Mathematics • Year 10 • Unit 4 • Lesson 5
Histograms and Grouped Data — Skill Drill
Build fluency with Lesson 5's three building blocks: class intervals (no gaps, no overlap, equal width), the histogram (bars touch because the horizontal axis is continuous), and the shape vocabulary (symmetric, positively skewed, negatively skewed). Practise building a grouped frequency table and drawing the matching histogram.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. The heights (cm) of 20 Year 10 students are recorded:
152, 168, 159, 164, 171, 158, 162, 175, 169, 161, 167, 153, 178, 165, 160, 172, 156, 170, 163, 166.
Build a grouped frequency table with class intervals of width 5 starting at 150-154, then describe the shape.
Step 1 — List the class intervals.
150-154, 155-159, 160-164, 165-169, 170-174, 175-179.
Reason: equal width (5), no overlap, no gaps — the three rules from Lesson 2/5.
Step 2 — Tally each value into one (and only one) class.
150-154: | 152, 153 → 2 155-159: | 156, 158, 159 → 3 160-164: | 160, 161, 162, 163, 164 → 5
165-169: | 165, 166, 167, 168, 169 → 5 170-174: | 170, 171, 172 → 3 175-179: | 175, 178 → 2
Reason: integer boundaries 150-154 / 155-159 (no overlap because boundaries are inclusive whole numbers).
Step 3 — Sum-check.
2 + 3 + 5 + 5 + 3 + 2 = 20 ✓ (matches 20 students).
Reason: total frequency = total data values.
Step 4 — Describe the shape.
Frequencies: 2, 3, 5, 5, 3, 2 — mirror-image around the middle.
Reason: a roughly symmetric distribution has left and right sides that match.
Answer: Grouped frequencies {150-154: 2, 155-159: 3, 160-164: 5, 165-169: 5, 170-174: 3, 175-179: 2}. Shape: symmetric (peak in the middle, equal tails).
2. We do — fill in the missing parts
Same idea as Section 1, but the working is faded. Fill in each blank. 4 marks
Problem. Test scores out of 100 for 25 students:
72, 81, 67, 90, 75, 84, 60, 73, 88, 79, 65, 92, 70, 77, 83, 68, 95, 80, 74, 86, 71, 78, 82, 76, 64.
Build a grouped frequency table with class intervals of width 10 starting at 60-69, then identify the modal class and shape.
Step 1 — Class intervals.
60-69, 70-79, _________, _________ (continue if needed).
Step 2 — Tally each class.
60-69: ____ 70-79: ____ 80-89: ____ 90-99: ____
Step 3 — Sum-check.
Sum = ____ (should be 25).
Step 4 — Identify the modal class.
Modal class = ____________ (the interval with the highest frequency).
Step 5 — Shape. The distribution is __________________________ (symmetric / positively skewed / negatively skewed).
3. You do — independent practice
Show your working under each problem. The first four are foundation. The middle two are standard. The last two are extension (use shape vocabulary and the lesson's misconception).
Foundation — read off and compute
3.1 For grouped classes 0-9, 10-19, 20-29, the class width is ____. 1 mark
3.2 A histogram has bars touching with class intervals 5-9, 10-14, 15-19. What is wrong (or right) about saying the class width is 4? 1 mark
3.3 A grouped frequency table has frequencies 4, 9, 12, 7, 3. State (a) total frequency, (b) the modal class if classes are 0-9, 10-19, 20-29, 30-39, 40-49. 1 mark
3.4 A distribution with a long tail to the right is called: (A) symmetric (B) negatively skewed (C) positively skewed (D) bimodal. 1 mark
Standard — build a grouped frequency table
3.5 The number of hours 18 students spent on homework last week:
2, 4, 6, 3, 8, 11, 9, 5, 7, 12, 14, 6, 10, 4, 8, 15, 3, 9.
Build a grouped frequency table with class width 3 starting at 0-2. State the modal class. 2 marks
3.6 A distribution has these grouped frequencies: 0-9: 2, 10-19: 5, 20-29: 9, 30-39: 11, 40-49: 8, 50-59: 4, 60-69: 1.
Sketch the histogram (label both axes), state the modal class, and describe the shape in one word. 2 marks
Extension — shape and misconceptions
3.7 A distribution of household incomes shows most households earning under $80,000 and a long tail of high earners extending to $500,000. State the skew direction, give one sentence explaining what the long tail means in plain English, and explain whether the mean or median would be larger. 2 marks
3.8 A student says: "A taller bar on a histogram means the class is wider." Using Lesson 5's misconceptions card, explain why this is wrong. State in one sentence what bar height actually represents. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (test scores)
Step 1: 60-69, 70-79, 80-89, 90-99.
Step 2: 60-69: 4 (60, 64, 65, 67, 68) → actually 5; let me recount. Values 60-69: 60, 64, 65, 67, 68 = 5. 70-79: 70, 71, 72, 73, 74, 75, 76, 77, 78, 79 = 10. 80-89: 80, 81, 82, 83, 84, 86, 88 = 7. 90-99: 90, 92, 95 = 3.
Step 3: Sum = 5 + 10 + 7 + 3 = 25 ✓.
Step 4: Modal class = 70-79 (frequency 10).
Step 5: Slightly positively skewed (peak at 70-79, longer tail to 90s than to 60s — actually here, the tail to the right is shorter; treat this as roughly symmetric with a slight pull toward the lower end → mildly negatively skewed). Accept "approximately symmetric" or "very slightly negatively skewed".
3.1 — Class width 0-9, 10-19, 20-29
Class width = 10 (from 0-9 the next class starts at 10, so the width spans 10 integers).
3.2 — Width of 5-9
The class width is 5, not 4. The class 5-9 contains integers 5, 6, 7, 8, 9 — that is 5 integers; equivalently, the next class starts at 10, so the width is 10 − 5 = 5.
3.3 — Total and modal class
(a) Total = 4 + 9 + 12 + 7 + 3 = 35.
(b) Modal class = 20-29 (highest frequency 12).
3.4 — Long tail to the right
(C) Positively skewed — peak on the left, long tail extending to the right.
3.5 — Homework hours grouped
Classes (width 3): 0-2, 3-5, 6-8, 9-11, 12-14, 15-17.
Frequencies: 0-2: 1 (2) 3-5: 4 (3, 4, 4, 5) 6-8: 5 (6, 6, 7, 8, 8) 9-11: 4 (9, 9, 10, 11) 12-14: 3 (12, 14, 15→actually 15 goes in 15-17). Recount: 12-14 = 2 (12, 14); 15-17 = 2 (15)→1. Let me redo carefully. Values: 2, 4, 6, 3, 8, 11, 9, 5, 7, 12, 14, 6, 10, 4, 8, 15, 3, 9.
0-2: 2 → 1. 3-5: 3, 4, 5, 4, 3 → 5. 6-8: 6, 8, 7, 6, 8 → 5. 9-11: 11, 9, 10, 9 → 4. 12-14: 12, 14 → 2. 15-17: 15 → 1. Sum = 1+5+5+4+2+1 = 18 ✓.
Modal classes: 3-5 and 6-8 (both with frequency 5 — bimodal).
3.6 — Sketch + shape
Histogram bars (heights): 2, 5, 9, 11, 8, 4, 1 above classes 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69. Bars touch. Vertical axis = Frequency. Horizontal axis = value.
Modal class = 30-39. Shape: approximately symmetric (peak in the middle, similar tails).
3.7 — Income distribution
Positively skewed. The long right tail means a small number of households earn extremely high incomes (e.g. CEO salaries) far above the typical household income. The mean would be larger than the median, because the few extreme high earners drag the mean upward while the median (middle value) stays close to the typical household.
3.8 — Taller bar ≠ wider class
Lesson 5 misconceptions card directly says this is wrong. Bar height = frequency (how many data values are in that class), while bar width = class width (the size of the interval, e.g. 10). A taller bar means more data values landed in that class, not that the class is wider.