Mathematics • Year 10 • Unit 4 • Lesson 5

Histograms and Grouped Data in the Real World

Apply histograms and grouped frequency tables to real Australian data: city rainfall totals, exam mark distributions, salary surveys, and Sydney commute times. Practise identifying the modal class, describing skew, and explaining why a histogram (not a bar chart) is the right tool.

Apply · Real-World Maths

1. Word problems

Each problem uses Lesson 5's ideas: grouped tables, histograms, modal class, and shape. Show your working — a final answer with no working only earns half marks.

1.1 — Year 10 exam marks. A Year 10 cohort of 30 students sits a maths exam (out of 100):
58, 67, 72, 49, 88, 73, 91, 65, 70, 82, 76, 60, 84, 79, 95, 55, 78, 81, 63, 71, 69, 87, 74, 80, 66, 85, 77, 92, 68, 75.

(a) Build a grouped frequency table using class width 10 starting at 40-49.
(b) Identify the modal class.
(c) Describe the shape (symmetric / positively skewed / negatively skewed). 3 marks

Stuck? Classes will be 40-49, 50-59, 60-69, 70-79, 80-89, 90-99. Tally each value into exactly one class.

1.2 — Sydney monthly rainfall. Sydney's recorded monthly rainfall totals (mm) for 24 months:
45, 78, 112, 32, 156, 88, 23, 67, 134, 91, 56, 18, 148, 72, 39, 105, 167, 84, 28, 119, 95, 41, 73, 102.

(a) Build a grouped frequency table using class width 30 starting at 0-29.
(b) Identify the modal class.
(c) Describe the shape. 3 marks

Stuck? Classes will be 0-29, 30-59, 60-89, 90-119, 120-149, 150-179. Width is 30 — check that each class spans exactly 30 integers.

1.3 — Salary survey. A survey of 40 graduates records their starting salaries (in thousands of dollars):
Class 50-59: 4, 60-69: 11, 70-79: 14, 80-89: 6, 90-99: 3, 100-109: 1, 110-119: 1.

(a) State the total number of graduates (sum-check).
(b) Identify the modal class.
(c) Describe the shape. Explain what the shape tells you about graduate salaries in plain English. 3 marks

Stuck? Long tail to the right = positive skew. The tail represents the small minority of grads on much higher salaries.

1.4 — Commute times in minutes. A survey of 50 Sydney commuters records one-way travel time (min):
Class 0-14: 4, 15-29: 12, 30-44: 18, 45-59: 9, 60-74: 5, 75-89: 2.

(a) Sum-check the total.
(b) Identify the modal class.
(c) Estimate the percentage of commuters whose journey takes 45 minutes or more. 3 marks

1.5 — Wrong display. A reporter publishes two charts about a Year 10 cohort. Chart A shows favourite subject (Maths, English, Science...) on a histogram with bars touching. Chart B shows test scores out of 100 on a bar chart with gaps between bars.

(a) For each chart, name the variable's data type (from Lesson 1).
(b) Explain why each chart is using the wrong display, and state the correct display in each case. 3 marks

2. Explain your thinking

This question is about communication, not just plotting. Use full sentences. 4 marks

2.1 A news article reports that "the average Australian household income is $130,000 per year" but a friend points out that "most Australians earn nowhere near that". Using Lesson 5's shape vocabulary, write a four-sentence reply that (i) names the skew direction likely present in the income distribution, (ii) draws a quick sketch of the histogram shape (described in words), (iii) explains why the mean exceeds the median in this case, and (iv) suggests which of mean or median is a better summary "typical" income for a news article.

Stuck? Income is the classic positively skewed example — most values clustered low, with a long tail of high earners that drags the mean up but leaves the median near the cluster.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Year 10 exam marks (n = 30)

(a) Frequencies — 40-49: 1 (49). 50-59: 2 (55, 58). 60-69: 6 (60, 63, 65, 66, 67, 68, 69) → let me recount: 60, 63, 65, 66, 67, 68, 69 = 7. 70-79: 9 (70, 71, 72, 73, 74, 75, 76, 77, 78, 79) → that's 10 values; checking original: 72, 73, 70, 76, 79, 78, 71, 74, 77, 75 = 10. 80-89: 7 (88, 82, 84, 81, 87, 80, 85) = 7. 90-99: 3 (91, 95, 92) = 3.
Sum: 1 + 2 + 7 + 10 + 7 + 3 = 30 ✓.
(b) Modal class = 70-79.
(c) Approximately symmetric with a very slight negative skew (a single mark at 49 forms the small left tail; the right tail to the 90s is comparable in length).

1.2 — Sydney rainfall (n = 24)

(a) Frequencies — 0-29: 3 (23, 18, 28). 30-59: 4 (45, 32, 56, 39, 41) = 5. 60-89: 6 (78, 88, 67, 72, 84, 73) = 6. 90-119: 5 (112, 91, 105, 119, 95, 102) = 6. 120-149: 2 (134, 148). 150-179: 2 (156, 167).
Recount sum: 3 + 5 + 6 + 6 + 2 + 2 = 24 ✓.
(b) Modal classes = 60-89 and 90-119 (both at frequency 6 — bimodal).
(c) Approximately symmetric with a slight positive skew (the lighter-rainfall and heavier-rainfall tails are both present, but the very-heavy 150-179 class extends further from the centre than the 0-29 class).

1.3 — Graduate salaries

(a) Total = 4 + 11 + 14 + 6 + 3 + 1 + 1 = 40 ✓.
(b) Modal class = 70-79 ($70,000-$79,999).
(c) Shape = positively skewed. Most graduates earn $60-90k, but a small tail of grads earns $100k+ — those few high earners pull the mean upward, while most start in the $60-80k range.

1.4 — Commute times (n = 50)

(a) 4 + 12 + 18 + 9 + 5 + 2 = 50 ✓.
(b) Modal class = 30-44 minutes.
(c) "45 min or more" = 9 + 5 + 2 = 16. Percentage = 16/50 × 100 = 32%.

1.5 — Wrong displays

(a) Chart A: favourite subject is categorical. Chart B: test scores are continuous numerical.
(b) Chart A is wrong because a histogram with bars touching falsely suggests subjects sit on a continuous number line; subjects are categories and should be on a bar chart with gaps. Chart B is wrong because gapped bars falsely suggest test scores are separate categories; they form a continuous scale and should be on a histogram with bars touching. This is exactly the HSC trap warned about in Lesson 1.

2.1 — Income distribution (sample response)

(i) Australian household income is positively skewed: most households cluster around the middle-to-lower end, with a long right tail. (ii) Sketched in words: the histogram has its tallest bars near the modal class (around $60k-$100k), then bars drop off sharply on the right but never quite reach zero, stretching out to a handful of very-high-income households over $500k or even $1M+. (iii) The mean exceeds the median because those few very high values are added in to the total when computing the mean, dragging the average upward, while the median (the middle household) sits near the typical-income cluster. (iv) For a news article, the median is a better summary of "typical" income because it is not distorted by the few extremely high earners.

Marking: 1 mark for naming positive skew, 1 for describing the shape clearly, 1 for explaining why mean > median, 1 for recommending the median (with reason).