Mathematics • Year 10 • Unit 4 • Lesson 3
Frequency Tables and Dot Plots — Mixed Challenge
Pull together every idea from Lesson 3: build frequency tables, read dot plots, identify clusters, gaps and outliers, and decide when a dot plot vs a bar chart fits the data. Spot a classmate's mistake, then design your own dot plot to match a given description.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 3. Decide whether to build a table, read a plot, or identify a shape feature before you start writing. 3 marks each
1.1 Build a frequency table for the values: 3, 4, 3, 5, 4, 3, 6, 4, 5, 3, 4, 3. State (a) total count, (b) mode, (c) the value of any clear cluster.
1.2 A dot plot shows: 10 → 3, 11 → 4, 12 → 6, 13 → 4, 14 → 2, 15 → 0, 16 → 1. Identify the mode, any gap, and any outlier.
1.3 A frequency table has frequencies 3, 8, ?, 4, 2, and the total is 24. Find the missing frequency.
1.4 A dot plot shows the number of laps swum by 18 students: 1 → 2, 2 → 4, 3 → 6, 4 → 4, 5 → 2. (a) Describe the shape in one word. (b) State the mode. (c) If one extra student is added who swam 20 laps, describe the effect on the shape.
1.5 A teacher claims their two data sets must be different because "their modes are different (5 and 7)". Construct two short example data sets, both with 8 values, both ranging from 1 to 10, where the modes differ but a quick look at the dot plot shows the rest of the shapes are very similar. Briefly explain why mode alone is not enough to tell two data sets apart.
1.6 A student wants to show "favourite colour for 30 students" on a dot plot above a number line. Explain in one or two sentences why a dot plot is the wrong choice here, and recommend a better display from Lesson 3.
2. Find the mistake
Another Year 10 student has tried to build a frequency table for a small data set and to use it. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — data: 4, 5, 4, 6, 4, 5, 7, 4, 5, 6 (10 values).
Line 1: Frequency table: {4: 4, 5: 3, 6: 2, 7: 1}.
Line 2: Sum-check: 4 + 3 + 2 + 1 = 11 ✓ (matches 10 values).
Line 3: Mode = 4 (highest frequency).
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including a corrected sum-check.
Stuck? 4 + 3 + 2 + 1 = 10, not 11. The MCQ rule from Lesson 3 says frequencies must sum to the total number of data values.3. Open-ended challenge — design the data
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design a data set of exactly 20 whole-number values between 0 and 10 (inclusive) that satisfies all four of the following:
- the mode is 4,
- there is a clear cluster between values 3 and 5,
- there is a gap at exactly one value between 0 and 10,
- there is one outlier (a single value far from the main cluster).
For your design:
(i) Write the 20 values.
(ii) Build the frequency table and confirm the sum-check equals 20.
(iii) Sketch a quick dot plot (use "x" characters above a number line 0 1 2 3 4 5 6 7 8 9 10).
(iv) Identify the cluster, the gap and the outlier explicitly.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Simple frequency table
{3: 5, 4: 4, 5: 2, 6: 1}. (a) Total = 12. (b) Mode = 3. (c) Cluster around values 3-4 (the bulk of the data).
1.2 — Read the dot plot
Mode = 12. Gap at 15 (no dots). Outlier = 16 (single dot far from the main cluster of 10-14).
1.3 — Missing frequency
Sum: 3 + 8 + ? + 4 + 2 = 24, so 17 + ? = 24, and ? = 7.
1.4 — Swim laps
(a) Symmetric (peak in the middle, mirror sides).
(b) Mode = 3 laps.
(c) Adding one student at 20 laps would create an outlier far above the cluster and add a large gap between values 5 and 20, making the distribution strongly right-skewed.
1.5 — Same shape, different mode
Sample set A: 1, 5, 5, 5, 5, 6, 7, 9 (mode 5).
Sample set B: 1, 5, 6, 7, 7, 7, 7, 9 (mode 7).
Both sets are roughly symmetric with one peak, both range from 1 to 9, and both have 8 values. Mode alone cannot distinguish them because two distributions can have the same overall shape while the most-frequent value sits in different places. Use the mode together with the mean, median, and shape description.
1.6 — Favourite colour on a dot plot
Colour is categorical data and a dot plot is for numerical data (above a number line). Colours have no numerical scale, so they cannot be placed on a number line. A better display from Lesson 3 (combined with Lesson 1) is a frequency table + bar chart.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The arithmetic is wrong: 4 + 3 + 2 + 1 = 10, not 11. Lesson 3's MCQ rule says frequencies must sum to the total number of data values — they do, but the student's stated sum is incorrect.
(c) Corrected: Sum-check = 4 + 3 + 2 + 1 = 10 ✓ (matches the 10 data values). The mode (4) is still correct.
3 — Open-ended challenge (sample solution)
Sample 20 values: 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 8, 8, 8, 10.
(ii) Frequency table: {2: 1, 3: 3, 4: 6, 5: 3, 6: 3, 7: 0, 8: 3, 9: 0, 10: 1}. Sum: 1 + 3 + 6 + 3 + 3 + 0 + 3 + 0 + 1 = 20 ✓.
(iii) Sketch:
x
x x
x x x
x x x x
x x x x x
x x x x x x x
─── ─── ─── ─── ─── ─── ─── ─── ─── ─── ───
0 1 2 3 4 5 6 7 8 9 10
(iv) Cluster: values 3-5 (the densest region with mode 4). Gap: value 7 (and value 9) with no dots. Outlier: value 10 (single dot far above the main cluster).
Marking: 1 mark for mode 4, 1 for valid cluster around 3-5, 1 for a gap somewhere between 0 and 10, 1 for an outlier with a sum-check that equals 20. Any valid design earns full marks.