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

2. 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

3. 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 ) FreqAGE Suppose the data are grouped so that those below 20 and above 69 are combined. 670+ 660 - 69 850 - 59 940 - 49 930 - 39 720 - 29 150 - 19 AGE (years) Freq (millions) To draw the diagram we must have an upper class value

4. 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 - 109 660 - 69 850 - 59 940 - 49 930 - 39 720 - 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

5. Histograms e.g. The projected population of the U.K. for 2005 ( by age ) 670 - 109 660 - 69 850 - 59 940 - 49 930 - 39 720 - 29 150 - 19 Freq (millions) AGE (years) If we use the data below to draw an age/frequency graph then it is very misleading as the 1 st 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

6. 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 - 109 660 - 69 850 - 59 940 - 49 930 - 39 720 - 29 150 - 19 Freq (millions) AGE (years) Class width 20 Since these are ages, the 1 st class, for example, has u.c.b. = 20 and the l.c.b. = 0, so the width is 20.

7. Histograms A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by age )  height = frequency width 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 - 109 660 - 69 850 - 59 940 - 49 930 - 39 720 - 29 20150 - 19 Class width Freq (millions) AGE (years) 40 10

8. Histograms 40670 - 109 10660 - 69 10850 - 59 10940 - 49 10930 - 39 10720 - 29 20150 - 19 Class width Freq (millions) AGE (years) Freq density 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 1 st class, freq. density = To draw the histogram, we need to find the width and height of each column. height = frequency width

9. Histograms 40670 - 109 10660 - 69 10850 - 59 10940 - 49 10930 - 39 10720 - 29 20150 - 19 Freq density Class width 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 1 st class, freq. density = To draw the histogram, we need to find the width and height of each column. height = frequency width 0 ·75

10. Histograms 40670 - 109 10660 - 69 10850 - 59 10940 - 49 10930 - 39 10720 - 29 20150 - 19 Freq density Class width 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 = frequency width 0 ·75 0 ·15 0 ·6 0 ·8 0 ·9 0 ·7

11. Histograms AGE (years) Freq (millions) Class width Freq density 0 - 1915200 ·75 20 - 297100 ·7 30 - 399100 ·9 40 - 499100 ·9 50 - 598100 ·8 60 - 696100 ·6 70 - 1096400 ·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. )

12. 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 and l.c.b. is lower class boundary  frequency density =

13. Histograms Exercise 95 components are tested until they fail. The table gives the times taken ( hours ) until failure. Time to failure (hours) 0-1920-2930-3940-4445-4950-5960-89 Number of components 581622181610 Find 3 things wrong with the histogram which represents the data in the table.

14. Histograms Answer: Time to failure (hours) 0-1920-2930-3940-4445-4950-5960-89 Number of components 581622181610 Frequency has been plotted instead of frequency density. There is no title. There are no units on the x -axis.

15. Histograms Time taken for 95 components to fail Incorrect diagram 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. 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 and l.c.b. is lower class boundary  frequency density =

19. Histograms 6 8 50 – 59 9 40 – 49 9 30 – 39 7 20 – 29 15 0 – 19 ( years ) FreqAGE 60 – 69 6 70 – 109 Class width Freq density 0·15 0·6 0·8 0·9 0·8 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.