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📖 Lesson 5 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Histograms and Grouped Data

Construct histograms for grouped data and interpret the shape of distributions including symmetry and skew.

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Core learning
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From the lesson
Worksheet

Worksheet

Use the worksheet to complete this lesson in your book or digitally.

Warm-up
Think First
+5 XP each

Q1 · If you measured the heights of 100 students, would you show every single height separately? What might work better?

Q2 · Why might grouping data into ranges like 150–160 cm and 160–170 cm help us see patterns more clearly?

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From the lesson
Intentions

Learning Intentions

Know

  • A histogram uses bars to show the frequency of data within equal intervals (classes). There are no gaps between bars.

Understand

  • How the shape of a histogram reveals whether data is symmetric, positively skewed or negatively skewed.

Can Do

  • Create frequency tables with class intervals and draw histograms from grouped data.
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From the lesson
Key Terms

Key Terms

Histogram — A display using bars of equal width to show frequencies of grouped data.
Class interval — A range of values used to group data, e.g. 10-19.
Frequency density — Frequency divided by class width (used when intervals are unequal).
Symmetric distribution — A distribution where the left and right sides are approximately mirror images.
Skew — Asymmetry in a distribution: positive skew has a tail to the right; negative skew to the left.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: A taller bar means the class interval is wider.

✓

Right: Bar height = frequency (or frequency density). Bar width = class width. A taller bar means more data in that interval, not a wider interval.

✗

Wrong: Positively skewed means the peak is on the right.

✓

Right: Positively skewed data has a longer tail to the RIGHT; the bulk of data is on the LEFT. The peak (mode) is on the left with a long right tail.

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From the lesson
Content

Histograms and Grouped Data

Work through the content, activities and worked examples below. Test your understanding with the questions in the Questions phase.

Remember To find the median: (1) Order the data, (2) Find the middle position using (n+1)/2, (3) If n is even, average the two middle values.
Exam Tip Always write data in ascending order before finding the median or mode. This prevents careless errors.
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From the lesson
Activity
✏ Activity 1 — Find Median and Mode

Find the median and mode for each data set:

  1. 3, 7, 8, 12, 15
  2. 5, 8, 8, 10, 12, 15
  3. 21, 23, 23, 25, 27, 27, 29
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From the lesson
Worked Example

Worked Example

Step-by-step
Find the median and mode of the following data set: 12, 15, 18, 21, 21, 24, 27, 30.
  1. 1
    Step 1: Order the data (already ordered): 12, 15, 18, 21, 21, 24, 27, 30.
  2. 2
    Step 2: n = 8 (even). Median position = (8+1)/2 = 4.5. Average the 4th and 5th values.
  3. 3
    Step 3: Median = (21 + 21) / 2 = 21.
  4. 4
    Step 4: Mode = 21 (appears twice, more than any other value).
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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First response. What new understanding do you have now?

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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From the lesson
Multiple Choice

Multiple Choice

Select the best answer for each question.

1 mark The key difference between a histogram and a bar chart is:

1 mark A distribution with a long tail to the right is called:

1 mark For grouped data with classes 0-9, 10-19, 20-29, the class width is:

1 mark In a symmetric distribution, the mean and median are:

1 mark Which shape indicates that most data is clustered at the lower end?

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks For the data set 5, 8, 12, 15, 18, 21, 24, 27, 30:
(a) Find the median.
(b) A value of 100 is added to the data set. Find the new median.
(c) Explain why the median is less affected by outliers than the mean.

2. 3 marks A data set has values: 4, 6, 6, 8, 8, 10, 12.
(a) Find the mode.
(b) Is this data set unimodal, bimodal or multimodal? Explain.

3. 2 marks A student says: "The median of 4, 6, 8, 10 is 7 because (4+10)/2 = 7." Explain the error and find the correct median.

Marking guidance: 1 mark each for MCQs. See mark allocations for each short answer question.

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