Mathematics • Year 8 • Unit 4 • Lesson 7
Histograms
Build fluency identifying histograms, finding modal class, estimating the mean from grouped data using midpoints, and describing distribution shape.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why we do it, not just what we do.
Problem. Heights of 30 students grouped: 140–<150 (f=3), 150–<160 (f=8), 160–<170 (f=12), 170–<180 (f=6), 180–<190 (f=1). Find the modal class and estimate the mean.
Step 1 — Identify the modal class (highest frequency).
Highest f = 12 → modal class = 160–<170.
Reason: the modal class is the interval with the tallest bar (highest count).
Step 2 — Find each class midpoint.
Midpoints: 145, 155, 165, 175, 185.
Reason: midpoint = (lower + upper) ÷ 2. We assume every value in the class sits at its midpoint.
Step 3 — Build the f × midpoint column and sum.
3×145 + 8×155 + 12×165 + 6×175 + 1×185 = 435 + 1240 + 1980 + 1050 + 185 = 4890
Step 4 — Divide by total n.
Estimated mean = 4890 ÷ 30 = 163 cm.
Answer: Modal class = 160–<170. Estimated mean = 163 cm.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Ages at a community centre: 10–<20 (f=5), 20–<30 (f=12), 30–<40 (f=18), 40–<50 (f=10), 50–<60 (f=5). Find the modal class and estimate the mean.
Step 1 — Modal class:
Highest f = ______ → modal class = ______–<______.
Step 2 — Midpoints:
______, ______, ______, ______, ______.
Step 3 — Build f × midpoint:
5×15 + 12×25 + 18×35 + 10×45 + 5×55 = ______ + ______ + ______ + ______ + ______ = ______
Step 4 — Divide by n:
Total n = ______ . Estimated mean = ______ ÷ ______ = ______ years.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — quick recall
3.1 State the KEY visual difference between a histogram and a bar chart. 1 mark
3.2 A frequency table shows: 10–<15 (f=4), 15–<20 (f=7), 20–<25 (f=11), 25–<30 (f=14), 30–<35 (f=9). What is the modal class? 1 mark
3.3 A histogram bar covers 50–<60 with height 9 on the frequency axis. How many data values are in this interval? 1 mark
3.4 Find the midpoint of each class interval: (a) 60–<80, (b) 25–<35, (c) 100–<120. 1 mark
Standard — two-step problems
3.5 A grouped table: 0–<10 (f=2), 10–<20 (f=5), 20–<30 (f=9), 30–<40 (f=8), 40–<50 (f=4), 50–<60 (f=2). (a) Find n. (b) State the modal class. (c) Estimate the mean using midpoints. 2 marks
3.6 A histogram has 5 bars of equal width 10. Bar heights (frequencies): 3, 7, 10, 8, 2 for intervals starting at 0. (a) Build the grouped frequency table. (b) Add a cumulative frequency column. (c) State the modal class. 2 marks
Extension — describe shape
3.7 A histogram of salaries shows most employees earning $40k–$60k, with a few earning over $200k. Describe the shape using one of: symmetric / skewed left / skewed right. Explain in one sentence which way the tail points. 2 marks
3.8 A student draws bars with gaps between them on a histogram of student weight (kg). (a) Why is this wrong? (b) What does putting gaps in a histogram tell the reader? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (community centre ages)
Step 1: Highest f = 18 → modal class = 30–<40.
Step 2: Midpoints = 15, 25, 35, 45, 55.
Step 3: 5×15 + 12×25 + 18×35 + 10×45 + 5×55 = 75 + 300 + 630 + 450 + 275 = 1730.
Step 4: n = 5+12+18+10+5 = 50. Estimated mean = 1730 ÷ 50 = 34.6 years.
3.1 — Histogram vs bar chart
Histogram bars touch (no gaps) because the data is continuous; bar chart bars have gaps between them because the categories are separate.
3.2 — Modal class
Highest f = 14 → modal class = 25–<30.
3.3 — Bar height = frequency
9 data values (with equal-width intervals, bar height equals frequency directly).
3.4 — Midpoints
(a) (60+80)÷2 = 70. (b) (25+35)÷2 = 30. (c) (100+120)÷2 = 110.
3.5 — Grouped table summary
(a) n = 2+5+9+8+4+2 = 30.
(b) Modal class = 20–<30 (f = 9).
(c) Midpoints 5, 15, 25, 35, 45, 55. Σ(f × mid) = 2×5 + 5×15 + 9×25 + 8×35 + 4×45 + 2×55 = 10 + 75 + 225 + 280 + 180 + 110 = 880. Mean ≈ 880 ÷ 30 ≈ 29.3.
3.6 — Reconstruct from histogram
(a) Table: 0–<10: 3; 10–<20: 7; 20–<30: 10; 30–<40: 8; 40–<50: 2.
(b) Cumulative: 3, 10, 20, 28, 30.
(c) Modal class = 20–<30 (f = 10).
3.7 — Salary shape
Skewed right — most data is bunched at the low end ($40k–$60k) and a few very high values pull the tail to the right.
3.8 — Gaps in a histogram
(a) Weight is continuous data — values flow continuously through the number line. Bars should touch to reflect this. (b) Gaps would tell the reader the data is categorical, which is incorrect — they would mistake the histogram for a bar chart.