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KS4 Mathematics

KS4 Mathematics. D4 Moving averages and cumulative frequency. D4 Moving averages and cumulative frequency. Contents. A. D4.2 Plotting moving averages. A. D4.1 Moving averages. D4.3 Cumulative frequency. A. D4.4 Using cumulative frequency graphs. A. D4.5 Box-and-whisker diagrams. A.

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KS4 Mathematics

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  1. KS4 Mathematics D4 Moving averages and cumulative frequency

  2. D4 Moving averages and cumulative frequency Contents • A D4.2 Plotting moving averages • A D4.1 Moving averages D4.3 Cumulative frequency • A D4.4 Using cumulative frequency graphs • A D4.5 Box-and-whisker diagrams • A

  3. 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: Stop complaining! 5 2 0 40 4 5 2 0 0 0 0 0 0 0 35 1 6 0 0 0 3 3 2 1 0 0 0 0 0 0 0 1 1 They agree to give Tabina a prize if she can stop complaining for a whole week. Should she get a prize?

  4. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Groups of seven There are lots of groups of seven days in the data. 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?

  5. 5 2 0 4 0 4 5 2 0 0 0 0 0 0 0 3 5 1 6 0 0 0 3 3 2 1 0 0 0 0 0 0 0 1 1 The moving average We could calculate the mean for every group of seven. • How could this help us decide whether Tabina should get a reward? • How many of the means will be 0? • What method would you use to calculate the means? The means of each group of seven are collectively called a seven-point moving average.

  6. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Calculating a seven-point moving average The means (to 2 decimal places) for each of the 29 groups of 7 are as follows: 2.86 2.43 2.14 2.14 1.57 1.57 1.00 0.29 0.00 0.43 1.14 1.29 2.14 2.14 2.14 2.14 2.14 1.86 2.00 1.29 1.29 1.29 1.29 0.86 0.43 0.14 0.00 0.14 0.29 What can the moving average tell us about the general pattern of Tabina’s behaviour and whether she should win the prize?

  7. Moving averages

  8. D4 Moving averages and cumulative frequency Contents D4.1 Moving averages • A • A D4.2 Plotting moving averages D4.3 Cumulative frequency • A D4.4 Using cumulative frequency graphs • A D4.5 Box-and-whisker diagrams • A

  9. 7 6 5 4 Number of complaints 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Days 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?

  10. 7 6 5 4 Number of complaints 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Days 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:

  11. Range Method Position of mean on graph 1st – 7th 2nd – 8th 3rd – 9th … 29th – 35th Plotting moving averages on a time series graph When we plot the moving average, each mean is plotted halfway along the group that it represents. For our seven-point moving average we would have: (1 + 7) ÷ 2 Day 4 (2 + 8) ÷ 2 Day 5 (3 + 9) ÷ 2 Day 6 … … (29 + 35) ÷ 2 Day 32

  12. Class A 29 28 28 21 29 28 29 27 30 28 Class B 28 25 27 30 30 26 28 29 29 26 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.

  13. 28 29 29 29 29 28 29 28 28 28 28 28 25 28 28 28 25 25 25 25 27 27 28 28 27 28 28 27 27 28 21 21 30 30 21 30 21 30 30 21 29 29 30 30 30 29 30 29 29 30 28 26 26 28 28 26 26 26 28 28 29 29 28 29 28 28 29 28 28 29 27 29 27 29 27 27 29 27 29 29 30 30 30 29 29 29 29 29 30 30 28 26 28 28 28 26 26 26 26 28 29 28 25 28 28 27 21 30 30 29 28 26 29 28 29 27 30 29 26 28 28 29 28 25 27 28 21 30 30 29 28 26 28 29 27 29 30 29 28 26 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 27.0 26.8 27.0 26.8 28.6 28.4 Class B Means for class B 28.0 27.6 28.2 28.6 28.4 27.6

  14. Range Method Position of mean on graph 1st – 5th 2nd – 6th 3rd – 7th … Plotting a five-point moving average Each mean is then plotted halfway along the group that it represents. For a five-point moving average we have: (1 + 7) ÷ 2 4 (2 + 8) ÷ 2 5 (3 + 9) ÷ 2 6 … …

  15. 30 29 28 27 Attendance 26 25 24 23 22 21 20 1 2 3 4 5 6 7 8 9 10 Weeks Time series for class A

  16. 30 29 28 27 Attendance 26 25 24 23 22 21 20 1 2 3 4 5 6 7 8 9 10 Weeks Five-point moving average for class A

  17. 30 29 28 27 26 Attendance 25 24 23 22 21 20 1 2 3 4 5 6 7 8 9 10 Weeks Time series for class B

  18. 30 29 28 27 26 Attendance 25 24 23 22 21 20 1 2 3 4 5 6 7 8 9 10 Weeks Five-point moving average for class B

  19. Size of moving average Method Position of first mean on graph 3 4 5 6 7 8 Plotting the means for other moving averages We can find the positions of other moving averages as follows: (3 + 1) ÷ 2 2 (4 + 1) ÷ 2 2.5 (5 + 1) ÷ 2 3 (6 + 1) ÷ 2 3.5 (7 + 1) ÷ 2 4 (8 + 1) ÷ 2 4.5

  20. D4 Moving averages and cumulative frequency Contents D4.1 Moving averages • A D4.2 Plotting moving averages • A D4.3 Cumulative frequency • A D4.4 Using cumulative frequency graphs • A D4.5 Box-and-whisker diagrams • A

  21. Choosing class intervals Imagine you are going to record how long each member of your class can keep their eyes open without blinking. 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?

  22. Keeping your eyes open You will also have to decide what size class intervals to use. 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

  23. Cumulative frequency graph of results

  24. Time in seconds Frequency Cumulative frequency Time in seconds 30 < t ≤ 35 9 35 < t ≤ 40 12 40 < t ≤ 45 24 45 < t ≤ 50 28 50 < t ≤ 55 16 55 < t ≤ 60 11 Cumulative frequency Cumulative frequency is a running total. It is calculated by adding up the frequencies up to that point. Here are the results of 100 people holding their breath: 9 0 < t ≤ 35 9 + 12 = 21 0 < t ≤ 40 21 + 24 = 45 0 < t ≤ 45 45 + 28 = 73 0 < t ≤ 50 73 + 16 = 89 0 < t ≤ 55 89 + 11 = 100 0 < t ≤ 60

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

  26. D4 Moving averages and cumulative frequency Contents D4.1 Moving averages • A D4.2 Plotting moving averages • A D4.4 Using cumulative frequency graphs D4.3 Cumulative frequency • A • A D4.5 Box-and-whisker diagrams • A

  27. Cumulative frequency graphs Here is the cumulative frequency table for 100 people holding their breath: We can plot a cumulative frequency graph as follows:

  28. 100 90 80 70 60 Cumulative frequency 50 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds Plotting a cumulative frequency graph 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 50th person. This gives us a median time of 47 seconds.

  29. The interquartile range Remember; the rangeis 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 rangeis 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

  30. 100 90 80 70 60 Cumulative frequency 50 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds Finding the interquartile range The cumulative frequency graph can be used to locate the upper and lower quartiles, and so find the interquartile range. The lower quartile is the time of the 25th person. 42 seconds The upper quartile is the time of the 75th person. 51 seconds The interquartile range is the difference between these two values. 51 – 42 = 9 seconds

  31. D4 Moving averages and cumulative frequency Contents D4.1 Moving averages • A D4.2 Plotting moving averages • A D4.5 Box-and-whisker diagrams D4.3 Cumulative frequency • A D4.4 Using cumulative frequency graphs • A • A

  32. 100 90 80 70 60 50 Cumulative frequency 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds 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. 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

  33. Minimum value Median Maximum value Lower quartile Upper quartile 30 42 47 51 60 A box-and-whisker diagram The corresponding box-and-whisker diagram is as follows:

  34. th 378 + 1 value ≈ 2 Lap times James takes part in karting competitions and his Dad records his lap times on a spreadsheet. One of the karting tracks is at Shenington. In 2004, 378 of James’ lap times were recorded there. 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? There are 378 lap times and so the median lap time will be the 190th value

  35. th 3 × th value ≈ 378 + 1 378 + 1 value ≈ 4 4 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 95th 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 284th value

  36. 400 350 300 250 200 Cumulative frequency 150 100 50 0 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 Lap times in seconds Lap times at Shenington karting circuit James’ lap times are displayed in the following cumulative frequency graph.

  37. Box and whisker plot for James’ race times Minimum value Maximum value Lower quartile Median Upper quartile 52 54 58 91 53 What conclusions can you draw about James’ performance?

  38. James’ lap times 52 54 58 91 53 Shabnum’s lap times 52 54 60 65 86 Comparing sets of data Here are box-and-whisker diagrams representing James’ lap times and Shabnum’s lap times. Who is better and why?

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