Session – 7Measures of Dispersions

The measures of Central Tendency alone will not exhibit variouscharacteristics of the frequency distribution having the same total frequency. Twodistribution can have the same mean but can differ significantly. We need to knowthe extent of variation or deviation of the values in comparison with the central valueor average referred to as the measures of central tendency.

Measures of dispassion are the ‘average of second order’. The are based on theaverage of deviations of the values obtained from central tendencies x , Me or z. Thevariability is the basic feature of the values of variables. Such type of variation ordispersion refers to the ‘lack of uniformity’.

Definition: A measure of dispersion may be defined as a statistics signifying theextent of the scatteredness of items around a measure of central tendency.

Absolute and Relative Measures of Dispersion:

A measure of dispersion may be expressed in an absolute form, or in a relativeform. It is said to be in absolute form when it states the actual amount by which thevalue of item on an average deviates from a measure of central tendency. Absolutemeasures are expressed in concrete units i.e., units in terms of which the data havebeen expressed e.g.: Rupees, Centimetres, Kilogram etc. and are used to describefrequency distribution.

A relative measures of dispersion is a quotient by dividing the absolutemeasures by a quality in respect to which absolute deviation has been computed. It isas such a pure number and is usually expressed in a percentage form. Relativemeasures are used for making comparisons between two or more distribution.

Thus, absolute measures are expressed in terms of original units and they arenot suitable for comparative studies. The relative measures are expressed in ratios orpercentage and they are suitable for comparative studies.

Measures of Dispersion Types

Following are the common measures of dispersions.

a. The Range

b. The Quartile Deviation (QD)

c. The Mean Deviation (MD)

d. The Standard Deviation (SD)

Range‘Range’ represents the differences between the values of the extremes’. The

range of any such is the difference between the highest and the lowest values in theseries.

The values in between two extremes are not all taken into consideration. Therange is an simple indicator of the variability of a set of observations. It is denoted by‘R’. In a frequency distribution, the range is taken to be the difference between thelower limit of the class at the lower extreme of the distribution and the upper limit ofthe distribution and the upper limit of the class at the upper extreme. Range can becomputed using following equation.

Range = Large value – Small value

valueSmallvalueeargL

valueSmallvalueeargLRangeoftCoefficien

Problems

1. Compute the range and also the co-efficient of range of the given series of statewhich one is more dispersed and which is more uniform.

Series – I – 9, 10, 15, 19, 21 Series – II – 1, 15, 24, 28, 29

R = LV – SV = 21 – 9 = 12 R = LV – SV = 29 – 1 = 28

CR =30

12

921

12

= 0.4 CR =

30

28

SVLV

R

= 0.933

Series I is les dispersed and more uniform

Series II is more dispersed and less uniform

Evaluating Criteria

i. Less the CR is less dispersion

ii. More the CR is less uniform

Range Merits

i. It is very simplest to measure.

ii. It is defined rigidly

iii. It is very much useful in Statistical Quality Control (SBC).

iv. It is useful in studying variation in price of shars and stocks.

Limitations

i. It is not stable measure of dispersion affected by extreme values.

ii. It does not considers class intervals and is not suitable for C.I. problems.

iii. It considers only extreme values.

2. Find range of Co-efficient of range from following data.

A: 10 11 12 13 14

B: 40 41 42 43 44

C: 100 101 102 103 104

Series - I Series – II Series – III

R =LV – 3m

= 14 – 10

= 4

CR =SVLV

R

=24

4

= 0.166

R = 44 - 40

= 4

CR =SVLV

R

=84

4

= 0.0476

R = 104 - 100

= 4

CR =SVLV

R

=204

4

= 0.0196

Series III is less dispersed and more uniform

Series I is more dispersed and less uniform

3. Compute range and coefficient of range for the following data.

x: 6 12 18 24 30 36 42

f: 20 130 16 14 20 15 40

Range = LV – SV = 42 – 6 = 36

CR =SVLV

R

=

48

36= 0.75

Quartile DeviationQuartile divides the total frequency in to four equal parts. The lower quartile

Q1 refers to the values of variates corresponding to the cumulative frequency N/4.

Upper quartile Q3 refers the value of variants corresponding to cumulativefrequency ¾ N.

Quartile deviation is defined as QD =2

1(Q3 – Q1). In this quartile Q2 as it

corresponds to the value of variate with cumulative frequency is equal to c.f. =2

N.

a) QD =2

1(Q3 – Q1)

b) Relative measure of dispersion coefficient of QD =13

13

QQ

QQ

Problems

1. Find quartile deviation and coefficient of quartile deviation for the given groupeddata also compute middle quartile.

Class f

1 – 10 3

11 – 20 16

21 – 30 26

31 – 40 31

41 – 50 16

51 – 60 8

f = N = 100

Class f Cf

1 – 10 3 3

11 – 20 16 19

21 – 30 26 45 Q1 Class

31 – 40 31 76 Q2 & Q3 Class

41 – 50 16 92

51 – 60 8 100

N = 100

Q14

N= 25

4

100

Q1 =

C

4

N

f

h

Q1 = 192526

105.20

Q1 = 22.80

Q2 =

C

2

N

f

h

Q2 = 455031

105.30

Q2 = 32.11

Q3 =

CN4

3

f

h

Q3 = 457531

105.30

Q3 = 40.17

QD =2

1(Q3 – Q1) = 0.5 (Q3 – Q1)

=2

1(40.17 – 22.80)

= 8.685

Coef. QD =13

13

QQ

QQ

=80.2217.40

80.2217.40

97.62

37.17

= 0.275

2. Find quartile deviation from the following marks of 12 students and alsoco-efficient of quartile deviation.

Sl. No. Marks

1. 25

2. 30

3. 37

4. 43

5. 48

6. 54

7. 61

8. 67

9. 72

10. 80

11. 84

12. 89

Q1 = 3.25th item

= 3rd item + 0.25 of item

= 37 + 0.25 (43 - 37)

Q1 = 38.5

Q3 =9.75th item

= 9 + 0.75rd item

= 72 + 0.75 (80- 72)

Q3 = 78

QD =2

1(Q3 – Q1)

=2

1(78 – 38.3)

QD = 19.75

Coef. QD =13

13

QQ

QQ

=5.3878

5.3878

= 0.339

3. Compute quartile deviation. and its Coefficient for the data given below:

x f Cf

58 15 15

59 20 35

60 32 67 Q1 Class

61 35 102

62 33 135

63 22 157 Q3

Class

64 20 177

65 10 187

65 8 195

N = 195

Q1 = size4

1n th

= size4

1195 th

Q1 = 48.78th size and corresponding to cf 67, which gives

Q1 = 60

Q3 = size1n4

3 th

= .size33.1461964

3 thth

It lies in 157, cf. Against cf 157 Q3 = 63

QD =2

1(Q3 – Q1)

=2

1(63 – 60)

QD = 1.5

Coef. QD =13

13

QQ

QQ

=123

3

6063

6063

= 0.024

Merits of Quartile Deviation

It is very easy to compute

It is not affected by extreme values of variable.

It is not at all affected by open and class intervals.

Demerits of Quartile Deviation

It ignores completely the portions below the lower quartile and above the upper ofquartile.

It is not capable for further mathematical treatment.

It is greatly affected by fluctuations in the sampling.

It is only the positional average but not mathematical average.