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.