1. Statistical Measures 1 Measure of Central Tendency

2. Intro 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. Grouped Data This data is grouped into 8 class intervals of width 4. The data is discrete. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps A group of University students took part in a sponsored race. The number of laps completed is given in the table below.

3. Ex 1 Discrete 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. Grouped Data 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps

4. mp x f midpoint(x) 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequencynumber of laps Grouped Data 3 8 13 18 23 28 33 38 6 72 195 360 391 700 66 38 Mean estimate = 1828/91 = 20.1 laps 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. Modal Class 26 - 30 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. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps The modal class is simply the class interval of highest frequency.

6. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps 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. Grouped Data The 46 th data value is in the 16 – 20 class The Median Class Interval is the class interval containing the median. (91+1)/2 = 46

7. Ex 2 Continuous (a) 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. Grouped Data midpoint(x) mp x f 250 - 60 440 - 50 530 - 40 720 - 30 1010 - 20 270 - 10 frequencyminutes late This data is grouped into 6 class intervals of width 10. The data is continuous. 5 15 25 35 45 55 135 150 175 180 110 < Slide 7

8. 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. Grouped Data midpoint(x) mp x f 250 - 60 440 - 50 530 - 40 720 - 30 1010 - 20 270 - 10 frequencyminutes late Mean estimate = 925/55 = 16.8 minutes < Slide 7 This data is grouped into 6 class intervals of width 10. The data is continuous.

9. Grouped Data 250 - 60 440 - 50 530 - 40 720 - 30 1010 - 20 270 - 10 frequencyminutes late Modal class = 0 - 10 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.

10. ( 55+1)/2 = 28 Grouped Data 250 - 60 440 - 50 530 - 40 720 - 30 1010 - 20 270 - 10 frequencyminutes late The 28 th data value is in the 10 - 20 class 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.