8, 22, 11, 18, 9, 14, 17. Their working is below. Exactly one line contains the error. Spot it, explain, and re-do. 3 marks
Student's working:
Line 1: n = 8 (even).
Line 2: The middle two values in the ORIGINAL (unordered) list are 18 and 9.
Line 3: Median = (18 + 9) ÷ 2 = 13.5.
(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 ordered list and the corrected median.
Stuck? Lesson 7 misconception: the median is the middle of the ORDERED list. You must sort the data first.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 8 numbers that satisfies ALL of the following Lesson 7 properties:
- median = 10,
- the data set is bimodal, with modes 6 and 14,
- at least one value is greater than 25 (so the data has a clear outlier),
- all values are positive whole numbers.
For your data set, show:
(i) the 8 values in ORDER,
(ii) the calculation that proves median = 10 (state which positions you averaged),
(iii) which values give you the two modes,
(iv) one sentence using the Lesson 7 Key Term "bimodal" to describe your set.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Median and mode of 6, 9, 4, 6, 12, 4, 8, 6, 10
Ordered: 4, 4, 6, 6, 6, 8, 9, 10, 12. n = 9, middle = 5th value. Median = 6. 6 appears 3 times. Mode = 6.
1.2 — Median of 2.5, 3.0, 3.5, 4.0, 4.5, 5.0
n = 6, middle = 3rd and 4th values. Median = (3.5 + 4.0) ÷ 2 = 3.75.
1.3 — Activities frequency table
Σf = 25, so median is the 13th value. Cumulative frequencies: 3, 9, 17, 22, 24, 25. The 13th value falls in the "x = 2" group (cumulative count reaches 17 at x = 2). Median = 2 activities. Highest frequency f = 8 at x = 2, so mode = 2 activities.
1.4 — Compare A and B
Set A: mean = 70 ÷ 5 = 14, median = 14.
Set B: mean = 152 ÷ 5 = 30.4, median = 14.
The mean changed dramatically (14 → 30.4) but the median is unchanged (14). The median is outlier-resistant — it only depends on the middle value's position, not on the extreme value's size.
1.5 — Most popular sport
The data is categorical (sport names, not numbers). Only the mode makes sense: soccer (38 students). Mean and median require numerical values — you can't add "soccer + netball" or order sports by size.
1.6 — Design symmetric vs skewed
(a) Set X: 2, 4, 6, 8, 10. Mean = 30 ÷ 5 = 6. Median (3rd value) = 6. Mean = median ✓ (symmetric).
(b) Set Y: 2, 4, 6, 8, 100. Mean = 120 ÷ 5 = 24. Median = 6. Mean (24) > median (6) ✓ (positively skewed by the outlier 100).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student took the middle two values of the UNORDERED list. The median is defined as the middle of the ORDERED list (Lesson 7 Key Terms). Skipping ordering is the #1 median error.
(c) Ordered: 8, 9, 11, 14, 15, 17, 18, 22. n = 8, middle = 4th and 5th values. Median = (14 + 15) ÷ 2 = 14.5.
3 — Open-ended challenge (sample solution)
Data set (ordered): 1, 6, 6, 8, 12, 14, 14, 30.
(ii) n = 8. Middle = 4th and 5th values = (8 + 12) ÷ 2 = 10 ✓
(iii) 6 appears twice and 14 appears twice — both with the highest frequency. Modes = 6 and 14 ✓
(iv) The data set is bimodal because two distinct values (6 and 14) share the highest frequency of two each, and the value 30 is a clear outlier.
Marking: 1 mark for 8 values all positive whole numbers, 1 for verified median = 10, 1 for bimodal modes 6 and 14, 1 for value > 25 and correct use of "bimodal" in sentence.