The document discusses how to draw a histogram when the class widths are unequal. It shows how to calculate frequency density for each class by dividing the frequency by the class width. This allows the heights of the bars to be represented as frequency density rather than frequency. The area of each bar then represents the frequency for that class. Calculating the frequency density is necessary to properly represent the data when class widths differ.

1. Histograms

2. This table shows
the height of 50
random
secondary
school students
in the UK
Height Frequency
140≤h<150 8
150≤h<155 9
155≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
Source: censusatschool.org.uk

3. Drawing a histogram
Height Frequency
140≤h<150 8
150≤h<155 9
155≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
We need to add
a class width
column to do
this...

4. Drawing a histogram
Height Frequency Class width
140≤h<150 8
150≤h<155 9
155≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
Like so...

5. Drawing a histogram
Giving...
Height Frequency Class width
140≤h<150 8 10
150≤h<155 9 5
155≤h<160 4 5
160≤h<165 9 5
165≤h<170 9 5
170≤h<185 11 15

6. To draw a histogram for data
with different class widths we
need to calculate the frequency
density for each class.

7. To calculate
frequency density...
Frequency
Class width

8. Work out the frequency
density column...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5
155≤h<160 4 5
160≤h<165 9 5
165≤h<170 9 5
170≤h<185 11 15

9. Giving...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 = 0.8
160≤h<165 9 5 9 ÷ 5 = 1.8
165≤h<170 9 5 9 ÷ 5 = 1.8
170≤h<185 11 15 11 ÷ 15 = 0.73

10. 185140 145 150 155 160 165 170 175 180
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Height (cm)
FrequencyDensity
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 = 0.8
160≤h<165 9 5 9 ÷ 5 = 1.8
165≤h<170 9 5 9 ÷ 5 = 1.8
170≤h<185 11 15 11 ÷ 15 = 0.73
Now use your values to
draw the histogram...
remember not to
leave any gaps
between the bars.

11. Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 = 0.8
160≤h<165 9 5 9 ÷ 5 = 1.8
165≤h<170 9 5 9 ÷ 5 = 1.8
170≤h<185 11 15 11 ÷ 15 = 0.73
Now use your values to
draw the histogram...
remember not to
leave any gaps
between the bars.
185140 145 150 155 160 165 170 175 180
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Height (cm)
FrequencyDensity

12. Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 = 0.8
160≤h<165 9 5 9 ÷ 5 = 1.8
165≤h<170 9 5 9 ÷ 5 = 1.8
170≤h<185 11 15 11 ÷ 15 = 0.73
Now use your values to
draw the histogram...
remember not to
leave any gaps
between the bars.
185140 145 150 155 160 165 170 175 180
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Height (cm)
FrequencyDensity

13. What, if anything, can we see
from this histogram?
Why?

14. If we look a bit closer there a
few other useful things about a
histogram.

15. 140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
For each histogram
bar the width is the
class width and the
length is the
frequency density.

16. 140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
With any rectangle if
we do width x length
we get the area of
the rectangle.

17. 140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
And for a histogram
the area of the bar is
equal to the
frequency for that
class.

18. How does this work?
Frequency
Density
=
Frequency
Class width
can be rearranged to...

19. Frequency
Density
=
Frequency
Class width
How does this work?
Frequency =
Frequency
Density
× Class width
can be rearranged to...

20. Frequency =
Frequency
Density
× Class width
width of the bar
length of the bar
area of the bar

21. KEY POINTS
• To draw a histogram for unequal class widths
you must calculate frequency density
•
• The area of a bar on a frequency data
histogram is equal to the frequency
•
Frequency
Density
=
Frequency
Class width
Frequency =
Frequency
Density
× Class width