1. Histograms: Unequal class Intervals
In a histogram the heights of the bars give the frequency. We could if we wished to, use the area of the bars to represent frequency, since the height of a bar is proportional to its area.
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100
The area of each bar is class width x height 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120 70
This second bar has twice the height and therefore twice the area of the first. 60
10 x 32
frequency 50 40
10 x 16
320 30 20
160 10 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

2. Histograms: Unequal class Intervals
It is sometimes desirable to combine two or more bars into a single, larger, class interval.
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100
We are going to combine the first 3 bars of this histogram into a single bar with a (0 - 30) class interval
When we do this we can no longer use the height of the bar to represent frequency. We must use its AREA 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120 70 60
Total area of bars = 240
Our single bar must also have this area.
frequency 50
These two bars are not that clear so we could combine them with the 3rd . 40
240 30 ? 30 20
160 10 60 20 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

3. Histograms: Unequal class Intervals
It is sometimes desirable to combine two or more bars into a single, larger, class interval.
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100
We are going to combine the first 3 bars of this histogram into a single bar with a (0 - 30) class interval
When we do this we can no longer use the height of the bar to represent frequency. We must use its AREA 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120 70 60
Total area of bars = 240
Our single bar must also have this area.
frequency 50 40
240 30 ? 30 20 10 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

4. Histograms: Unequal class Intervals
It is sometimes desirable to combine two or more class intervals to give a clearer picture. This is usually done with low frequency adjacent bars.
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100
Repeating this for the last 4 bars gives: 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120 70
Total area of bars = 640 60
Our single bar must also have this area.
frequency 50 40
360
640 80
120 ? 30 20
140 80 10 60 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

5. Histograms: Unequal class Intervals
It is sometimes desirable to combine two or more class intervals to give a clearer picture. This is usually done with low frequency adjacent bars.
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100
Repeating this for the last 3 bars gives: 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120 70
Total area of bars = 640 60
Our single bar must also have this area.
frequency 50 40
640 80
120 ? 30 20 10 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

6. Histograms: Unequal class Intervals
Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR, Not the height
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100 90 80
Speed (s)
Frequency
0 < s < 10 2
10 < s < 20 6
20 < s < 30 16
30 < s < 40 32
40 < s < 50 50
50 < s < 60 78
60 < s < 70 96
70 < s < 80 82
80 < s < 90 36
90 < s < 100 14
100 < s < 110 8
110 < s < 120
The frequency table needs sorting out! 70
This has to go
We now give the vertical axis a new label called frequency density.
frequency 60
frequency 50
frequency density 40 30 20 10 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

7. Histograms: Unequal class Intervals
Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100 90 80
Speed (s)
Frequency
0 < s < 30
240
30 < s < 40
320
40 < s < 50
500
50 < s < 60
780
60 < s < 70
960
70 < s < 80
820
80 < s < 120
640
Speed (s)
Frequency
0 < s < 30
240
30 < s < 40
40 < s < 50
50 < s < 60
60 < s < 70
70 < s < 80
80 < s < 90
640 70 60 50
frequency density 40
320
500
780
960
820 30 20 10
640
240 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

8. Histograms: Unequal class Intervals
Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100 90 80
Speed (s)
Frequency
0 < s < 30
240
30 < s < 40
320
40 < s < 50
500
50 < s < 60
780
60 < s < 70
960
70 < s < 80
820
80 < s < 120
640
This needs tidying up and we need a column for frequency density 70 60 50
frequency density
Which bar has the higher frequency? 40
320
500
780
960
820 30 20 10
640
240 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

9. Histograms: Unequal class Intervals
Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height
Histograms: Unequal class Intervals
Recorded Speeds of Cars on a Motorway
100 90 80
Speed f fd
0 - 30
240 8
320 32
500 50
780 78
960 96
820 82
640 16
Why is it called frequency density?
The situation is similar to the normal use of the term density. 70 60 50
In this case its how much area you get per standard class width.
frequency density 40
320
500
780
960
820 30 16
Remember: frequency density is just the height of each bar. 20 8 10
640
240 10 20 30 40 50 60 70 80 90
100
110
120
Speed mph

10. Worksheet