Measures of Central Tendency

1. Measures of Central Tendency Mean Median and Mode Compiled and Modified by Jigar Mehta (for not-for-profit educational purpose only)

2. 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

3. Large quantities of data can be much more easily viewed and managed if placedin 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. minutes late (x) frequency 0 < x ≤ 10 27 10 < x ≤ 20 10 20 < x ≤ 30 7 30 < x ≤ 40 5 40 < x ≤ 50 4 50 < x ≤ 60 2 Averages from Grouped Data Data is grouped into 6 class intervals of width 10.

4. 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. 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. minutes Late frequency midpoint(x) fx 0 < x ≤ 10 27 10 < x ≤ 20 10 20 < x ≤ 30 7 30 < x ≤ 40 5 40 < x ≤ 50 4 50 < x ≤ 60 2 Averages from Grouped Data 5 135 150 15 175 25 35 175 45 180 55 110 Mean estimate = 925/55 = 16.8 minutes

5. The Modal Class 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. Modal class = 0 - 10 minutes late frequency 0 < x ≤ 10 27 10 < x ≤ 20 10 20 < x ≤ 30 7 30 < x ≤ 40 5 40 < x ≤ 50 4 50 < x ≤ 60 2 Averages from Grouped Data

6. The Median Class Interval The Median Class Interval is the class interval containing the median. 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. minutes late frequency 0 < x ≤ 10 27 10 < x ≤ 20 10 20 < x ≤ 30 7 30 < x ≤ 40 5 40 < x ≤ 50 4 50 < x ≤ 60 2 The 28th data value is in the 10 - 20 class Averages from Grouped Data Total =55 So, 27+1+27

7. 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. 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 Averages from Grouped Data Data is grouped into 8 class intervals of width 4.

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

9. 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. 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 Averages from Grouped Data

10. 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.  number of laps frequency (x) 1 - 5 2 Total is 91. So, 45+1+45 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1 The 46th data value is in the 16 – 20 class Averages from Grouped Data

11. Worksheet 1 minutes Late frequency midpoint(x) fx 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2 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.

12. Worksheet 2 number of laps frequency midpoint(x) fx 1 - 5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1 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.