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

Measures of Central Tendency

Mean Median and Mode

Compiled and Modified by Jigar Mehta(for not-for-profit educational purpose only)

Types of DataTypes 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 ofa person.

Shoe size You might have a 6½ or a 7but 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 youwill at some point every height on the way

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.

250 < x ≤ 60

440 < x ≤ 50

530 < x ≤ 40

720 < x ≤ 30

1010 < x ≤ 20

270 < x ≤ 10

frequencyminutes late (x)Data is grouped into 6 class intervals of width 10.

Averages from Grouped DataAverages 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) fx

250 < x ≤ 60

440 < x ≤ 50

530 < x ≤ 40

720 < x ≤ 30

1010 < x ≤ 20

270 < x ≤ 10

frequencyminutes 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

15

2535

45

55

135

150

175

175

180

110

Mean estimate = 925/55 = 16.8 minutes

55f 925f x

Averages from Grouped DataAverages 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

250 < x ≤ 60

440 < x ≤ 50

530 < x ≤ 40

720 < x ≤ 30

1010 < x ≤ 20

270 < x ≤ 10

frequencyminutes late

The modal class is simply the class interval of highest frequency.

Modal class = 0 - 10

Averages from Grouped DataAverages from Grouped Data

Total =55So,

27+1+27

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.

250 < x ≤ 60

440 < x ≤ 50

530 < x ≤ 40

720 < x ≤ 30

1010 < x ≤ 20

270 < x ≤ 10

frequencyminutes late

The 28th data value is in the 10 - 20 class

Averages from Grouped DataAverages from Grouped Data

Data is grouped into 8 class intervals of width 4.

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.

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequency (x)

number of laps

Averages from Grouped DataAverages from Grouped Data

fxmidpoint(x)

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequencynumber 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

8

1318

23

2833

38

6

72195360

39170066

381828fx 91f

Mean estimate = 1828/91 = 20.1 laps

Averages from Grouped DataAverages from Grouped Data

Modal Class 26 - 30

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.

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequency (x)

number of laps

Averages from Grouped DataAverages from Grouped Data

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 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.

Total is 91. So,

45+1+45

The 46th data value is in the 16 – 20 class

Averages from Grouped DataAverages 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) fx

250 - 60

440 - 50

530 - 40

720 - 30

1010 - 20

270 - 10

frequencyminutes Late

Averages from Grouped DataAverages from Grouped DataWorksheet 1

fxmidpoint(x)

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequencynumber 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.

Averages from Grouped DataAverages from Grouped DataWorksheet 2