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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 D4.3 Cumulative frequency D4 Moving averages and cumulative frequency D4.5 Box-and-whisker diagrams 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 Imagine you are going to record how long each member of your class can keep their eyes open without blinking. Choosing class intervals How could this information be recorded? What practical issues might arise? Time is an example of continuous data. You will have to decide how accurately to measure the times: to the nearest tenth of a second? to the nearest second? to the nearest five seconds?

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© Boardworks Ltd 2008 4 of 38 You will also have to decide what size class intervals to use. Keeping your eyes open When continuous data is grouped into class intervals it is important that no values are missed out and that there are no overlaps. For example, you may decide to use class intervals with a width of 5 seconds. If everyone keeps their eyes open for more than 10 seconds, the first class interval would be more than 10 seconds, up to and including 15 seconds. This is usually written as 10 < t ≤ 15, where t is the time in seconds. The next class interval would be _________.15 < t ≤ 20

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© Boardworks Ltd 2008 5 of 38 Cumulative frequency graph of results

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© Boardworks Ltd 2008 6 of 38 Cumulative frequency Cumulative frequency is a running total. It is calculated by adding up the frequencies up to that point. Cumulative frequency 1650 < t ≤ 55 1155 < t ≤ 60 930 < t ≤ 35 1235 < t ≤ 40 2440 < t ≤ 45 2845 < t ≤ 50 Time in secondsFrequencyTime in seconds 89 + 11 = 100 73 + 16 = 89 45 + 28 = 73 21 + 24 = 45 9 + 12 = 21 9 0 < t ≤ 55 0 < t ≤ 60 0 < t ≤ 35 0 < t ≤ 40 0 < t ≤ 45 0 < t ≤ 50 Here are the results of 100 people holding their breath:

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© Boardworks Ltd 2008 7 of 38 100 people took part in the experiment. Finding averages using cumulative frequency From the table, how could you find exact values or estimates for: the mean? the mode/ modal group? the median? To find a more accurate value for the median, a cumulative frequency graph can be used. the range?

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© Boardworks Ltd 2008 8 of 38 Contents A A A A A D4.4 Using cumulative frequency graphs D4 Moving averages and cumulative frequency D4.5 Box-and-whisker diagrams D4.3 Cumulative frequency D4.2 Plotting moving averages D4.1 Moving averages

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© Boardworks Ltd 2008 9 of 38 Cumulative frequency graphs Here is the cumulative frequency table for 100 people holding their breath: Time in secondsCumulative frequency 0 < t ≤ 359 0 < t ≤ 4021 0 < t ≤ 4545 0 < t ≤ 5073 0 < t ≤ 5589 0 < t ≤ 60100 We can plot a cumulative frequency graph as follows:

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© Boardworks Ltd 2008 10 of 38 Plotting a cumulative frequency graph Time in seconds Cumulative frequency 30354045505560 10 20 30 40 50 60 70 80 90 100 0 The upper boundary for each class interval is plotted against its cumulative frequency. A smooth curve is then drawn through the points. We can use the graph to estimate the median by finding the time for the 50 th person. This gives us a median time of 47 seconds.

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© Boardworks Ltd 2008 11 of 38 The interquartile range Remember; the range is a measure of spread. It is the difference between the highest value and the lowest value. When the range is affected by outliers it is often more appropriate to use the interquartile range. The interquartile range is the range of the middle 50% of the data. The lower quartile is the data item ¼ of the way along the list. The upper quartile is the data item ¾ of the way along the list. interquartile range = upper quartile – lower quartile

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© Boardworks Ltd 2008 12 of 38 Finding the interquartile range Time in seconds Cumulative frequency 30354045505560 10 20 30 40 50 60 70 80 90 100 0 The lower quartile is the time of the 25th person. The upper quartile is the time of the 75th person. The interquartile range is the difference between these two values. 51 – 42 = 9 seconds The cumulative frequency graph can be used to locate the upper and lower quartiles, and so find the interquartile range. 42 seconds 51 seconds

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