1. Histograms
Objectives:
A Grade Construct and interpret a histogram with unequal
class intervals
Prior knowledge:
Draw a frequency diagram
Rearrange equations
Be able to calculate the area of a rectangle

2. Summary of column graphs so far
Histograms
Summary of column graphs so far
A Bar Chart
Sales of Paddling Pools per quarter
A Frequency Diagram 2 4 6 8 10
Time taken to complete a Table Test
Time in Minutes 1 3 5
Frequency
Spot the differences from a bar chart to a frequency diagram:
There are no gaps between the bars
The horizontal axis has a continuous scale as it represents
continuous data
The width of the bars is the full width of the class interval
The class intervals (the width of the bars) is the same for all bars

3. Histograms
A histogram is a frequency diagram with unequal class intervals.
The area of the bar represents the frequency
Why have different class intervals?
Sometimes data will be collected where there may be very little
data in a class interval, and yet in another class interval of the
same width.
e.g. a survey was conducted to find the ages of surfers

4. Notice how the class intervals are different
Histograms
e.g. a survey was conducted to find the ages of 60 surfers, the results
are in the table below:
Age
Frequency
 0 < x ≤ 15 4 
 15 < x ≤ 20
 14
 20 < x ≤ 25
 12
 25 < x ≤ 30  8
 30 < x ≤ 40
40 < x ≤ 70
 10
Notice how the class intervals are different
If you kept the class interval of 5 (as for ages 15 to 30) you may
find several empty class intervals between 40 and 70
because the class width is different if we used frequency for the
y-axis the sizes of the bars would represent the frequencies
disproportionately.

5. We use frequency density to show the height.
Histograms
We use frequency density to show the height.
The frequency is the area of the bar
The width of the bar is the class width
Area of a rectangle = length × width
frequency = frequency density × class width
Frequency =
AREA
Frequency density
rearranging this equation
frequency density = frequency
class width
Class width

6. Draw a histogram to represent this data
Histograms
Draw a histogram to represent this data
Age
Frequency
 0 < x ≤ 15 5 
 15 < x ≤ 20
 15
 20 < x ≤ 25
 10
 25 < x ≤ 30  8
 30 < x ≤ 40
 12
40 < x ≤ 70
Class Width
Frequency Density 15 1 3 5 3 5 2 5
1.6 10
1.2 1 3 30 3 2 1
Frequency density
Age (years)

7. From this histogram you could be asked further questions:
Histograms
From this histogram you could be asked further questions:
Estimate how many people between the ages of 35 and 55 surf. 3 2 1
Frequency density
Age (years)
Identify the area of the
bars to which this applies.
Remember:
frequency is the area
1.2 35 55 1 3
Find the area separately of each bar 5 15
5 × 1.2 = 6
15 × 1 = = 5 3
Estimate of the frequency = = 11

8. Temperature (nearest oC)
Histograms
Draw Histograms for these tables:
Temperature (nearest oC)
Frequency
Class Interval
Frequency Density
10 ≤ t < 11 5
11 ≤ t < 12 7
12 ≤ t < 15 18
15 ≤ t < 16 4
16 ≤ t < 17 3
17 ≤ t < 20 6
Temperature oC
Frequency Density 10 8 6 4 2 1 3 5 7 6 4 3 2
Height (cm)
Frequency Density 50 40 30 20 10
Height (cm)
Frequency
Class Interval
Frequency Density
151 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12 2 1 4 32 43 47 24 12

9. Histograms Draw Histograms for these tables: 25 20 15 10 5 25 20 15 10
Time (mins)
Frequency Density 25 20 15 10 5
Time (nearest minute)
Frequency
Class Interval
Frequency Density
0 ≤ t < 8 72
9 ≤ t < 12 84
12 ≤ t < 15 54
15 ≤ t < 20 35 8 4 3 5 9 21 18 7
Age (years)
Frequency Density 25 20 15 10 5
Age (nearest year)
Frequency
Class Interval
Frequency Density
11 ≤ a < 14 51
14 ≤ a < 16 36
16 ≤ a < 17 12
17 ≤ a < 20 20 3 2 1 17 18 12
6.7

10. Temperature (nearest oC)
Histograms
Draw Histograms for these tables:
Temperature (nearest oC)
Frequency
Class Interval
Frequency Density
10 ≤ t < 11 5
11 ≤ t < 12 7
12 ≤ t < 15 18
15 ≤ t < 16 4
16 ≤ t < 17 3
17 ≤ t < 20 6
Height (cm)
Frequency
Class Interval
Frequency Density
152 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12

11. Histograms Draw Histograms for these tables: Time (nearest minute)
Frequency
Class Interval
Frequency Density
0 ≤ t < 8 72
9 ≤ t < 12 84
12 ≤ t < 15 54
15 ≤ t < 20 35
Age (nearest year)
Frequency
Class Interval
Frequency Density
11 ≤ a < 14 51
14 ≤ a < 16 36
16 ≤ a < 17 12
17 ≤ a < 20 20