Mathematics • Year 7 • Unit 4 • Lesson 10
The Mode and Range
Build fluency with two simple but essential tools. Mode = the VALUE that appears most often (one mode, two = bimodal, all-equal = no mode). Range = Maximum − Minimum (a measure of SPREAD, not centre). Then choose between mean, median and mode for any scenario.
1. I do — fully worked example
Read every line. Each step shows the question to ask and the reason for the answer.
Problem. Find the mode(s) and range of: 5, 8, 3, 8, 7, 5, 8, 3, 5.
Step 1 — Tally each value's frequency.
3 → 2 times 5 → 3 times 7 → 1 time 8 → 3 times
Reason: Go through the list value by value. Every value must be counted exactly once.
Step 2 — Find the highest frequency.
Highest count = 3, shared by 5 and 8 (a tie).
Reason: when two or more values tie for the highest frequency, the dataset is "bimodal" or "multimodal".
Step 3 — Report the mode(s) as VALUES (not counts).
Modes = 5 AND 8. This dataset is bimodal.
Reason: the mode is the data VALUE, not the count of times it appears. Mode ≠ 3.
Step 4 — Calculate the range = Max − Min.
Max = 8, Min = 3. Range = 8 − 3 = 5.
Reason: range tells us the SPREAD — the data spans 5 units.
Answer: Modes = 5 and 8 (bimodal). Range = 5.
2. We do — fill in the missing steps
Find the mode and range from this frequency table. Fill in each blank. 5 marks
Given frequency table (Score x out of 5):
Score (x): 1 2 3 4 5
Freq (f): 2 5 9 7 1
Step 1 — Find the largest frequency in the f-column: Largest f = ___.
Step 2 — Look LEFT to find the corresponding value: Mode = _____ (the SCORE that occurred most often).
Step 3 — Identify the minimum and maximum SCORES (x), not frequencies: Min x = ___, Max x = ___.
Step 4 — Calculate the range: Range = ___ − ___ = ___.
Step 5 — Interpret in context (one sentence): "The most popular score was _____, and the spread of scores was ___ marks."
3. You do — independent practice
Eight graduated problems. Show working — final-answer-only earns half marks.
Foundation — find mode and range
3.1 Find the mode of: 2, 5, 7, 2, 9, 2, 4. 1 mark
3.2 Find the range of: 12, 5, 18, 7, 22, 3, 15. 1 mark
3.3 Find the mode AND range of: 6, 8, 6, 10, 6, 4, 8. 2 marks
3.4 A dataset has no mode. Describe in one sentence what that means about the values. 1 mark
Standard — bimodal and choosing the measure
3.5 Find ALL modes of: 3, 5, 3, 7, 5, 9, 5, 3, 8. State whether it is unimodal, bimodal or multimodal. 2 marks
3.6 For each variable, state whether the MEAN, MEDIAN or MODE is the best measure of centre, and a one-line reason: (a) favourite school subject, (b) Year 7 student heights in cm, (c) annual incomes in a suburb with a few mansions. 3 marks
Extension — push your thinking
3.7 Scores: 50, 52, 54, 56, 100. (a) Calculate the range. (b) Now remove the value 100 and recalculate the range. (c) By how many marks did the outlier inflate the range, and what does this tell you about range as a measure of spread? 3 marks
3.8 A shoe shop's recent sales include sizes: 7, 8, 7, 9, 7, 8, 10, 7, 8. Use the data to answer: (a) what is the mode? (b) which measure (mean, median or mode) should the shop manager rely on when deciding which sizes to RE-STOCK most? Explain in one sentence. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Frequency table (We do)
Step 1: largest f = 9.
Step 2: Mode = 3 (the score with frequency 9).
Step 3: Min x = 1, Max x = 5.
Step 4: Range = 5 − 1 = 4.
Step 5: "The most popular score was 3, and the spread of scores was 4 marks."
3.1 — Mode
2 appears 3 times; all other values appear once. Mode = 2.
3.2 — Range
Max = 22, Min = 3. Range = 22 − 3 = 19.
3.3 — Mode and range of {6,8,6,10,6,4,8}
Tally: 4→1, 6→3, 8→2, 10→1. Mode = 6. Range = 10 − 4 = 6.
3.4 — No mode
"No mode" means every value in the dataset appears the same number of times — no single value (or set of values) is the most frequent. Example: {2, 4, 6, 8} — each appears once.
3.5 — All modes of {3,5,3,7,5,9,5,3,8}
Tally: 3→3, 5→3, 7→1, 8→1, 9→1. Highest count = 3, tied by 3 and 5. Modes = 3 and 5 (bimodal).
3.6 — Best measure of centre
(a) Favourite subject — Mode. Categorical data; mean and median are impossible.
(b) Student heights — Mean. Continuous numerical data, roughly symmetric, no extreme outliers.
(c) Annual incomes with mansions — Median. Skewed by high-income outliers; the median is resistant.
3.7 — Outlier inflates range
(a) Range = 100 − 50 = 50.
(b) Without 100: Max = 56, Min = 50. Range = 56 − 50 = 6.
(c) The outlier inflated the range by 44 marks. The range uses only the two extremes, so even one outlier dominates it — the range is NOT robust to outliers, and should always be reported with a note when outliers are present.
3.8 — Shoe shop re-stocking
(a) Tally: 7→4, 8→3, 9→1, 10→1. Mode = size 7.
(b) The mode — re-stocking is about meeting the most demand for the most popular size, and "mode" is exactly the measure for "most popular". Mean would suggest a fractional size (about 8) that doesn't even exist; median (8) would also under-stock the bestseller (size 7).