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Statistical Measures 1

Statistical Measures 1. Measure of Central Tendency. Intro. Grouped Data.

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Statistical Measures 1

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  1. Statistical Measures 1 Measure of Central Tendency

  2. Intro Grouped Data Large quantities of data can be much more easily viewed and managed if placedingroupsin afrequency table. Grouped data does not enable exact values for the mean, median and mode to be calculated. Alternate methods of analyising the data have to be employed. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. number of laps frequency (x) This data is grouped into 8 class intervals of width 4. The data is discrete. 1 - 5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1

  3. Ex 1 Discrete Grouped Data Example 1. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the median class interval. number of laps frequency (x) 1 - 5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1

  4. Grouped Data Estimating the Mean:An estimate for the mean can be obtained by assuming that each of the raw data values takes the midpoint value of the interval in which it has been placed. number of laps frequency midpoint(x) mp x f 3 1 - 5 2 6 8 6 – 10 9 72 13 11 – 15 15 195 18 360 16 – 20 20 23 21 – 25 17 391 700 28 26 – 30 25 66 33 31 – 35 2 38 38 36 - 40 1 Mean estimate = 1828/91 = 20.1 laps

  5. Grouped Data Example 1. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the median class interval. number of laps frequency (x) 1 - 5 2 6 – 10 9 11 – 15 15 Modal Class 26 - 30 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1 The modal class is simply the class interval of highest frequency.

  6. Grouped Data Example 1. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class.  (c) Determine the median class interval. number of laps frequency (x) 1 - 5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 (91+1)/2 = 46 26 – 30 25 31 – 35 2 36 - 40 1 The 46th data value is in the 16 – 20 class  The Median Class Interval is the class interval containing the median.

  7. Ex 2 Continuous (a) minutes late frequency midpoint(x) mp x f 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2 Grouped Data < Slide 7 Example 2. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. (a) Calculate an estimate for the mean number of minutes late. (b) Determine the modal class. (c) Determine the class interval containing the median. This data is grouped into 6 class intervals of width 10. The data is continuous. 5 135 150 15 175 25 35 175 45 180 55 110

  8. minutes late frequency midpoint(x) mp x f 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2 Grouped Data < Slide 7 Example 2. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. (a) Calculate an estimate for the mean number of minutes late. (b) Determine the modal class. (c) Determine the class interval containing the median. This data is grouped into 6 class intervals of width 10. The data is continuous. Mean estimate = 925/55 = 16.8 minutes

  9. Modal class = 0 - 10 Grouped Data Example 2. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. (a) Calculate an estimate for the mean number of minutes late.  (b) Determine the modal class. (c) Determine the class interval containing the median. minutes late frequency 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2

  10. minutes late frequency 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2 The 28th data value is in the 10 - 20 class Grouped Data Example 2. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. (a) Calculate an estimate for the mean number of minutes late.  (b) Determine the modal class.  (c) Determine the class interval containing the median. (55+1)/2 = 28

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