Mathematics • Year 10 • Unit 4 • Lesson 3
Frequency Tables and Dot Plots in the Real World
Apply frequency tables and dot plots to real Australian contexts: PE shuttle-run scores, canteen orders, classroom test marks and a coach comparing two teams. Read off the mode, spot clusters/gaps/outliers, and explain what the data is saying.
1. Word problems
Each problem uses Lesson 3's ideas: build a frequency table, read a dot plot, identify mode/cluster/gap/outlier. Show your working — a final answer with no working only earns half marks.
1.1 — PE shuttle-run levels. In a Year 10 PE class, 22 students complete the beep test and record the level they reach:
6, 7, 8, 7, 9, 6, 7, 8, 10, 7, 8, 7, 6, 9, 8, 7, 6, 7, 8, 7, 11, 7.
(a) Build a frequency table.
(b) State the mode.
(c) How many students reached at least level 8? 3 marks
1.2 — Canteen orders. A canteen records the number of items bought per student in one lunch break. The dot plot (dots above each value) shows:
0 → 2, 1 → 11, 2 → 14, 3 → 6, 4 → 1, 5 → 0, 6 → 1.
(a) Build the frequency table.
(b) State the mode.
(c) Identify the outlier and explain why it qualifies as one. 3 marks
1.3 — Spelling test scores. Mr Patel marks 30 spelling tests out of 10:
8, 9, 7, 10, 8, 6, 9, 8, 7, 10, 8, 9, 7, 8, 10, 9, 8, 6, 9, 8, 7, 9, 10, 8, 8, 9, 7, 10, 9, 8.
(a) Build the frequency table.
(b) State the mode.
(c) What percentage of students scored 9 or 10? 3 marks
1.4 — Two football teams. A coach builds dot plots of goals scored per game across a 12-game season for two teams.
Tigers (per game): 1 → 3, 2 → 4, 3 → 3, 4 → 1, 5 → 0, 6 → 1.
Sharks (per game): 0 → 2, 1 → 2, 2 → 3, 3 → 3, 4 → 2.
(a) State the mode for each team.
(b) Which team has an outlier? Identify it and say why.
(c) Briefly describe one difference in shape between the two distributions. 3 marks
1.5 — Choosing a display. A teacher has two small data sets to display in class:
(A) Favourite breakfast cereal among 28 students (Weet-Bix, Cornflakes, Oats, Other).
(B) Number of pencils each of 20 students has on their desk.
(a) Which data set is best shown as a frequency table + bar chart, and which as a frequency table + dot plot?
(b) Justify each choice in one sentence, referencing Lesson 3's reasons for using a dot plot. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A Year 10 student looks at a dot plot of "number of pets per student" in their class and says: "The tallest stack is at value 1, so the average number of pets is 1." Using Lesson 3's misconceptions card, explain in 3 to 5 sentences (i) what the tallest stack actually tells us, (ii) why this is not the same as the average (mean), (iii) what extra calculation the student would have to do to find the mean, and (iv) one situation where the mode and mean would be very different.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Beep test levels
(a) Frequencies: {6: 4, 7: 9, 8: 5, 9: 2, 10: 1, 11: 1}. Sum-check: 4 + 9 + 5 + 2 + 1 + 1 = 22 ✓.
(b) Mode = level 7.
(c) "At least 8" = 5 + 2 + 1 + 1 = 9 students.
1.2 — Canteen orders
(a) Frequencies: {0: 2, 1: 11, 2: 14, 3: 6, 4: 1, 5: 0, 6: 1}. Total = 35 students.
(b) Mode = 2 items.
(c) Outlier = 6 items. It sits far from the main cluster (1-3) with a clear gap at value 5, so the single dot at 6 is unusually high compared to the rest.
1.3 — Spelling test scores
(a) {6: 2, 7: 4, 8: 10, 9: 8, 10: 6}. Sum-check: 2 + 4 + 10 + 8 + 6 = 30 ✓.
(b) Mode = 8.
(c) Scoring 9 or 10 = 8 + 6 = 14 students. Percentage = 14 ÷ 30 × 100 = 46.7% (to 1 d.p.).
1.4 — Tigers vs Sharks
(a) Tigers mode = 2 goals; Sharks mode = 2 goals or 3 goals (bimodal).
(b) Tigers have an outlier: 6 goals in one game, with a gap at 5. The Sharks distribution sits between 0 and 4 with no gaps.
(c) The Tigers distribution is right-skewed (long tail to high scores with one outlier), while the Sharks distribution is roughly even/flat from 0 to 4.
1.5 — Choosing a display
(a) (A) Cereal preference → frequency table + bar chart (categorical data).
(B) Pencils per desk → frequency table + dot plot (small numerical data set).
(b) Cereal is categorical, so a bar chart with gapped bars is correct. Pencils per desk is small and numerical, which is exactly Lesson 3's stated use case for a dot plot ("most useful for small numerical data sets").
2.1 — Tallest stack ≠ mean (sample response)
(i) The tallest stack tells us the mode — the most-frequent value (here, 1 pet). (ii) That is not the same as the mean (average), because the mean depends on every value in the data set, not just on which value appears most often. (iii) To find the mean, the student would have to multiply each value by its frequency, add all those products, and divide by the total number of students: mean = Σ(value × frequency) ÷ Σ(frequency). (iv) The mode and mean can differ a lot when there is an outlier — e.g. if one student owns 8 pets, that single value pulls the mean above 1 even though the mode is still 1. Lesson 3 misconceptions card calls this out directly.
Marking: 1 mark per sub-question (i)-(iv).