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.