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Example 1. 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. Grouped Data midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes Late 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. 5 135 15 150 25 175 35 175 45 180 55 110 Mean estimate = 925/55 = 16.8 minutes

Large quantities of data can be much more easily viewed and managed if placed in groups in a frequency 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. Example 1. 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. Grouped Data 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late Data is grouped into 6 class intervals of width 10.

The modal class is simply the class interval of highest frequency. Example 1. 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. Grouped Data The Modal Class 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late The modal class is simply the class interval of highest frequency. Modal class = 0 - 10

The 28th data value is in the 10 - 20 class Example 1. 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. Grouped Data The Median Class Interval The Median Class Interval is the class interval containing the median. 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late (55+1)/2 = 28 The 28th data value is in the 10 - 20 class

Example 2. 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 class interval containing the median. Grouped Data 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency (x) number of laps Data is grouped into 8 class intervals of width 4.

Grouped Data Mean estimate = 1828/91 = 20.1 laps Example 2. mp x f midpoint(x) 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency number of laps Example 2. 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 class interval containing the median. Grouped Data 3 6 8 72 13 195 18 360 23 391 28 700 33 66 38 38 Mean estimate = 1828/91 = 20.1 laps

Grouped Data Example 2. Modal Class 26 - 30 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 class interval containing the median. 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency (x) number of laps Modal Class 26 - 30

Grouped Data Example 2. (91+1)/2 = 46 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency (x) number of laps Example 2. 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 class interval containing the median. Grouped Data The 46th data value is in the 16 – 20 class  (91+1)/2 = 46

Averages From Grouped Data Example 1. 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. Averages From Grouped Data midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes Late 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. 15 25 35 45 55 135 150 175 180 110 Mean estimate = 925/55 = 16.8 minutes Modal Class (55 +1 )/2 = 28 Median Class Interval

Grouped Data Example 1. mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 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. midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes Late

Grouped Data Example 2. 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 class interval containing the median. mp x f midpoint(x) 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency number of laps