1. Cumulative frequency (thousands) Distances travelled to work (Nottingham) x x x x x x Distance (km) Teach GCSE Maths Diagrams Cumulative Frequency

2. "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" © Christine Crisp Cumulative Frequency Diagrams Data from the Office for National Statistics which is included in this presentation is reproduced under the terms of the click-use licence.

3. This data set gives the distances people in Nottingham travel to work. “What percentage travel more than 25 km ?” The word “accumulate” means collect: cumulative frequencies are frequencies collected up. We can use a cumulative frequency diagram to answer questions such as: “How many people travel less than 15 km ?” or Distance (km) Frequency (thousands) 0 x < 2 35 2 x < 5 49 5 x < 10 46 10 x < 20 24 20 x < 30 9 30 x < 40 3 40 x < 60 2 Total < < < < < < < Source: 2001 Census Office for National Statistics 168

4. Distance (km) Frequency (thousands) Cumulative frequency 0 x < 2 35 2 x < 5 49 5 x < 10 46 10 x < 20 24 20 x < 30 9 30 x < 40 3 40 x < 60 2 Total This data set gives the distances people in Nottingham travel to work. < < < < < < < Source: 2001 Census Office for National Statistics  84 130 154 163 166 168 35      Copy the 1 st frequency Add the 2 nd Add the 3 rd and so on... Check the last one equals the total frequency. 168

5. This data set gives the distances people in Nottingham travel to work. Before we plot the points we must think what the cumulative frequencies tell us. e.g. This one... 40 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < < 84 130 154 163 166 168 35 2 5 10 20 30 40 60 Source: 2001 Census Office for National Statistics

6. This data set gives the distances people in Nottingham travel to work. Before we plot the points we must think what the cumulative frequencies tell us. e.g. This one... tells us there are 84 000 people who travel less than 5 km 40 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < < 84 130 154 163 166 168 35 2 5 10 20 30 40 60 Source: 2001 Census Office for National Statistics

7. How far do these people travel? This data set gives the distances people in Nottingham travel to work. Before we plot the points we must think what the cumulative frequencies tell us. 40 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < < 84 130 154 163 166 168 35 2 5 10 20 30 40 60 Source: 2001 Census Office for National Statistics

8. Each point on a cumulative frequency diagram is plotted at the upper class boundary. This data set gives the distances people in Nottingham travel to work. Before we plot the points we must think what the cumulative frequencies tell us. Ans: Less than 30 km. 40 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < < 84 130 154 163 166 168 35 2 5 10 20 30 40 60 How far do these people travel? We need to plot 163 thousand against 30 to show that these people travel less than 30 km. Source: 2001 Census Office for National Statistics

9. This data set gives the distances people in Nottingham travel to work. 60 Cumulative frequency (thousands) Distances travelled to work (Nottingham) x x x x x x x 40 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < 84 130 154 163 166 168 35 5 10 20 30 40 60 5 10203040 Distance (km) 2 35 84 130 154 163 166 168 We draw a smooth curve and extend it to the x -axis at the lower class boundary of the 1 st class. 0 x < << 2 Source: 2001 Census Office for National Statistics

10. Cumulative frequency (thousands) 100 50 200 150 604020103050 Distances travelled to work (Nottingham) x x x x x x x Distance (km) How many people travel less than 15 km.? Ans: About 145 thousand. Watch the units on the axis. It’s easy to say 145 ! (60, 168)

11. Cumulative frequency (thousands) 100 50 200 150 604020103050 Distances travelled to work (Nottingham) x x x x x x x Distance (km) How many people travel more than 25 km? The number on the y -axis is 160. What is the answer ? Be careful again ! Cumulative frequencies give “less than” so we need the top part of the axis to get “more than”. The number travelling more than 25 km is about 8 000. (60, 168)

12. Cumulative frequency (thousands) 100 50 200 150 604020103050 Distances travelled to work (Nottingham) x x x x x x x Distance (km) 8 000 people travel more than 25 km Percentage travelling more than 25 km 8 000 168 000  100 = 4·8 % ( 1 d.p. ) = (60, 168)

13. SUMMARY A cumulative frequency table collects up the frequencies. Each cumulative frequency shows how many of the variable are less than the upper class boundary. Cu. freqs. are plotted against upper class boundaries. 5 x < Cumulative frequency x < 130 10 upper class boundary For a greater than question subtract the cumulative frequency from the total frequency. 130 are less than 10 e.g.

14. The exercise that follows needs squared paper. The diagram shows a scale you could use. A sheet suitable for photocopying is at the end of the presentation. Age (years) Ages of Ilkley Residents Cumulative Frequency (thousands)

15. Exercise 1.The table gives the ages of residents in Ilkley ( W Yorkshire ). 1 14Total 3 2 2 2 41 33 Cumulative frequency ( 000 ’s ) Frequency ( 000 ’s ) Age (years) < 0 x < 20 < 20 x < 30 < 30 x < 40 < 40 x < 50 < 50 x < 60 < 60 x < 80 < 80 x < 100 (a) Complete the table. (b) Draw a cumulative frequency diagram. Use your graph to estimate (c) the number of children under 10, (d) the percentage who are over 70. Source: 2001 Census, Office for National Statistics

16. Cumulative frequency ( 000 ’s ) 4 3 Age (years) < 0 x < 20 < 20 x < 30 < 30 x < 40 < 40 x < 50 < 50 x < 60 < 60 x < 80 < 80 x < 100 6 8 13 14 10 Solution: Age (years) Cumulative Frequency (thousands) Ages of Ilkley Residents If you continued your curve after this point, it must not go up, which would show more people, or dip down, which would give fewer people.

17. Age (years) Cumulative Frequency (thousands) Ages of Ilkley Residents (a)The number of children under 10: (b)The number over 70 : 17·1% ( 3 s.f. ) Solution: 14  11·6 = 2·4, so the number older than 70 is 2 400. 1·5 thousand or 1500. The percentage is 2400 14000  100 =

19. Age (years) Ages of Ilkley Residents Cumulative Frequency (thousands)