MEASURES OF LOCATION

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LECTURE 4

• MEASURES OF LOCATION

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• Mean
• Median
• Mode
• Untabulated Tabulated
• Ungrouped
  Grouped

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UNTABULATED DATA

• Mean - simple average
• Example 1
• 10 22 31 9 24 27 29 9 23 12
• Mean
• 10 22 31 9 24 27 29 9 23 12
• 10
• 196
• 10
• 19.6

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• Median - value in the middle when numbers
  have been listed either in ascending
  or descending order
• Example 1 (cont)
• Arranged in ascending order
• 9 9 10 12 22 23 24 27 29 31
• Median (n1)/2
• where n is the number of value s
• (101) /2
• 5.5

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• Median is between the numbers 22 and 23. Take
  average
• ( 2223)/2
• 22.5
• Mode - most frequent
• Example 1 (cont)
• It is obvious 9 occurs twice. The rest of the
  numbers occur only once. Therefore mode 9

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TABULATED DISCRETE DATA

• Example 2 Number of working days lost by
  employees in the last quarter
• Number of days Number of employees

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MEAN

• To obtain mean multiply the number of lost
  days (x) by frequency() and obtain sum of x.
  Divide sum of x by the number included (n)
• x x
• 0 410 0
• 1 430 430
• 2 290 580
• 3 180 540
• 4 110 440
• 5 20 100
• 1440 2090
• Mean 2090 1440 1.451 days lost

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Mean

• Obtain the highest frequency
• Example 2 (cont)
• The highest frequency is 430.
• Number of lost days associated with this
  frequency is 1
• Therefore the mode is 1.

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Median

• To find median obtain cumulative frequency
• x cumulative frequency
• 0 410 410
• 1 430 410 430 840
• 2 290 840 290 1130
• 3 180 1130 180 1310
• 4 110 1310 110 1420
• 5 20 1420 20 1440
• Median (n 1) 2
• (1440 1) 2
• 720.5
• Between 720 and 721 1 day lost

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TABULATED CONTINUOUS DATA

• Example 3
• Expenditure Household
• under 5 2
• 5 but under 10 6
• 10 but under 15 8
• 15 but under 20 12
• 20 but under 30 10
• 30 but under 40 4
• 40 or more 2
• 44

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• Mean (sum of x) n
• For continuous data use mid point.
• Expenditure Mid-point Household
• (x) () x
• 0 but under 5 2.50 2 5.00
• 5 but under 10 7.50 6 45.00
• 10 but under 15 12.50 8 100.00
• 15 but under 20 17.50 12 210.00
• 20 but under 30 25.00 10 250.00
• 30 but under 40 35.00 4 140.00
• 40 but under 50 45.00 2
  90.00
• 44 840.00
• Mean 840 44 19.09
• Assumed boundary

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Median

• Step 1 - find cumulative frequencies.
• Example 3 (cont)
• Expenditure Household Cumulative
  frequency()
• under 5 2 2
• 5 but under 10 6 8
• 10 but under 15 8 16
• 15 but under 20 12 28
• 20 but under 30 10 38
• 30 but under 40 4 42
• 40 or more 2 44
• Median 44 2
• 22

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• The median is 22nd respondent, under 15 but
  under 20 category.
• Median l i n 2 - F
• where l lower boundary of the median
• i width of the median group
• n number of values
• F cumulative frequency
• frequency in the median group
• Median 15 5 44 2 - 16
• 12
• 17.50

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• Mode.
• Mode l m - m -1 x i
• 2 m - m-1 - m1
• where l - lower boundary of the mode group
• m - scaled frequency of the modal group
• m-1 scaled frequency of the pre-modal group
• m1 - post modal group
• i - width of the modal group

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• Example 3 (cont)
• Mode 15 12-8 x5
• 2 x 12 - 8-5
• 15 4 x 5
• 11
• 16.82

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Histogram of distribution
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shape of distribution small sample
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shape of distribution Large sample
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Relationship between mean, median and mode
Mean Median Mode
Normal Distribution
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Positively skewed distribution
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Negatively skewed distribution
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Do question 6 in page 119 Break 20 minutes
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• MEASURE OF DISPERSION
• Standard deviation
• It measures difference from the mean
• a larger value indicates a larger measure
  of overall variation

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• Standard Deviation
• Undulated data
• Example 4
• Number of cars entering Tesco car park in 10
  minutes interval
• 10, 22, 31, 9, 24, 27, 29, 9, 23, 12
• See formula sheet for Standard Deviation formula

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• x (x-x) (x-x)2
• 10 -9.6 92.16
• 22 2.4 5.76
• 31 11.4 129.96
• 9 -10.6 112.36
• 24 4.4 19.36
• 27 7.4 54.76
• 29 9.4 88.36
• 9 -10.6 112.36
• 23 3.4 11.56
• 12 -7.6 57.76
• 196 684.40
• Mean 19.6 cars

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• Standard Deviation 684.4 10
• 8.27cars
• Do question 6 in page 140
• Independent study
• Do questions
• 7 in page 119
• 7 in page 140