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Boardworks Ltd 2005 1 of 38 D4 Moving averages and cumulative frequency KS4 Mathematics Slide 2 Boardworks Ltd 2005 2 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 Slide 3 Boardworks Ltd 2005 3 of 38 They agree to give Tabina a prize if she can stop complaining for a whole week. 5204045200000003516000332100000001152040452000000035160003321000000011 Stop complaining! Tabinas 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? Slide 4 Boardworks Ltd 2005 4 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? 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 Slide 5 Boardworks Ltd 2005 5 of 38 We could calculate the mean for every group of seven. The moving average 1100000 00123300061530 00000025404025 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? Slide 6 Boardworks Ltd 2005 6 of 38 Calculating a seven-point moving average 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 1100000 00123300061530 00000025404025 The means (to 2 decimal places) for each of the 29 groups of 7 are as follows: 0.290.140.000.140.430.861.29 2.001.862.14 1.291.14 0.430.000.291.001.57 2.14 2.432.86 What can the moving average tell us about the general pattern of Tabinas behaviour and whether she should win the prize? Slide 7 Boardworks Ltd 2005 7 of 38 Moving averages Slide 8 Boardworks Ltd 2005 8 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 Slide 9 Boardworks Ltd 2005 9 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 Tabinas behaviour? Slide 10 Boardworks Ltd 2005 10 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: Slide 11 Boardworks Ltd 2005 11 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: (29 + 35) 2 (3 + 9) 2 (2 + 8) 2 (1 + 7) 2 32 6 5 4 When we plot the moving average, each mean is plotted halfway along the group that it represents. Slide 12 Boardworks Ltd 2005 12 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 2830272928292128 29 Class B 2629 282630 272528 Slide 13 Boardworks Ltd 2005 13 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 2830272928292128 29 Means for class A 28.428.626.827.026.827.0 2830272928292128 292830272928292128 292830272928292128 292830272928292128 292830272928292128 292830272928292128 29 Class B 2629 282630 272528 Means for class B 27.628.428.628.227.628.0 2629 282630 2725282629 282630 2725282629 282630 2725282629 282630 2725282629 282630 2725282629 282630 272528 Slide 14 Boardworks Ltd 2005 14 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 6 5 4 Each mean is then plotted halfway along the group that it represents. Slide 15 Boardworks Ltd 2005 15 of 38 Time series for class A Attendance Weeks 20 21 22 23 24 25 26 27 28 29 30 12345678910 Slide 16 Boardworks Ltd 2005 16 of 38 Five-point moving average for class A Attendance Weeks 20 21 22 23 24 25 26 27 28 29 30 12345678910 Slide 17 Boardworks Ltd 2005 17 of 38 Time series for class B Attendance Weeks 20 21 22 23 24 25 26 27 28 29 30 12345678910 Slide 18 Boardworks Ltd 2005 18 of 38 Five-point moving average for class B Attendance Weeks 20 21 22 23 24 25 26 27 28 29 30 12345678910 Slide 19 Boardworks Ltd 2005 19 of 38 5 6 7 Method 8 4 3 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) 2 4.5 2.5 2 Slide 20 Boardworks Ltd 2005 20 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 Slide 21 Boardworks Ltd 2005 21 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? Slide 22 Boardworks Ltd 2005 22 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 Slide 23 Boardworks Ltd 2005 23 of 38 Cumulative frequency graph of results Slide 24 Boardworks Ltd 2005 24 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: Slide 25 Boardworks Ltd 2005 25 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? Slide 26 Boardworks Ltd 2005 26 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 Slide 27 Boardworks Ltd 2005 27 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 5066 0 < t 5585 0 < t 60100 We can plot a cumulative frequency graph as follows: Slide 28 Boardworks Ltd 2005 28 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. Slide 29 Boardworks Ltd 2005 29 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 of the way along the list. The upper quartile is the data item of the way along the list. interquartile range = upper quartile lower quartile Slide 30 Boardworks Ltd 2005 30 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 Slide 31 Boardworks Ltd 2005 31 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 Slide 32 Boardworks Ltd 2005 32 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 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 Slide 33 Boardworks Ltd 2005 33 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 Slide 34 Boardworks Ltd 2005 34 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 378 + 1 2 th value 190 th value Slide 35 Boardworks Ltd 2005 35 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 Slide 36 Boardworks Ltd 2005 36 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 Slide 37 Boardworks Ltd 2005 37 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 Slide 38 Boardworks Ltd 2005 38 of 38 Comparing sets of data Here are box-and-whisker diagrams representing James lap times and Shabnums lap times. Who is better and why? 52 53 545891 James lap times 5260546586 Shabnums lap times