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Lesson 56 of 75
Introduction

Histogram reconstruction

Objective. Recover frequencies from density and class width.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Histogram reconstruction recovers missing frequencies or class widths using the rule that frequency density = frequency ÷ class width, so bar height represents density, not frequency itself. Given the area of a bar (frequency density × class width), you can rearrange to find frequency = frequency density × class width. If one bar's frequency and class width are known, you can find the scale linking area to frequency, then apply it to other bars. For a class of width 10 with frequency density 3, frequency = 3 × 10 = 30. Always read frequency density carefully from the vertical axis before multiplying.

Worked example

A histogram bar for the class 20–35 has frequency density 4. Find the frequency for this class.

20354
Bar chart of the values 20, 35, 4
  1. 1Class width = 35 − 20 = 15
  2. 2Use frequency = frequency density × class width
  3. 3Frequency = 4 × 15
  4. 4Frequency = 60

Answer: 60

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Common mistake. A very common error is reading bar height as frequency directly, ignoring unequal class widths, which badly misrepresents groups with wider intervals. Always multiply height by width to get frequency.

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