Measures of Central Tendency Mean Median and Mode
- Slides: 12
Measures of Central Tendency Mean Median and Mode Compiled and Modified by Jigar Mehta (for not-for-profit educational purpose only)
Types of Data Discrete 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 that can have any value within a range For example Time A person running a 100 m race could finish at any time between 10 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. Data is grouped into 6 class intervals of width 10. 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 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 0 < x ≤ 10 27 5 10 < x ≤ 20 10 15 20 < x ≤ 30 7 30 < x ≤ 40 5 25 35 40 < x ≤ 50 4 45 180 50 < x ≤ 60 2 55 110 midpoint(x) fx 135 150 175 Mean estimate = 925/55 = 16. 8 minutes
Averages from Grouped Data 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. 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 Modal class = 0 - 10
Averages from Grouped Data 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 Total =55 So, 27+1+27 The 28 th 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. Data is grouped into 8 class intervals of width 4. 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 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 1 -5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1 midpoint(x) 3 8 13 18 23 28 33 38 fx 6 72 195 360 391 700 66 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. 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 Modal Class 26 - 30
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. 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 Total is 91. So, 45+1+45 The 46 th data value is in the 16 – 20 class
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. minutes Late frequency 0 - 10 27 10 - 20 10 20 - 30 7 30 - 40 5 40 - 50 4 50 - 60 2 midpoint(x) fx
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. number of laps frequency 1 -5 2 6 – 10 9 11 – 15 15 16 – 20 20 21 – 25 17 26 – 30 25 31 – 35 2 36 - 40 1 midpoint(x) fx
- In quartiles central tendency median is
- Central tendency and variation
- Measures of central tendency and variation
- Measures of central tendency
- Measure of central tendency grouped data
- Objectives of measures of central tendency
- Measures of central tendency range
- Measures of central tendency of ungrouped data
- Objectives of measures of central tendency
- Measures of central tendency
- How to work out cumulative frequency
- Measures of central tendency
- Advantages of central tendency