Transcript “Teach A Level Maths” Statistics 1 Histograms © Christine Crisp Histograms e.g. The projected population of the U.K.

“Teach A Level Maths”
Statistics 1
Histograms
© Christine Crisp
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
AGE
Freq
( years )
(millions)
0–9
7
10 – 19
8
20 – 29
7
30 – 39
9
40 – 49
9
50 – 59
8
60 – 69
6
70 – 79
4
80 – 89
2
90+
0
Source: USA IDB
Suppose the data are grouped so
that those below 20 and above 69
are combined.
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
AGE
Freq
( years )
(millions)
0–9
7
10 – 19
8
20 – 29
7
30 – 39
9
40 – 49
9
50 – 59
8
60 – 69
6
70 – 79
4
80 – 89
2
90+
0
Source: USA IDB
Suppose the data are grouped so
that those below 20 and above 69
are combined.
AGE
Freq
(years) (millions)
0 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70+
15
7
9
9
8
6
6
To draw the diagram we must
have an upper class value
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
AGE
Freq
( years )
(millions)
0–9
7
10 – 19
8
20 – 29
7
30 – 39
9
40 – 49
9
50 – 59
8
60 – 69
6
70 – 79
4
80 – 89
2
90+
0
Source: USA IDB
Suppose the data are grouped so
that those below 20 and above 69
are combined.
AGE
Freq
(years) (millions)
0 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 109
15
7
9
9
8
6
6
I chose a sensible figure
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
If we use the data below to draw an age/frequency graph
then it is very misleading as the 1st and last bar dominate
AGE
Freq
(years) (millions)
0 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 109
15
7
9
9
8
6
6
18^|
Y
16
14
12
10
8
6
4
2
X->
0
10
20
30
40
50
60
70
80
90
100
-2
Bar1 1 should represent
just over twice as many people
as bar
2 but it appears
to be aboutby4 areas
times as many
So frequencies
are represented
Histograms
A histogram shows frequencies as areas.
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.).
AGE
Class
Freq
(years) (millions) width
0 - 19
15
20
20 - 29
7
30 - 39 for 9example,
Since these are ages, the 1st class,
49 the9 width is 20.
has u.c.b.= 20 and the l.c.b.=40
0,- so
50 - 59
8
60 - 69
6
70 - 109
6
Histograms
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.).
AGE
Class
Freq
(years) (millions) width
Area of a rectangle
= width  height 0 - 19
So,
20 - 29
frequency = width  height 30 - 39
40 - 49
 height = frequency
50 - 59
width
60 - 69
70 - 109
15
7
9
9
8
6
6
20
10
10
10
10
10
40
Histograms
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.).
height = frequency
width
AGE
Class
Freq
(years) (millions) width
0 - 19
20 - 29
The height is called
30 - 39
the frequency density
40 - 49
50 - 59
60 - 69
e.g. For the 1st class,
freq. density = 70 - 109
15
7
9
9
8
6
6
20
10
10
10
10
10
40
Freq
density
Histograms
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.).
height = frequency
width
AGE
Class
Freq
(years) (millions) width
0 - 19
15
20 - 29
7
The height is called
30 - 39
9
the frequency density
40 - 49
9
50 - 59
8
60 - 69
6
e.g. For the 1st class,
15
 0  756
freq. density = 70 - 109
20
20
10
10
10
10
10
40
Freq
density
0 ·75
Histograms
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.).
 height = frequency
width
The height is called
the frequency density
We can now draw the
histogram.
AGE
Class
Freq
(years) (millions) width
0 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 109
15
7
9
9
8
6
6
20
10
10
10
10
10
40
Freq
density
0 ·75
0 ·7
0 ·9
0 ·9
0 ·8
0 ·6
0 ·15
Histograms
AGE
Class
Freq
(years) (millions) width
0 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 109
15
7
9
9
8
6
6
20
10
10
10
10
10
40
Freq
density
The projected population of the U.K.
for 2005 ( by age )
0 ·75
0 ·7
0 ·9
0 ·9
0 ·8
0 ·6
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. )
Histograms
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
 frequency density =
class width
fluffyducks
float
clear w ater
Histograms
Exercise
95 components are tested until they fail. The table
gives the times taken ( hours ) until failure.
Time to
failure (hours)
Number of
components
0-19
5
Find 3 things wrong
with the histogram
which represents the
data in the table.
20-29 30-39 40-44 45-49 50-59 60-89
8
16
22
18
16
10
Histograms
Answer:
Time to
failure (hours)
Number of
components
0-19
20-29 30-39 40-44 45-49 50-59 60-89
5
• Frequency has been
plotted instead of
frequency density.
• There is no title.
• There are no units on
the x-axis.
8
16
22
18
16
10
Histograms
Incorrect
diagram
Time taken for 95 components to fail
Correct
diagram
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.
Histograms
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
 frequency density =
class width
Histograms
e.g.
AGE
Freq
0 – 19
15
7
9
9
8
6
6
( years )
20 – 29
30 – 39
40 – 49
50 – 59
60 – 69
70 – 109
Class
Freq
width density
20
10
10
10
10
10
40
The projected population of the U.K.
for 2005 ( by age )
0·75
0·8
0·9
0·9
0·8
0·6
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. )