HistogramsHistograms
© Christine Crisp
““Teach A Level Teach A Level Maths”Maths”
Statistics 1Statistics 1
Histograms
Suppose the data are grouped so that those below 20 and above 69 are combined.
e.g. The projected population of the U.K. for 2005 ( by age )
Source: USA IDB
090+
280 – 89
470 – 79
660 – 69
850 – 59
940 – 49
930 – 39
720 – 29
810 – 19
70 – 9
(millions)
( years )
FreqAGE
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
Source: USA IDB
090+
280 – 89
470 – 79
660 – 69
850 – 59
940 – 49
930 – 39
720 – 29
810 – 19
70 – 9
(millions)
( years )
FreqAGESuppose the data are grouped so that those below 20 and above 69 are combined.
670+660 - 69850 - 59940 - 49930 - 39720 - 29
150 - 19
AGE(years)
Freq(millions)
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 )
Source: USA IDB
Suppose the data are grouped so that those below 20 and above 69 are combined.
I chose a sensible figure
670 - 109660 - 69850 - 59940 - 49930 - 39720 - 29
150 - 19
Freq(millions)
AGE(years)
Source: USA IDB
090+
280 – 89
470 – 79
660 – 69
850 – 59
940 – 49
930 – 39
720 – 29
810 – 19
70 – 9
(millions)
( years )
FreqAGE
Histograms
e.g. The projected population of the U.K. for 2005 ( by age )
670 - 109660 - 69850 - 59940 - 49930 - 39720 - 29
150 - 19
Freq(millions)
AGE(years)
-2
2
4
6
8
10
12
14
16
18
10 20 30 40 50 60 70 80 90 1000X->
|̂Y
If we use the data below to draw an age/frequency graph then it is very misleading as the 1st and last bar dominate
So frequencies are represented by
areas
Bar1 1 should represent just over twice as many people as bar 2 but it appears to be about 4 times as many
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.).
670 - 109660 - 69850 - 59940 - 49930 - 39720 - 29
150 - 19
Freq(millions)
AGE(years)
Classwidth
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.
Histograms
A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by
age )
height= frequencywidth
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.).
Area of a rectangle = width height
To draw the histogram, we need to find the width and height of each column.
So, frequency= width height
670 - 109660 - 69850 - 59940 - 49930 - 39720 - 29
20150 - 19
Classwidth
Freq(millions)
AGE(years)
401010101010
Histograms
40670 - 10910660 - 6910850 - 5910940 - 4910930 - 3910720 - 2920150 - 19
Classwidth
Freq(millions)
AGE(years)
Freqdensity
A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by
age )
The height is called the frequency
density
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.).
e.g. For the 1st class,freq. density =
To draw the histogram, we need to find the width and height of each column.
height= frequencywidth
Histograms
40670 - 10910660 - 6910850 - 5910940 - 4910930 - 3910720 - 2920150 - 19
Freqdensity
Classwidth
Freq(millions)
AGE(years)
A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by
age )
The height is called the frequency
density
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.).
e.g. For the 1st class,freq. density =
To draw the histogram, we need to find the width and height of each column.
75020
15
height= frequencywidth 0 ·75
Histograms
40670 - 10910660 - 6910850 - 5910940 - 4910930 - 3910720 - 2920150 - 19
Freqdensity
Classwidth
Freq(millions)
AGE(years)
A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by
age )
The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.).
We can now draw the histogram.
To draw the histogram, we need to find the width and height of each column.
The height is called the frequency
density
height= frequencywidth 0 ·75
0 ·150 ·60 ·80 ·90 ·90 ·7
Histograms
AGE(years)
Freq(million
s)
Classwidth
Freqdensity
0 - 19 15 20 0 ·7520 - 29 7 10 0 ·730 - 39 9 10 0 ·940 - 49 9 10 0 ·950 - 59 8 10 0 ·860 - 69 6 10 0 ·6
70 - 109 6 40 0 ·15
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. )
Histograms
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
andl.c.b. is lower class boundary
frequency density = width class
frequency
waterclear
float ducksfluffy
HistogramsExercis
e95 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 16 10
Find 3 things wrong with the histogram which represents the data in the table.
HistogramsAnswer:
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 16 10
• Frequency has been plotted instead of frequency density.
• There is no title.
• There are no units on the x-axis.
Histograms
Time taken for 95 components to fail
Incorrect diagram
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
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
andl.c.b. is lower class boundary
frequency density = width class
frequency
Histograms
6
850 – 59
940 – 49
930 – 39
720 – 29
150 – 19
( years )Freq
AGE
60 – 69
670 – 109
Class width
Freq densit
y
0·15
0·6
0·8
0·9
0·9
0·8
0·75
40
10
10
10
10
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.