Types of Data Discrete Data Data that can only have a specific value (often whole numbers) For example Number of people You cannot have ½ or ¼ of a person. Shoe size You might have a 6½ or a 7 but not a size 6.23456 Continuous Data Data that can have any value within a range For example Time A person running a 100m race could finish at any time between10 seconds and 30 seconds with no restrictions Height As you grow from a baby to an adult you will at some point every height on the way

Averages from Grouped Data 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. 2 50 < x ≤ 60 4 40 < x ≤ 50 5 30 < x ≤ 40 7 20 < x ≤ 30 10 10 < x ≤ 20 27 0 < x ≤ 10 frequency minutes late (x) Data is grouped into 6 class intervals of width 10.

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. midpoint(x) mp x f 2 50 < x ≤ 60 4 40 < x ≤ 50 5 30 < x ≤ 40 7 20 < x ≤ 30 10 10 < x ≤ 20 27 0 < x ≤ 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

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

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

Averages from 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. 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.

Averages from Grouped Data 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. 3 6 8 72 13 195 18 360 23 391 28 700 33 66 38 38 Mean estimate = 1828/91 = 20.1 laps

Averages from 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. 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

Averages from 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 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. The 46th data value is in the 16 – 20 class  (91+1)/2 = 46

Averages from Grouped Data Worksheet 1 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. 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

Averages from Grouped Data Worksheet 2 Averages from 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