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© Boardworks Ltd of 38 D4 Moving averages and cumulative frequency KS4 Mathematics

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

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© Boardworks Ltd of 38 They agree to give Tabina a prize if she can stop complaining for a whole week Stop complaining! Tabina’s friends claim that she is always complaining and decide to keep a record of how many times she is heard complaining every day for five weeks. These are the results: Should she get a prize?

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© Boardworks Ltd of 38 There are lots of groups of seven days in the data. Groups of seven Is it fair to consider only Monday to Sunday? What if you included Sunday to Saturday, Tuesday to Monday, Wednesday to Tuesday and so on?

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© Boardworks Ltd of 38 We could calculate the mean for every group of seven. The moving average The means of each group of seven are collectively called a seven-point moving average. 1)How could this help us decide whether Tabina should get a reward? 2)How many of the means will be 0? 3)What method would you use to calculate the means?

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© Boardworks Ltd of 38 Calculating a seven-point moving average The means (to 2 decimal places) for each of the 29 groups of 7 are as follows: What can the moving average tell us about the general pattern of Tabina’s behaviour and whether she should win the prize?

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© Boardworks Ltd of 38 Moving averages

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

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© Boardworks Ltd of 38 A graph showing number of complaints each day This graph shows the number of times Tabina complains each day. How well does this graph illustrate the general trend in Tabina’s behaviour?

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© Boardworks Ltd of 38 A graph showing number of complaints each day A line graph that shows how a value changes over time is called a time series. To smooth out the fluctuations in this time series we can plot the moving average:

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© Boardworks Ltd of 38 Method 29 th – 35 th … 3 rd – 9 th 2 nd – 8 th 1 st – 7 th Position of mean on graphRange Plotting moving averages on a time series graph For our seven-point moving average we would have: ( ) ÷ 2 … (3 + 9) ÷ 2 (2 + 8) ÷ 2 (1 + 7) ÷ 2 32 … When we plot the moving average, each mean is plotted halfway along the group that it represents.

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© Boardworks Ltd of 38 Comparing sets of data Here are the attendance records for two hip hop dance classes of 30 students over ten weeks. Draw line graphs for each class to represent the changes in attendance. Class A Class B

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© Boardworks Ltd of 38 Calculating a five-point moving average We can smooth out the fluctuations for each graph by calculating a five-point moving average. Class A Means for class A Class B Means for class B

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© Boardworks Ltd of 38 Plotting a five-point moving average Method … 3 rd – 7 th 2 nd – 6 th 1 st – 5 th Position of mean on graphRange For a five-point moving average we have: … (3 + 9) ÷ 2 (2 + 8) ÷ 2 (1 + 7) ÷ 2 … Each mean is then plotted halfway along the group that it represents.

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© Boardworks Ltd of 38 Time series for class A Attendance Weeks

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© Boardworks Ltd of 38 Five-point moving average for class A Attendance Weeks

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© Boardworks Ltd of 38 Time series for class B Attendance Weeks

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© Boardworks Ltd of 38 Five-point moving average for class B Attendance Weeks

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© Boardworks Ltd of Method Position of first mean on graph Size of moving average Plotting the means for other moving averages We can find the positions of other moving averages as follows: 3(5 + 1) ÷ 2 3.5(6 + 1) ÷ 2 4(7 + 1) ÷ 2 (8 + 1) ÷ 2 (4 + 1) ÷ 2 (3 + 1) ÷

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

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© Boardworks Ltd 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 ≤ < t ≤ < t ≤ < t ≤ < t ≤ < t ≤ 50 Time in secondsFrequencyTime in seconds = = = = = < 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 of 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 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 of 38 Cumulative frequency graphs Here is the cumulative frequency table for 100 people holding their breath: Time in secondsCumulative frequency 0 < t ≤ < t ≤ < t ≤ < t ≤ < t ≤ < t ≤ We can plot a cumulative frequency graph as follows:

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© Boardworks Ltd of 38 Plotting a cumulative frequency graph Time in seconds Cumulative frequency 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 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 of 38 Finding the interquartile range Time in seconds Cumulative frequency 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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© Boardworks Ltd 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 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 median, the lower quartile and the upper quartile. These values can be found from a cumulative frequency graph. Time in seconds Cumulative frequency 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 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 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? One of the karting tracks is at Shenington. In 2004, 378 of James’ lap times were recorded. There are 378 lap times and so the median lap time will be the th value ≈ 190 th value

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© Boardworks Ltd 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 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 th value ≈ 3 ×

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© Boardworks Ltd 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

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© Boardworks Ltd 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 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? James’ lap times Shabnum’s lap times

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