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© Boardworks Ltd 2008 1 of 38 D4 Moving averages and cumulative frequency Maths Age 14-16

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© Boardworks Ltd 2008 2 of 38 Contents A A A A A D4.5 Box-and-whisker diagrams D4 Moving averages and cumulative frequency D4.3 Cumulative frequency D4.2 Plotting moving averages D4.1 Moving averages D4.4 Using cumulative frequency graphs

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© Boardworks Ltd 2008 3 of 38 A box-and-whisker diagram A box-and-whisker diagram, or boxplot, can be used to illustrate the spread of the data in a given distribution using the highest and lowest values, the median, the lower quartile and the upper quartile. These values can be found from a cumulative frequency graph. Time in seconds Cumulative frequency 30354045505560 10 20 30 40 50 60 70 80 90 100 0 For example, for this cumulative frequency graph showing the results of 100 people holding their breath, Minimum value = 30 Lower quartile = 42 Median = 47 Upper quartile = 51 Maximum value = 60

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© Boardworks Ltd 2008 4 of 38 A box-and-whisker diagram The corresponding box-and-whisker diagram is as follows: 30 Minimum value 42 Lower quartile 47 Median 51 Upper quartile 60 Maximum value

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© Boardworks Ltd 2008 5 of 38 Lap times James takes part in karting competitions and his Dad records his lap times on a spreadsheet. The track is 1108 metres long. James’ fastest time in a race was 51.8 seconds. In which position in the list would the median lap time be? In 2004, 378 of James’ lap times were recorded. There are 378 lap times and so the median lap time will be the 378 + 1 2 th value ≈ 190 th value

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© Boardworks Ltd 2008 6 of 38 Lap times In which position in the list would the lower quartile be? There are 378 lap times and so the lower quartile will be the 378 + 1 4 th value ≈ 95 th value In which position in the list would the upper quartile be? There are 378 lap times and so the upper quartile will be the 284 th value 378 + 1 4 th value ≈ 3 ×

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© Boardworks Ltd 2008 7 of 38 Lap times at Shenington karting circuit James’ lap times are displayed in the following cumulative frequency graph. Lap times in seconds Cumulative frequency 525456586062646668707274767880828486889092 0 50 100 150 200 250 300 350 400

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© Boardworks Ltd 2008 8 of 38 Box and whisker plot for James’ race times What conclusions can you draw about James’ performance? 52 Minimum value 53 Lower quartile 54 Median 58 Upper quartile 91 Maximum value

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© Boardworks Ltd 2008 9 of 38 Comparing sets of data Here are box-and-whisker diagrams representing James’ lap times and Shabnum’s lap times. Who is better and why? 52 53 545891 James’ lap times 5260546586 Shabnum’s lap times

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The projected population of the U.K. for 2005 ( by age ) The boxOne whisker The other whisker The box can be any depth. median minimum age maximum age.

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