Mathematics • Year 7 • Unit 4 • Lesson 10
The Mode and Range — Mixed Challenge
Combine every Unit 4 skill: identify the mode (including bimodal and no-mode), calculate the range, choose between mean / median / mode for any data type, and judge how outliers distort the range. Then spot one plausible student error, and design your own summary report.
1. Mixed problems — apply every skill
Each question uses a different idea from the lesson. Show working. 2 marks each
1.1 For data {12, 15, 18, 12, 20, 12, 15, 18}: find the mode and the range.
1.2 A dataset is {2, 4, 6, 8, 10}. State the mode (be careful!). Then state the range.
1.3 Decide whether MEAN, MEDIAN or MODE is the best measure of centre for each, with a one-line reason: (a) eye colour of 30 students; (b) age in years at a primary-school parents' meeting (mix of teachers, parents and one great-grandparent aged 92); (c) goals per game by a soccer team over 12 matches.
1.4 A class scored: 50, 50, 52, 55, 56, 58, 60. (a) Find the range. (b) The teacher then adds a missing score of 95. Recalculate the range. (c) State by how much the range changed and explain in one sentence why it changed by so much.
1.5 The mode of {3, 5, 7, x, 9, 5, 11} is 5. What can you conclude about the value of x? List ALL possible answers.
1.6 For data {4, 7, 10, 13, 16, 19}: (a) find the mean, (b) find the median, (c) find the mode, (d) find the range. Which two of these four numbers tell you about CENTRE, and which one tells you about SPREAD?
2. Find the mistake
Another Year 7 student answered the prompt: "Find the mode of: 4, 4, 7, 7, 7, 9." Their working has exactly one error. Spot it, explain why it's wrong, then write the correct solution. 3 marks
Student's working:
Line 1: Tally: 4 appears 2 times.
Line 2: 7 appears 3 times.
Line 3: 9 appears 1 time.
Line 4: The highest frequency is 3.
Line 5: So the mode = 3. [Answer: 3]
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the correct conclusion (the right mode).
Stuck? Revisit lesson § "Common Pitfalls" — "Confusing mode value with its frequency". The mode is the VALUE that appears most, not the count.3. Open-ended challenge — design a class data summary report
This question has many correct answers. Show your work clearly. 4 marks
3.1 You are asked to write a one-page "Class Stats" report for your Year 7 cohort. Collect (or invent) data for three variables:
- (A) one categorical variable (e.g. favourite subject, eye colour);
- (B) one discrete numerical variable (e.g. number of siblings, number of pets);
- (C) one continuous numerical variable (e.g. height in cm, daily screen time in minutes).
For EACH variable: (i) state the variable and provide 10+ invented values, (ii) identify the most appropriate measure of centre — mean, median, or mode — with a one-line reason, (iii) calculate that measure AND the range (where meaningful). End by writing ONE sentence about what your report tells the reader.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Mode and range
Tally: 12→3, 15→2, 18→2, 20→1. Mode = 12. Max = 20, Min = 12. Range = 8.
1.2 — {2, 4, 6, 8, 10}
Every value appears exactly once → no mode. Range = 10 − 2 = 8.
1.3 — Best measure of centre
(a) Eye colour — Mode. Categorical; the only measure that works.
(b) Ages with a 92-year-old at a primary parents' meeting — Median. The great-grandparent is an outlier that would inflate the mean.
(c) Goals per game — Mean (or median). Discrete numerical, usually no extreme outliers; mean is the standard sporting average.
1.4 — Range with outlier
(a) Max = 60, Min = 50. Range = 10.
(b) Adding 95: Max = 95, Min = 50. Range = 45.
(c) The range jumped by 35 marks because the range depends on only the two extreme values — one new high value of 95 immediately raised the maximum.
1.5 — Finding x for mode 5
Currently 5 appears twice; every other listed value appears once. For mode = 5, no other value can tie or exceed two appearances. So x can equal 5 (making 5 appear 3 times, even more clearly the mode) OR x can be any value not already appearing twice (i.e. not 5… wait — if x = 5, mode is definitely 5; if x is any unique new value like 6 or 100, then 5 still has the unique highest frequency of 2). So x ∈ {5, or ANY value not equal to 3, 7, 9 or 11 — because if x equalled one of those, that value would tie 5 at frequency 2, making the data bimodal}. The strictly safe answer: x = 5 (most clearly preserves a unique mode of 5). Acceptable: x = any value that isn't 3, 5, 7, 9 or 11 also keeps mode = 5 uniquely.
1.6 — {4, 7, 10, 13, 16, 19}
(a) Σ = 69, n = 6. Mean = 69 ÷ 6 = 11.5.
(b) Median: sorted already, n = 6, middle = 3rd & 4th = 10 and 13. Median = (10+13)÷2 = 11.5.
(c) Every value appears once → no mode.
(d) Range = 19 − 4 = 15.
Mean (11.5) and Median (11.5) describe the CENTRE. Range (15) describes the SPREAD. Mode would also be a centre measure, but here there isn't one.
2 — Find the mistake
(a) The mistake is on Line 5.
(b) The student confused the FREQUENCY (3 — how many times the value appeared) with the MODE (the VALUE that appeared most). The highest frequency is 3, but that count belongs to the value 7.
(c) Correct conclusion: Mode = 7 (the value 7 has the highest frequency of 3).
3 — Class stats report (sample answer)
(A) Favourite subject (categorical). Values from 10 students: Maths, Maths, English, PE, Science, Maths, English, PE, Maths, History.
(i)+(ii) Use the mode — only mode works for categories.
(iii) Mode = Maths (4 votes). Range is not meaningful for categorical data.
(B) Number of siblings (discrete numerical). Values: 0, 1, 2, 1, 0, 1, 3, 1, 2, 0.
(i)+(ii) Use the mode or median — small whole-number range, the mean (1.1) is awkward (nobody has 1.1 siblings).
(iii) Mode = 1 sibling. Range = 3 − 0 = 3.
(C) Height in cm (continuous numerical). Values: 142, 148, 150, 152, 154, 155, 158, 160, 165, 172.
(i)+(ii) Use the mean — continuous, roughly symmetric, no extreme outliers.
(iii) Σ = 1556. Mean = 1556 ÷ 10 = 155.6 cm. Range = 172 − 142 = 30 cm.
Summary sentence: A typical Year 7 student in our class likes Maths most, has 1 sibling, and is about 155.6 cm tall.
Marking: 1 mark for variable + 10 values, 1 for correct measure choice with reason, 1 for calculations, 1 for summary sentence.