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

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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 < x < x < x < x < x < x < 60 2 Total < < < < < < < Source: 2001 Census Office for National Statistics 168

Distance (km) Frequency (thousands) Cumulative frequency 0 x < x < x < x < x < x < x < 60 2 Total This data set gives the distances people in Nottingham travel to work. < < < < < < < Source: 2001 Census Office for National Statistics       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

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 x < 30 x < 20 x < 10 x < 5 x < 2 x < 0 x < Cu. freq. ( ‘ 000s) Distance (km) < < < < < < < Source: 2001 Census Office for National Statistics

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 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) < < < < < < < Source: 2001 Census Office for National Statistics

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) < < < < < < < Source: 2001 Census Office for National Statistics

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) < < < < < < < 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

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) < < < < < < Distance (km) 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

Cumulative frequency (thousands) 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)

Cumulative frequency (thousands) 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 (60, 168)

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

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 < upper class boundary For a greater than question subtract the cumulative frequency from the total frequency. 130 are less than 10 e.g.

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)

Exercise 1.The table gives the ages of residents in Ilkley ( W Yorkshire ). 1 14Total 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

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 < 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.

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 ·5 thousand or The percentage is  100 =

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