Histograms

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Notes

Frequency Density

**Frequency density** = frequency ÷ class width.
Used for grouped data, especially when class intervals are unequal.
Class width = upper bound − lower bound of an interval.
Frequency density measures how dense data is within its interval.
Always calculate frequency densities before drawing a histogram.

Drawing Histograms

Histograms are for **continuous data** grouped into class intervals.
The **area** of each bar represents the frequency, not the height.
Bars have no gaps between them (continuous scale).
Horizontal axis: class intervals; vertical axis: frequency density.
To draw a bar, calculate frequency density = frequency ÷ class width.
Height of $bar =$ frequency density; width = class width.
Often exam questions ask to complete a partially drawn histogram.

Interpreting Histograms

Frequency = frequency density × class width (area of bar).
The vertical axis may be labelled 'frequency density' or just 'frequency' if widths are equal.
To estimate frequency for part of an interval, find area of that part.
Estimates assume data is evenly distributed within the interval.
Histograms can compare distributions only if scales and intervals match.

Common Calculations

Given frequency and class width, find frequency density: $fd = f \div w$ .
Given frequency density and class width, find frequency: $f = fd \times w$ .
Given frequency and frequency density, find class width: $w = f \div fd$ .
Always show working for method marks in exams.

Worked Example: Frequency Density Table

Add columns for class width and frequency density.
Example: interval $20 \le s < 40$ , frequency 5, width 20 → $fd = 0.25$ .
Example: interval $40 \le s < 50$ , frequency 15, width 10 → $fd = 1.5$ .
Check calculations carefully to avoid errors.

Worked Example: Completing a Histogram

Missing bars: compute frequency density for each missing interval.
Draw bar from start to end of interval at correct height.
Example: interval $60 \le x < 70$ , frequency 8, width 10 → $fd = 0.8$ .
Example: interval $80 \le x < 100$ , frequency 2, width 20 → $fd = 0.1$ .

Worked Example: Interpreting to Find Frequency

Use histogram bar height (fd) and width to find frequency.
Example: bar height 0.6, width 15 → frequency $= 0.6 \times 15 = 9$ .
To estimate frequency for part of an interval, use linear interpolation.
Example: for $13 \le w < 15$ , $fd = 4$ , width 2 → estimated frequency $= 8$ .

Exam Tips

Always write down frequency densities before drawing.
The frequency density axis scale may not be 1 unit per square.
Method marks are awarded for showing use of frequency density.
Answers from histograms are estimates because data distribution within intervals is unknown.

Histogram Example with Frequency Density

Histogram with Missing Bar (Interpreting)

Histogram with Missing Bar (Drawing)

Frequency Density vs Frequency Bar Chart Comparison

Practice questions

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1.What is the formula for frequency density?
Easy
- AFrequency density = frequency / class width
- BFrequency density = class width / frequency
- CFrequency density = frequency × class width
- DFrequency density = total frequency / number of classes
2.In a histogram, what does the area of a bar represent?
Easy
- AFrequency
- BFrequency density
- CClass width
- DTotal frequency
3.A frequency table has a class interval $20 < t \le 50$ with frequency 30. What is the frequency density?
Medium
- A1
- B0.5
- C2
- D30
4.The frequency density for a class interval $10 < h \le 20$ is 3. The class width is 10. What is the frequency?
Medium
- A30
- B3
- C0.3
- D13
5.A histogram has a bar for the interval $0 < t \le 10$ with height 2.4 cm. The frequency for this interval is 12. What is the scale for frequency density?
Hard
- A1 $cm = 0.5$ frequency density
- B1 $cm = 2$ frequency density
- C1 $cm = 5$ frequency density
- D1 $cm = 12$ frequency density
6.The class widths in a histogram are 5, 10, 20, and 5. Which class interval will have the tallest bar if all frequencies are equal?
Medium
- AThe interval with width 5
- BThe interval with width 10
- CThe interval with width 20
- DAll bars will have the same height
7.In a histogram, the bars are drawn with no gaps because:
Easy
- AThe data is continuous
- BThe data is discrete
- CThe data is categorical
- DThe data is grouped
8.The frequency density for a class interval is 5 and the class width is 4. What is the frequency?
Medium
- A20
- B1.25
- C9
- D5

Summary

## Histograms (IGCSE Maths Extended) ### Key Formula - **Frequency density** = frequency / class width - **Frequency** = frequency density × class width ### Drawing Histograms 1. Calculate class width (upper bound − lower bound). 2. Compute frequency density for each class. 3. On the graph: horizontal axis = class intervals (continuous, no gaps); vertical axis = frequency density. 4. Draw bars with height = frequency density, width = class width. ### Interpreting Histograms - Area of bar = frequency. - To find frequency from histogram: frequency = frequency density × class width. - To estimate frequency for part of an interval: find area of that part. - Estimates are needed because data within an interval may not be evenly distributed. ### Common Exam Tips - Always write down frequency densities before drawing. - Check vertical scale carefully (may not be 1 unit per square). - For incomplete histograms, use given bar heights to find missing frequencies or densities. ### Example | Class | Frequency | Class Width | Frequency Density | |-------|-----------|-------------|-------------------| | 0–10 | 20 | 10 | 2 | | 10–20 | 30 | 10 | 3 | | 20–30 | 10 | 10 | 1 | Bar heights are 2, 3, 1 respectively.

Key terms

Frequency density: Frequency divided by class width; used as the height of bars in a histogram.
Class width: The difference between the upper and lower boundaries of a class interval.
Histogram: A graphical representation of continuous grouped data where the area of each bar is proportional to the frequency.
Continuous data: Data that can take any value within a range; used in histograms with no gaps between bars.

Slides

What is a Histogram?

Histograms are used for continuous data grouped into class intervals.
The area of each bar represents the frequency.
Bars are drawn touching each other (no gaps).
The vertical axis shows frequency density, not frequency.

What does the area of a bar in a histogram represent?

Frequency
Frequency density
Class width
Total frequency

Frequency Density Formula

Frequency density = frequency ÷ class width.
Class width = upper bound − lower bound of the interval.
Use frequency density when class intervals are unequal.

Frequency density = ____ ÷ class width.

Drawing a Histogram

First calculate frequency density for each class interval.
Draw bars with width equal to the class interval on the x-axis.
Height of each bar equals the frequency density on the y-axis.
Bars should touch each other.

For a class interval $20 < t \le 30$ with frequency 15, what is the frequency density?

Interpreting a Histogram

To find frequency from a histogram: frequency = frequency density × class width.
To estimate frequency for part of an interval, find the area of that part.
Estimates assume data is evenly distributed within the interval.

Which of these are true about histograms?

The height of a bar is always the frequency.
The area of a bar represents the frequency.
Bars are drawn with gaps between them.
Frequency density = frequency ÷ class width.

Worked Example: Drawing

Given: interval $60 \le x < 70$ , frequency 8, class width 10 → frequency density $= 0.8$ .
Given: interval $80 \le x < 100$ , frequency 2, class width 20 → frequency density $= 0.1$ .
Draw bars with these heights on the histogram.

For interval $40 < v \le 50$ with frequency 120 and class width 10, frequency density = ____.

Worked Example: Interpreting

From a histogram, bar height $(frequency density) = 0.6$ , class width $= 15$ → frequency $= 0.6 \times 15 = 9$ .
To estimate number in part of an interval, multiply frequency density by the width of that part.
This gives an estimate because actual distribution within the interval is unknown.

A bar has frequency density 4 and class width 2. What is the frequency?

Key Points to Remember

Always calculate frequency densities before drawing.
The y-axis may be labelled 'frequency density' or just have a scale.
If class widths are equal, the y-axis may be labelled 'frequency'.
Use area = frequency to solve problems.

Which statements are correct?

Frequency density is always labelled on the y-axis.
Frequency = frequency density × class width.
Histograms are used for discrete data.
Bars in a histogram touch each other.

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