Mathematics • Year 10 • Unit 4 • Lesson 5
Histograms and Grouped Data — Mixed Challenge
Pull together every idea from Lesson 5 — and the rest of Unit 4: class intervals (no gaps / no overlap / equal width), histogram construction, the modal class, skew direction, and the misconception that "taller bar = wider class". Spot a Year 10 mistake, then design your own grouped data set that produces a target shape.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 5 (and revisits earlier Unit 4 lessons). Decide whether the question is about building a table, drawing a histogram, identifying skew, or fixing a class-interval error before you start writing. 3 marks each
1.1 Build a grouped frequency table with class width 5 starting at 20-24 for: 23, 28, 31, 25, 22, 36, 29, 34, 27, 33, 26, 38, 30, 24, 35.
1.2 A histogram shows class intervals 10-19, 20-29, 30-39, 40-49 with frequencies 6, 14, 10, 3. (a) Identify the modal class. (b) Describe the shape in one word. (c) State whether the mean would be larger or smaller than the median.
1.3 A frequency table has classes 0-9, 10-19, 20-29, 30-39, 40-49 with frequencies 2, 4, ?, 4, 2 and total frequency 18. Find the missing frequency and describe the shape.
1.4 A student claims that the class width of intervals 0-9, 10-19, 20-29 is 9 (because 9 − 0 = 9). Show the correct calculation and explain in one sentence why "9" is the wrong answer.
1.5 Lesson 5 says positively skewed data has the peak on the LEFT and the long tail on the RIGHT. Give one real Australian example (other than household income) where you'd expect a positively skewed distribution. Sketch the histogram in words and explain why the skew direction makes sense.
1.6 A bar chart and a histogram look almost the same — but the difference is not just aesthetic. Using the Lesson 3 callout ("Histogram bars TOUCH because the horizontal axis represents continuous numerical data. Bar chart bars do not touch because the categories are separate"), explain in your own words when each one is appropriate, and give one example of each.
2. Find the mistake
Another Year 10 student has tried to build class intervals for a grouped frequency table and start a histogram. 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 — group data of heights between 150 cm and 189 cm using class width 10:
Line 1: Class intervals: 150-160, 160-170, 170-180, 180-190.
Line 2: Each class has width 10 (e.g. 160 − 150 = 10).
Line 3: Tally height 160 cm into the 160-170 class.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected class intervals (and explain how to handle 160 cm correctly).
Stuck? Lesson 2 misconception card warned about overlapping classes: where does 160 cm go — 150-160 or 160-170? The classes overlap.3. Open-ended challenge — design data for a target shape
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design two grouped frequency tables, each based on a real-world context of your choice, that satisfy all of the following:
- Table A's histogram is symmetric with at least 5 class intervals.
- Table B's histogram is positively skewed with at least 5 class intervals (peak on the left, long tail to the right).
- Both tables have a total frequency between 30 and 50.
- Both tables use equal class width, with no overlap and no gaps.
For each table:
(i) State the context (e.g. heights of students, response times in a survey).
(ii) Write the class intervals and frequencies.
(iii) Confirm the sum equals your stated total.
(iv) State the modal class.
(v) Sketch a quick rough histogram in words (e.g. "bars heights 3, 7, 12, 7, 3").
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Build grouped table (width 5)
Classes: 20-24, 25-29, 30-34, 35-39.
Frequencies: 20-24: 3 (23, 22, 24). 25-29: 5 (28, 25, 29, 27, 26). 30-34: 4 (31, 34, 33, 30). 35-39: 3 (36, 38, 35).
Sum-check: 3 + 5 + 4 + 3 = 15 ✓.
1.2 — Read histogram (6, 14, 10, 3)
(a) Modal class = 20-29.
(b) Positively skewed (peak on the left, tail to the right).
(c) The mean would be larger than the median, because the tail of higher values pulls the average up while the median stays near the cluster.
1.3 — Missing frequency
Sum: 2 + 4 + ? + 4 + 2 = 18, so 12 + ? = 18, and ? = 6. Frequencies 2, 4, 6, 4, 2 are symmetric (mirror image around the middle class 20-29).
1.4 — Class width of 0-9, 10-19, 20-29
Class width = 10 − 0 = 10 (or equivalently the next class starts at 10, so each class spans 10 integers). The student computed 9 − 0 = 9 by counting the largest value in the class, not the width to the next class. Lesson 5 callout: when classes are 0-9 the next class starts at 10, so the width is 10.
1.5 — Real positively-skewed example
Wait time at a Sydney GP clinic. Most patients wait under 20 minutes, but a small number of patients (with complex cases ahead of them) wait 60+ minutes. The histogram has tall bars at 0-9, 10-19, 20-29 and then bars trail off across 30-39, 40-49, 50-59, 60+. Skew is positive because most data sits low and a few extreme high values stretch the right tail. (Other valid examples: rainfall on a dry month, social-media post likes per video, fire-call response times.)
1.6 — Histogram vs bar chart
A histogram is for continuous numerical data: the horizontal axis is a number line, so the bars touch (no real "gap" between a height of 159.9 cm and 160.0 cm). Example: heights of 100 students. A bar chart is for categorical data: each category sits at a separate position with a visible gap, because there is no "between" two categories. Example: favourite sport (AFL, NRL, soccer, cricket).
2 — Find the mistake
(a) The mistake is on Line 1.
(b) The classes overlap at 160, 170, 180 — where does a height of exactly 160 cm belong, 150-160 or 160-170? Lesson 2 misconceptions card: class intervals must have no overlap and no gaps.
(c) Corrected intervals: 150-159, 160-169, 170-179, 180-189 (each is width 10 and uses inclusive integer boundaries with no overlap). A height of 160 cm now sits clearly in the 160-169 class.
3 — Open-ended challenge (sample solution)
Table A — symmetric. Context: heights (cm) of 36 Year 10 students.
Classes: 150-159, 160-169, 170-179, 180-189, 190-199.
Frequencies: 4, 9, 10, 9, 4.
Sum-check: 4 + 9 + 10 + 9 + 4 = 36 ✓. Modal class = 170-179.
Sketch (in words): bars rise 4 → 9 → 10 (peak) → 9 → 4. Symmetric mirror image.
Table B — positively skewed. Context: number of TikTok videos watched per day by 40 students.
Classes (width 20): 0-19, 20-39, 40-59, 60-79, 80-99.
Frequencies: 16, 12, 7, 3, 2.
Sum-check: 16 + 12 + 7 + 3 + 2 = 40 ✓. Modal class = 0-19.
Sketch: bars 16 → 12 → 7 → 3 → 2 (peak on left, long right tail).
Marking: 2 marks per table (1 for correctly-shaped frequencies, 1 for a valid context with equal-width non-overlapping classes and a confirmed sum). Any valid design earns full marks.