1. “Teach A Level Maths” Statistics 1
Histograms
© Christine Crisp

2. Statistics 1 AQA EDEXCEL MEI/OCR OCR
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3. A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
Suppose the data are grouped so that those below 20 and above 69 are combined.
Source: USA IDB
90+ 2
80 – 89 4
70 – 79 6
60 – 69 8
50 – 59 9
40 – 49
30 – 39 7
20 – 29
10 – 19
0 – 9
(millions)
( years )
Freq
AGE

4. To draw the diagram we must have an upper class value
A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
Suppose the data are grouped so that those below 20 and above 69 are combined.
Source: USA IDB
90+ 2
80 – 89 4
70 – 79 6
60 – 69 8
50 – 59 9
40 – 49
30 – 39 7
20 – 29
10 – 19
0 – 9
(millions)
( years )
Freq
AGE 6
70+ 8 9 7 15
0 - 19
AGE
(years)
Freq
(millions)
To draw the diagram we must have an upper class value

5. A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
Suppose the data are grouped so that those below 20 and above 69 are combined.
Source: USA IDB
90+ 2
80 – 89 4
70 – 79 6
60 – 69 8
50 – 59 9
40 – 49
30 – 39 7
20 – 29
10 – 19
0 – 9
(millions)
( years )
Freq
AGE 6 8 9 7 15
0 - 19
Freq
(millions)
AGE
(years)
I chose a sensible figure
Source: USA IDB

6. A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
To draw the histogram, we need to find the width and height of each column.
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). 6 8 9 7 15
0 - 19
Freq
(millions)
AGE
(years)
Class
width 20
Since these are ages, the 1st class, for example,
has u.c.b.= 20 and the l.c.b.= 0, so the width is 20.

7. A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
To draw the histogram, we need to find the width and height of each column.
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). 6 8 9 7 20 15
0 - 19
Class
width
Freq
(millions)
AGE
(years)
Area of a rectangle
= width  height
So,
frequency = width  height 40 10
 height = frequency
width

8. The height is called the frequency density
A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
To draw the histogram, we need to find the width and height of each column.
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). 40 6 10 8 9 7 20 15
0 - 19
Class
width
Freq
(millions)
AGE
(years)
Freq
density
height = frequency
width
The height is called the frequency density
e.g. For the 1st class,
freq. density =

9. The height is called the frequency density
A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
To draw the histogram, we need to find the width and height of each column.
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). 40 6 10 8 9 7 20 15
0 - 19
Freq
density
Class
width
(millions)
AGE
(years)
height = frequency
width
0 ·75
The height is called the frequency density
e.g. For the 1st class,
freq. density =

10. The height is called the frequency density
A histogram shows frequencies as areas.
e.g. The projected population of the U.K. for 2005 ( by age )
To draw the histogram, we need to find the width and height of each column.
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). 40 6 10 8 9 7 20 15
0 - 19
Freq
density
Class
width
(millions)
AGE
(years)
 height = frequency
width
0 ·75
0 ·15
0 ·6
0 ·8
0 ·9
0 ·7
The height is called the frequency density
We can now draw the histogram.

11. The projected population of the U.K. for 2005 ( by age )
(years)
Freq
(millions)
Class
width
density
0 - 19 15 20
0 ·75 7 10
0 ·7 9
0 ·9 8
0 ·8 6
0 ·6 40
0 ·15
Notice that the frequencies for the last 2 classes are the same.
On the histogram the areas showing these classes are the same.
If we had plotted frequency on the y-axis, the diagram would be very misleading. ( It would suggest there are 6 million in each age group 70 – 79, 80 – 89, 90 – 99 and 100 – 109. )

12. SUMMARY
Histograms are used to display grouped frequency data.
Frequency is shown by area.
The y-axis is used for frequency density.
Class width is given by
u.c.b. – l.c.b.
where, u.c.b. is upper class boundary and
l.c.b. is lower class boundary
frequency density =

13. Exercise
95 components are tested until they fail. The table gives the times taken ( hours ) until failure.
Time to failure (hours)
0-19
20-29
30-39
40-44
45-49
50-59
60-89
Number of components 5 8 16 22 18 10
Find 3 things wrong with the histogram which represents the data in the table.

14. Answer:
Time to failure (hours)
0-19
20-29
30-39
40-44
45-49
50-59
60-89
Number of components 5 8 16 22 18 10
Frequency has been plotted instead of frequency density.
There is no title.
There are no units on the x-axis.

15. Incorrect diagram
Time taken for 95 components to fail
Correct diagram

17. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

18. SUMMARY
Frequency is shown by area.
The y-axis is used for frequency density.
Histograms are used to display grouped frequency data.
Class width is given by
u.c.b. – l.c.b.
where, u.c.b. is upper class boundary and
l.c.b. is lower class boundary
frequency density =

19. The projected population of the U.K. for 2005 ( by age )6 8
50 – 59 9
40 – 49
30 – 39 7
20 – 29 15
0 – 19
( years )
Freq
AGE
60 – 69
70 – 109
Class width
Freq density
0·15
0·6
0·8
0·9
0·75 40 10 20
The projected population of the U.K. for 2005 ( by age )
Notice that the frequencies for the last 2 classes are the same.
On the histogram the areas showing these classes are the same.
If we had plotted frequency on the y-axis, the diagram would be very misleading. ( It would suggest there are 6 million in each age group 70 – 79, 80 – 89, 90 – 99 and 100 – 109. )
e.g.