Measures of skewness-measure of central tendency, Mathematics

Measures Of Skewness

- These are numerical values such assist in evaluating the degree of deviation of a frequency distribution from the general distribution.

- Given are the normally used measures of skewness.

1. Coefficient Skewness

=  3 * {(mean - median)/standard deviation}

2. Coefficient of skewness

= (mean - median)/standard deviation

NB: These 2 coefficients above are also termed as Pearsonian measures of skewness.

1. Quartile Coefficient of skewness

=  (Q3 + Q1 - 2Q2)/(Q3 + Q1)

Whereas: Q1 = 1st quartile

            Q2 = 2nd quartile

            Q3 = 3rd quartile

NB: The Pearsonian coefficients of skewness generally range between -ve (negative) 3 and +ve (positive) 3. These are extreme value that is +ve (positive) 3 and -ve(negative)  3 which hence indicate that a given frequency is negatively skewed and the amount of skewness is quite high.

Correspondingly if the coefficient of skewness is +ve or positive it can be concluded that the amount of skewness of deviation from the general distribution is quite high and the degree of frequency distribution also is positively skewed.

Illustration

The given information was acquired from an NGO which was providing small loans to some small scale business enterprises in year 1996 the loans are in the form of thousands of Kshs.

Loans

Units (f)

Midpoints(x)

x-a=d

d/c= u

fu

Fu2

UCB

cf

46 - 50

32

48

-15

-3

-96

288

50.5

32

51 - 55

62

53

-10

-2

-124

248

55.5

94

56 - 60

97

58

-5

-1

-97

97

60.5

191

61 -65

120

63       (A)

0

0

0

0

0

0

66 -70

92

68

5

+1

92

92

70.5

403

71 -75

83

73

10

+2

166

332

75.5

486

76 - 80

52

78

15

+3

156

468

80.5

538

81 - 85

40

83

20

+4

160

640

85.5

57.8

86 - 90

21

88

25

+5

105

525

90.5

599

91 - 95

11

93

30

+6

66

396

95.5

610

Total

610

 

 

 

428

3086

 

 

Required

By using the Pearsonian measure of skewness, estimate the coefficients of skewness and thus comment briefly on the nature of the distribution of the loans.

Arithmetic mean = Assumed mean +1090_Measures Of Skewness.png

= 63 +  {(428 * 5)/610}

= 66.51

This is very significant to note that the method of acquiring arithmetic mean or any other statistic by  misusing assumed mean (A) from X and then dividing by c can be a bit confusing, if this is the case then just employ the straight forward method of:

                         476_Measures Of Skewness 1.png

=5 × 1335_Measures Of Skewness 2.png

= 10.68

The Position of the median lies m =  (n + 1)/2

=   (610 + 1)/2= 305.5

= 60.5 +  {(305.5 - 191 )/120} × 5

= 60.5 +  (114.4/120) × 5

Median = 65.27

Hence the Pearsonian coefficient

            = 3 * {(66.51 - 64.27)/10.68}

            = 0.348

Posted Date: 2/18/2013 1:24:55 AM | Location : United States







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