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
Histograms
Summary of column graphs so far
A Frequency Diagram
A Bar Chart
Time taken to complete a Table Test
Sales of Paddling Pools per quarter
100
6
Frequency
80
60
40
5
4
3
2
20
1
0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
0
2
4
6
Time in Minutes
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
8
10
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
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
25 < x ≤ 30
30 < x ≤ 40
12
40 < x ≤ 70
10
8
12
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.
Histograms
Frequency density
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
=
AREA
Class width
frequency = frequency density × class width
rearranging this equation
frequency density = frequency
class width
Histograms
Draw a histogram to represent this data
Age
Frequency Class Width Frequency Density
1
0 < x ≤ 15
5
15
3
15 < x ≤ 20
15
5
3
20 < x ≤ 25
10
2
5
25 < x ≤ 30
8
5
1.6
30 < x ≤ 40
40 < x ≤ 70
1.2
10
30
12
10
1
3
Frequency density
3
2
1
0
10
20
30
40
Age (years)
50
60
70
Histograms
From this histogram you could be asked further questions:
Estimate how many people between the ages of 35 and 55 surf.
Frequency density
3
Identify the area of the
bars to which this applies.
2
Remember:
frequency is the area
1
0
10
20
30
40
Age (years)
35
50
55
60
70
1.2
1
3
5
15
Find the area separately of each bar
5 × 1.2 = 6 15 × 1 = = 5 Estimate of the frequency = 6 + 5 = 11
3
Histograms
Draw Histograms for these tables:
10
Frequency
Class
Interval
Frequency
Density
10 ≤ t < 11
5
1
5
11 ≤ t < 12
7
1
7
12 ≤ t < 15
18
3
6
15 ≤ t < 16
4
1
4
16 ≤ t < 17
3
1
3
2
17 ≤ t < 20
6
3
2
0
10 11 12 13 14 15 16 17 18 19 20
Frequency Density
Temperature
(nearest oC)
8
6
4
Temperature oC
Frequency
Density
Frequency
151 ≤ h < 153
64
2
32
153 ≤ h < 154
43
1
43
154 ≤ h < 155
47
1
47
155 ≤ h < 159
96
4
24
159 ≤ h < 160
12
1
12
50
40
Frequency Density
Height (cm)
Class
Interval
30
20
10
0
151 152 153 154 155 156 157 158 159 160
Height (cm)
Histograms
Draw Histograms for these tables:
25
Frequency
Class
Interval
Frequency
Density
0≤t<8
72
8
9
9 ≤ t < 12
84
4
21
12 ≤ t < 15
54
3
18
15 ≤ t < 20
35
5
7
20
Frequency Density
Time
(nearest
minute)
15
10
5
0
0
2
4
6
8
10
12
14 16
18
Time (mins)
25
Frequency
Class
Interval
Frequency
Density
11 ≤ a < 14
51
3
17
14 ≤ a < 16
36
2
18
16 ≤ a < 17
12
1
12
17 ≤ a < 20
20
3
6.7
20
Frequency Density
Age (nearest
year)
15
10
5
0
11
12
13 14
15
16
Age (years)
17
18
19
20
20
Histograms
Draw Histograms for these tables:
Temperature
(nearest oC)
Frequency
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
152 ≤ h < 153
64
153 ≤ h < 154
43
154 ≤ h < 155
47
155 ≤ h < 159
96
159 ≤ h < 160
12
Class
Interval
Frequency
Density
Class
Interval
Frequency
Density
Histograms
Draw Histograms for these tables:
Time
(nearest
minute)
Frequency
0≤t<8
72
9 ≤ t < 12
84
12 ≤ t < 15
54
15 ≤ t < 20
35
Age (nearest
year)
Frequency
11 ≤ a < 14
51
14 ≤ a < 16
36
16 ≤ a < 17
12
17 ≤ a < 20
20
Class
Interval
Class
Interval
Frequency
Density
Frequency
Density