Measures of the Shape of Distribution

The frequency distribution can be described by the following four characteristics:

  1. The ‘central value’ in the distribution around which the observations tend to lie, which is described by the ‘measures of central tendency’,
  2. The ‘dispersion’ i.e., the extent to which the observations are spread out from the central value, which is described by the ‘measures of dispersion’,
  3. The manner in which the observations are distributed around the central values, i.e., whether the distributions is ‘symmetrical’ or ‘skewed’, which is described by the ‘measures of skewness’, and
  4. The ‘peakedness’ or ‘flatness’ of the distribution which is measured relative to a distribution known as ‘normal distribution’, which is described by the ‘measures of kurtosis’.

All these four characteristics can be described by what are known as ‘moments’.

Moments:

Moments are the AM of the powers to which the deviations are raised.  Thus, the mean of the first power of the deviations from mean is the ‘first moment about mean’, the mean of the second power (squares) of the deviations from mean is the ‘second moment about mean’, and so on.

(a) Moments about Mean:

Symbolically, the first four ‘moments about mean’ (denoted by m1, m2, m3 and m4) are defined as:

Grouped Data

Ungrouped Data

(b) Moments about an Arbitrary Value A:

The first four ‘moments about an arbitrary value A’ are defined as follows: (Raw moment)

Grouped Data

Ungrouped Data

(c) Moments about Zero / Origin:

The first four ‘moments about zero / origin’ are defined as follows: (Raw moment)

Grouped Data

Ungrouped Data

It should be noted here that in ‘moments about zero’, the arbitrary value is assumed to be 0.  Further, the 1st moment about 0 is the AM.

Example:

Class Interval

Frequency

10-19

5

20-29

8

30-39

13

40-49

19

50-59

23

60-69

15

70-79

7

80-89

5

90-99

3

100-109

2

Total

100

Calculate:

(a)    Moments about origin

(b)   Moments about mean

(c)    Moments about the value 4

Solution:

(a)   Moments about origin:

C.I

f

x

fx

x2

fx2

x3

fx3

x4

fx4

10-19

5

14.5

72.5

210.25

1051.25

3048.62

15243.12

44205.06

221025.31

20-29

8

24.5

196

600.25

4802

14706.13

117649

360300.06

2882400.5

30-39

13

34.5

448.5

1190.25

15473.25

41063.62

533827.13

1416695.06

18417035.81

40-49

19

44.5

845.5

1980.25

37624.75

88121.13

1674301.37

3921390.06

74506411.19

50-59

23

54.5

1253.5

2970.25

68315.75

161878.62

3723208.38

8822385.06

202914856.44

60-69

15

64.5

967.5

4160.25

62403.75

268336.13

4025041.87

17307680.06

259615200.94

70-79

7

74.5

521.5

5550.25

38851.75

413493.62

2894455.38

30805275.06

215636925.44

80-89

5

84.5

422.5

7140.25

35701.25

603351.13

3016755.62

50983170.06

254915850.31

90-99

3

94.5

283.5

8930.25

26790.75

843908.62

2531725.88

79749365.06

239248095.19

100-109

2

104.5

209

10920.25

21840.5

1141166.13

2282332.25

119251860.06

238503720.12

Total

100

 

5220

 

312855

 

20814540

 

1506861521.25

(b)   Moments about mean:

C.I

f

x

10-19

5

14.5

-37.7

-188.5

1421.29

7106.45

-3048.62

-15243.1

2020065.26

10100326.3

20-29

8

24.5

-27.7

-221.6

767.29

6138.32

-14706.13

-117649.04

588733.94

4709871.52

30-39

13

34.5

-17.7

-230.1

313.29

4072.77

-41063.62

-533827.06

98150.62

1275958.06

40-49

19

44.5

-7.7

-146.3

59.29

1126.51

-88121.13

-1674301.47

3515.30

66790.7

50-59

23

54.5

2.3

52.9

5.29

121.67

161878.62

3723208.26

27.98

643.54

60-69

15

64.5

12.3

184.5

151.29

2269.35

268336.13

4025041.95

22888.66

343329.9

70-79

7

74.5

22.3

156.1

497.29

3481.03

413493.62

2894455.34

247297.34

1731081.38

80-89

5

84.5

32.3

161.5

1043.29

5216.45

603351.13

3016755.65

1088454.02

542270.1

90-99

3

94.5

42.3

126.9

1789.29

5367.87

84908.62

254725.86

3201558.70

9604676.1

100-109

2

104.5

52.3

104.6

2735.29

5470.58

1141166.13

2282332.26

7481811.38

14963622.76

Total

100

 

 

0

 

40371

 

13855498.65

 

43338570.36

 

Where

(c)   Moments about the value 4:

C.I

f

x

D =

(x – A)

fD

D2 =

(x – A)2

fD2

D3 =

(x – A)3

fD3

D4 =

(x – A)4

fD4

10-19

5

14.5

10.5

52.5

110.25

551.25

1157.62

5788.1

12155.06

60775.3

20-29

8

24.5

20.5

164

420.25

3362

8615.12

68920.96

176610.06

1412880.48

30-39

13

34.5

30.5

396.5

930.25

12093.25

28672.62

672744.06

865365.06

11249745.78

40-49

19

44.5

40.5

364.5

1640.25

31164.75

66430.12

1262172.28

2690420.06

51117981.14

50-59

23

54.5

50.5

1161.5

2550.25

58655.75

128787.62

2962115.26

6503775.06

149586826.38

60-69

15

64.5

60.5

907.5

3660.25

54903.75

221445.12

3321676.8

13397430.06

200961450.9

70-79

7

74.5

70.5

493.5

4970.25

34791.75

350402.62

2452818.34

24703385.06

172923695.42

80-89

5

84.5

80.5

402.5

6480.25

32401.25

521660.12

2608300.6

41993640.06

209968200.3

90-99

3

94.5

90.5

271.5

8190.25

24570.75

741217.62

2223652.86

67080195.06

201240585.18

100-109

2

104.5

100.5

201

10100.25

20200.5

1015075.12

2030150.24

102015050.06

204030100.12

Total

100

 

 

4415

 

2272495

 

17608339.5

 

1202552241

Symmetry and Skewness:

  1. A frequency distribution is said to be symmetrical if the values equidistant from a central maximum have the same frequency.
  2. In a symmetrical distribution, a deviation below the mean is equal to the corresponding deviation above the mean.  This is called ‘symmetry’.
  3. ‘Skewness’ is the lack of ‘symmetry’ in a distribution around some central value, i.e., mean, median or mode.  It is, thus, the ‘degree of asymmetry’.
  4. When a distribution departs from symmetry, the mean, median and mode are pulled apart and one tail becomes longer than the other.
  5. If the frequency curve has a longer tail to the right, the distribution is to be positively skewed.
  6. If the frequency curve has a longer tail to the left, the distribution is said to be negatively skewed:

Measures of Skewness:

  1. To measure skewness is to measure the extent to which and also the direction in which the distribution (or curve) is non-symmetrical or skewed.
  2. In a symmetrical distribution, the mean, median and mode coincide.  In a skewed distribution, these values pulled apart.
  3. There three measure of skewness:

(a)    Absolute measure of skewness, i.e., the difference between the mean and mode.  In a moderately skewed distribution, the empirical relation between * , ,  is:

Therefore  or

(b)   The difference between the distances (or differences) of Q3 and Q2, and Q2 and Q1:

or

(c)    The third order moment about mean:

 or

Relative Measures of Skewness:

There are three relative measures of skewness:

(a)   Pearson’s 1st and 2nd Coefficients of Skewness:

1.      If we divide the absolute measure of skewness (i.e., the difference between *  and  by SD, we get a relative measure of skewness:

2.      If we employ the empirical relation between * ,  and , we get the following alternate formula:

3.      If the above formulae give positive results, it means the distribution is positively skewed and vice versa.  For a symmetrical distribution, the measure will be equal to 0.

(b)   Bowley’s Measure of Skewness:

1.      This measure of skewness is true for a symmetrical distribution.  In a symmetrical distribution, the quartiles (i.e., Q1, Q2, Q3 and Q4) are equidistant from the median, i.e., (Q3 – Q2) = (Q2 – Q1):

2.      This measure will always be equal to zero for a symmetrical distribution.  It is positive for positively skewed distribution and negative for negatively skewed distribution.  This measure varies between – 1 to + 1.

(c)    Moment Coefficient of Skewness:

1.      In a symmetrical distribution, the sum of odd powers of deviations from mean is zero.  Thus, the odd order moments about mean, i.e., m1, m3, etc., in a symmetrical distribution are zero.

2.      This measure of skewness is true in case of a skewed distribution:

3.      Or, alternatively:

For a symmetrical distribution α3 and β1 will be 0.

Example:

C.I

f

x

fx

10-19

5

14.5

72.5

20-29

8

24.5

196

30-39

13

34.5

448.5

40-49

19

44.5

845.5

50-59

23

54.5

1253.5

60-69

15

64.5

967.5

70-79

7

74.5

521.5

80-89

5

84.5

422.5

90-99

3

94.5

283.5

100-109

2

104.5

209

Total

100

 

5220

Calculate:

(a)    Pearson’s first and second coefficient of skewness,

(b)   Quartile coefficient of skewness, and

(c)    Moment coefficient of skewness.

Solution:

(a)   Pearson’s first and second coefficient of skewness:

 ------------- Pearson’s first coefficient of skewness

 --- Pearson’s second coefficient of skewness

Comments: Negatively skewed distribution.

(b)   Quartile coefficient of skewness:

Comments: Asymmetrical distribution – negatively skewed distribution.

(c)    Moment coefficient of skewness:

Kurtosis:

  1. Kurtosis is the degree of peakedness of a distribution usually taken relative to a normal distribution.
  2. A distribution having a relatively high peak is called ‘leptokurtic’.
  3. A distribution which plat topped is called ‘platykurtic’.
  4. A normal distribution which is neither very peaked nor very flat-topped is also called ‘mesokurtic’.

Measures of Kurtosis:

  1. Two frequency distribution both symmetrical having same means and SDs may be different in ‘flatness’ of the top of their curves.  The flatness of the top of a frequency curve is called the ‘kurtosis’.
  2. It is measured by a quantity denoted by β2 where:

  1. If β2 = 3, it is mesokurtic or normal,

If β2 > 3, it is leptokurtic, and

If β2 < 3, it platykurtic.

Note: μ, β and α are used for population data, and , b and a for sample data.

Example:

Given (moments about mean): m1 = 0, m2 = 1.25, m3 = 0.267, m4 = 2.264

Find the coefficient of kurtosis β2 and comment on the flatness of the distribution.

Solution:

Since β2 is less than 3, therefore, the distribution is Platykurtic.

Relation between Moments:

or

We have

Since ,

and

Example:

Convert the following moments about origin into moment about mean:

Solution:

Top

Home Page