J. K. SHAH CLASSES Measures of Dispersion€¦ · Different central tendency and different dispersion Types of Measures of Dispersion There are two types of measures of dispersion,

J. K. SHAH CLASSES Measures of Dispersion

3. Measures of Dispersion

(Average of Second Order)

MEASURES OF DISPERSION

Introduction:

• Dispersion is defined as deviation or scattering of values from their central values i.e,

average (Mean, Median or Mode but preferably Mean or Median)

• Dispersion discovers variability in uniformity.

• In other words, dispersion measures the degree or extent to which the values of a

variable deviate from its average

• Dispersion indicates the degree of heterogeneity between the variables and as

heterogeneity increases dispersion increases

• If all values are equal then any measure of dispersion is always zero

• All measures of dispersion are positive

• All measures of dispersions are independent of the change of origin but dependent on

the change of scale

• All pre requisites of a good measure of central tendency are equally applicable for

good measure of dispersion

• Two distributions may have;

i. Same central tendency and same dispersion

ii. Different central tendency but same dispersion

iii. Same central tendency but different dispersion

iv. Different central tendency and different dispersion

Types of Measures of Dispersion

There are two types of measures of dispersion,

Absolute Measure Relative Measure

a. These measures of dispersion will

have the same units as those of the

variables

a. These are usually expressed as ratios

or percentages and hence unit free

b. Absolute measures are related to the

distribution itself.

b. Relative measures are used

i) to compare variability

between two or more

series.

ii) To check the relative

accuracy of the data

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MEASURES OF DISPERSION (AVERAGE OF SECOND ORDER)

A good measure of dispersion should obey conditions similar to those for a satisfactory

average and are as follows :

i. It should be rigidly defined.

ii. It should be based on all observations.

iii. It should be readily comprehensible.

iv. It should be fairly easily calculated.

v. It should affected as little as possible by fluctuations of sampling;

vi. It should readily lend itself to algebraic treatment and

vii. It should be east affected by the presence by extreme values

Measure of Dispersion

Absolute Relative

Range Quartile Mean Standard Coefficient Coefficient Coefficient Coefficient

Deviation Deviation Deviation of of of of

or Or Range Quartile Mean Variation

Semi Inter Mean Deviation Deviation

Quartile Absolute

Range Deviation

RANGE

• It is the quickest measure, of finding out Dispersion

• It does not depend on all observations

• It’s a crude method of finding out dispersion and most unreliable

• Range is unaffected by the presence of frequency

• Range is independent of the change of origin but dependent on change of scale

• If y=a±bx

R(y)=|b| ×R(x)

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Calculation Of Range:

• For simple series and simple Frequency Distribution :

Range = Highest Value – Lowest Value (H – L).

• For grouped frequency distribution:

o Range = Upper boundary of last class – Lower boundary of 1st class

o Range = Upper Limit of last class – Lower limit of 1st class + 1

• Co-efficient of Range (Relative Range)= H - L x 100H + L

Quartile Deviation or Semi-inter quartile Range:

• QD is defined as the half of the range between the quartiles

• It is based on the upper and the lower Quartile and covers 50% of the observations.

• It does not depend on all observations

• For distributions with the Open Ends Q.D is the best measure of dispersion

• QD is independent of the change of Origin but dependent on the change of Scale.

• If y=a±bx

QD( y)=|b| ×QD(x)

• Quartile Deviation (QD) = 3 1Q - Q

2, Where Q3 is the upper quartile and Q1 is the lower

quartile.

• Co-efficient of QD(Relative Measure) =

3 1

3 1

2 2

Q Q

Q QQD 2 x 100 = x 100 = x 100Median Q 2Q

−−

• For symmetrical distribution; 1 3

2

Q QQ

2

+= , i.e., median is the average of two extreme

quartiles.

Thus coefficient of QD for symmetrical distribution =

3 1

3 1

3 1 3 1

Q Q

Q Q2 x 100 = x 100Q Q Q Q

2

−−

+ +

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Mean Deviation / Mean Absolute Deviation

• It is based on all observations and hence it provides much better dispersion than

Range and Quartile Deviation

• Mean deviation of a set of values of a variance is defined as the AM of the Absolute

Deviation taken about Mean, Median or Mode.(Preferably AM or Median)

• Absolute Deviation implies Deviation without any regard to sign

• If nothing is specified Mean Deviation will imply Deviation about AM only.

• Since sum of Deviations is least when Deviations are taken about Median hence MD

about Median will have the least value.

• MD is the independent of the change of origin but dependent on the change of scale

• If y=a±bx

MD( y)=|b| ×MD(x)

• Formula to calculate Mean Deviation:

Simple Series Simple / Grouped Frequency

Distribution

x xMD =

n

−∑

f x xMD =

f

−∑∑

xMD =

n

M−∑ x

MD = f

f M−∑∑

Where n = number of observation

∑f=N = Total frequency

x =A.M

M = Median

X=Either actual values of the variables or mid values if it a group frequency

distributions

o Coefficient of MD(Relative Measure) = MD

x 100Mean/Median

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Standard Deviation

• It is the best measure and the most commonly used Measure of Dispersion.

• It takes into consideration the magnitude of all the observations and gives the

minimum value of dispersion possible.

• SD has all the pre-requisites of a good measure of dispersion, except the fact that it

gets unduly affected by the presence of extreme values,

• It is also known as Root Mean Square Deviation about mean.

• It is denoted by σ

• SD2 = Variance= σ2

• If all observations are equal variance =SD=0

• SD is the independent of the change of origin but dependent on the change of scale

• If If y=a±bx

SD( y)=|b| ×SD(x)

V(y)=b2×v(x)

Definition of SD:

• SD of a set of values of a variable is defined as the positive Square Root of the AM of

the Square of Deviations of the values from their AM( x )

• Thus, SD is also known as Root -Mean –Square- Deviations(RMSD)

Calculation of SD

Simple Series(Without Frequency) Simple /Grouped Frequency Distribution

i)( )2x x

nσ

−=∑

i)( )2f x x

fσ

−=∑∑

ii)σ =

22x x

n n

−

∑ ∑ ii)σ =

22

fx fx

f f

−

∑ ∑∑ ∑

iii)

22d d

xi

n nσ

= − ×

∑ ∑ iii)

22

fd fd

f fx iσ

= − ×

∑ ∑∑ ∑

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• Where, x A

di

−= ,

x= mid-values if it is a grouped frequency distribution or original values if it is a

discrete series

A = Assumed Mean i.e., a value arbitrarily chosen from mid-values or any other

value.

i = class width or any arbitrary value

Note1 : Use form i) when you find that x is whole number

Note2 : Use form ii) when the value of the variable x are small

Note3 : Use Form iii) when you find that the values of x are large x is not a whole

number( usually to be used for grouped frequency distribution)

Useful Results:

• SD of two numbers is the half of their absolute difference(Range), i.e., if numbers are

a and b, then SD = a b

2

−

• Variance of first “n” natural numbers (1, 2, 3, …., n) is 2n 1

12

−

• Sum of the aquares of observations ∑x2=n(σ2+ x 2)

Formula for combined or composite or pooled S.D. of two groups

Group I Group II

Numbers n1 n2

Mean 1x `

2x

Standard Deviation 1σ

2σ

• Step 1 – Find Combined Mean: 1 1 2 2

1 2

n x n xx

n n

+=

+

• Step 2 – Find Deviations : 1 1 2 2 d x x d x x= − = −

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Step 3 – Use Formula: 2 2 2 2

2 1 1 2 2 1 1 2 2

1 2

n n n d n d

n n

σ σσ

+ + +=

+

• Coefficient of Variation (C.V)(Relative Measure) = SD

x 100= 100Mean x

σ×

• C.V is the best relative measure of dispersion

• C.V is used to compare variability or consistency between 2 or more series

• More C.V implies more variability indicating thereby less stability or consistency and vice versa.

• Regarding choice of an item always choose that item which has less C.V, because the item with lower C.V is more stable.

Range

1. Find the range and the coefficient of range of the weights of 10 students from the

following data: 41, 20, 15, 65, 73, 84, 53, 35, 71, 55.

(a) 69, 69.69 (b) 68, 69.09 (c) 69, 66.09 (d) None of these

2. Find the Range of the daily wages (in `̀̀̀) of 10 persons: 24, 18, 25, 16, 20, 28, 22, 17,

21, 27.

a) `̀̀̀ 11 (b) ` ` ` ` 12 (c) `̀̀̀ 13 (d) `̀̀̀ 14

3. Find out the range of the following data:

Height No. of students

60 – 62 8

63 – 65 27

66 – 68 42

69 – 71 18

72 – 74 5

(a) 14 (b) 15 (c) 14.5 (d) 15.5

4. If the relationship between x and y is given by 3x + 2y = 13, and the range of x is 12,

what would be the range of y.

(a) 15 (b) 16 (c) 17 (d) 18

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5. If the range of x is 2, what would be the range of – 3x + 50?

(a) 6 (b) 5 (c) 3 (d) 2

6. If RX and RY denotes range of x and y respectively where x and y are related by

3x + 2y + 10 = 0, what would be the relation between RX and RY?

a) RX = RY

b) 2RX = 3RY

c) 3RX = 2RY

d) RX = 2RY

Quartile Deviation

7. For a symmetrical distribution Q1 = 20 and Q3 = 40; the median is equal to:

a) 25 (b) 30 (c) 35 (d) None of the above

8. Find the quartile deviation of the following distribution:

Height Number of students

60 4

62 10

64 18

66 26

68 20

70 12

72 5

a) 1.5

b) 2.5

c) 2

d) 3

9. In a frequency distribution, the three quartiles are 25, 50 and 75. What is the value of quartile deviation and its coefficient?

a) 25, 0.667

b) 25, 0.533

c) 25, 0.50

d) 25, 0.475

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10. If the QD of x is 8 and 3x + 6y = 15, what is the QD of y.

(a) 1 (b) 2 (c) 3 (d) 4

11. If the quartile deviation of x is 3 and 3x + 6y = 20, what is the quartile deviation of y?

a) 3

b) 3.5

c) 2

d) 1.5

Mean Deviation

12. The marks obtained by 10 students in an examination were as follows: 70, 65, 68, 70,

75, 73, 80, 70, 83, 86. Find the deviation about the mean.

a) 5.6,

b) 6.5

c) 5.5

d) 6.6

13. Find the mean deviation about A.M. of the first ten natural numbers.

a) 3

b) 4

c) 3.5

d) 2.5

14. The mean deviation about the AM for the following data is:

Daily Wages No. of workers

8 – 11 5

12 – 15 11

16 – 19 20

20 – 23 10

24 – 27 4

a) 3

b) 3.21

c) 4

d) 4.56

15. The M.D. of the variates 10, 15, 20, 25, …., 85 over A.M. is …… nearly

(a) 20 (b) 42 (c) 40.7 (d) 38.7

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16. Find the mean deviation about median for the following data: 8, 15, 53, 49, 19, 62, 7,

15, 95, 77.

a) 26

b) 27

c) 27.2

d) 27.8

17. Find the mean deviation from arithmetic mean and median for the following set of

Data : 20, 23, 33, 44, 46, 50, 50. Which mean deviation is smaller for the set? What

general principle does it illustrate?

18. If a relationship between x and y is given by: 2x + 3y = 10 and MD of x is 15, what is

the MD of y?

(a) 30 (b) 20 (c) 10 (d) 15

19. If x and y are related as 4x + 3y + 11 = 0 and mean deviation of x is 4.80, what is the

mean deviation of y?

a) 3.60

b) 6.40

c) 4.60

d) 7.20

Standard Deviation

20. Find the standard deviation of 16, 13, 17 and 22.

(a) 3.24 (b) 3.20 (c) 3 (d) None of these

21. If n = 10, AM = 12, Sum of square of numbers = 1530, find the coefficient of variation.

(a) 20% (b) 25% (c) 30% (d) None of these

22. What is the standard deviation of the two values 10 and 7?

a) 17 (b) 3 (c) 1.5 (d) none of the above

23. Standard Deviation of two quantities is 3. If one of the number is 7, find the other.

a) 1

b) 13

c) either a) or b)

d) none of the above

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24. What is the standard deviation of six numbers: 7, 7, 7, 9, 9, 9.

a) 1

b) 2

c) 0

d) 0.5

25. For 2 values of a variable x, the mean and the standard deviation are 10 and 2

respectively. What are the values of x?

a) 7, 13

b) 8, 12

c) 10, 10


26. For a set of ungrouped values the following sums are found: N = 15, X∑ = 480, 2X∑ =

15735. Find the mean and standard deviation of the values.

a) 30, 3

b) 31, 5

c) 32, 5

d) 32, 3

27. The variance of first n positive integers is:

a) 21

12

n −

b) 21

6

n −

c) 21

12

n +

d) None of the above

28. The standard deviation of 1, 2, 3, ….., n is 14 ; find n.

a) 11

b) 12

c) 13

d) 14

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29. After settlement the average weekly wage in a factory has increased from 8000 to

12000 and the standard deviation had increased from `̀̀̀ 150 to 175. After settlement

the wage has become higher and more uniform. Do you agree?

(a) Yes (b) No (c) Cannot say (d) Data Insufficient

30. The arithmetic mean of the runs scored by three bastman, Amit, Sumeet and Kapil in

the series are 50, 48, and 12 respectively. The standard deviations of their runs are

respectively 15, 12 and 2. Who is the most consistent of the three? If one of the three

is to be selected, who will be selected?

(a) Amit (b) Sumeet (c) Kapil (d) All

31. An analysis of the monthly wages paid to workers in two firms A and B, belonging to

the same industry, gives the following results :

Firm A Firm B

Number of wage-earners 586 648

Average monthly wage `̀̀̀ 52.5 `̀̀̀ 47.5

Variance of the distribution of wage 100 121

I) In which firm is the total expenditure on wage is more?

II) In which firm is the variation in wage more?

(a) A, B (b) B, A (c) A, A (d) B, B

32. For a set of 100 observation, taking assumed mean as 4, the sum of the deviations is

-11 cm, and the sum of the squares of these deviation is 275 cm2. The coefficient of

variation is :

(a) 41.13% (b) 40.13% (c) 42.13% (d) none of these

33. Mean and Variance of two series is given below. Which series is more stable?

A: Mean = 53; Variance = 7

B: Mean = 105; Variance = 4

a) Series B

b) Series A

c) Both are equally stable

d) None of the above

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34. If the SD of x is 3, what is the variance of (5 – 2x)?

a) 1

b) 6

c) 9

d) 36

35. If the mean and SD of x are a and b respectively, then the SD of (x – a) / b is?

a) ab

b) 1

c) – 1

d) a/b

36. If the two variables x and y are related as: 2x + 3y = 10 and the standard deviation of

x is 15, then what is the standard deviation of y?

(a) 15 (b) 10 (c) 20 (d) None of these

37. If x and y are related by 3y = 7x – 9 and SD of y is 7, then what is the variance of x?

a) 7

b) 8

c) 9

d) 16

38. The S.D. of 15 items is 7.9 and if each item is decreased by 1, then standard

deviation will be :

(a) 5 (b) 7 (c) 91/15 (d) 7.9

39. If each observation of a raw data, whose variance is σσσσ2 is multiplied by λλλλ then the

variance of the new set is :

(a) σσσσ2 (b) λλλλ2 σσσσ2 (c) λλλλ + σσσσ2 (d) λλλλ2 + σσσσ2

40. The mean and S.D. for p, q and 2 are 3 and 1 respectively. Find the value of p.q.

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41. From the following data, calculate the value of mean and standard deviation.

Class Interval Frequency

0 – 10 4

10 – 20 10

20 – 30 16

30 – 40 12

40 – 50 8

a) 27, 11

b) 25, 11

c) 27, 11.66

d) 25, 11.66

Combined SD, Miscellaneous Properties

42. The AM and the SD of a set of 9 items are 43 and 5 respectively. If an item of value

63 is added to the set, find the mean and the standard deviation of all the 10 items.

43. A sample of size 15 has mean 3.5 and standard deviation 3.0. Another sample of size

22 has mean 4.7 and standard deviation 4.0. If the two samples are pooled together,

find the standard deviation of the combined sample.

a) 2.68

b) 4

c) 3.68

d) 3.5

44. For a group of 50 observations, arithmetic mean and standard deviation of weight

were found to be 60 kgs. And 4 kgs respectively. It was later found that the weight of

one of the children was wrongly noted as 59 kgs. Instead of 64 kgs. Find the correct

values of mean and standard deviation.

45. For a group of 40 observations, the arithmetic mean was 36 and standard deviation

was 2.5. Two more items were added to this group with values 39 and 40. What will

be the arithmetic mean and standard deviation for the new group of 42 observations?

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Theoretical Aspects

Introduction:

46. Dispersion discovers

a) Uniformity in variability

b) Variability in variability

c) Variability in uniformity

d) None of these

47. The word dispersion is used to denote the ___________ in the data.

a) degree of similarity

b) degree of homogeneity

c) degree of heterogeneity


48. If all the observations are equal, the dispersion would be:

a) – 1

b) 1

c) 0

d) data insufficient

49. The wider the ________ from one observation to another, the larger will be the

dispersion.

a) discrepancy

b) similarity

c) jump


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50. All the _______ measures of dispersion are defined as the ratio of an absolute

measure of dispersion to the corresponding measure of central tendency, and the

ratio is expressed as a percentage.

a) absolute

b) relative

c) perfect


51. When we compare two or more distributions, we consider:

a) relative measures of dispersion

b) absolute measures of dispersion

c) neither a) nor b)

d) either a) or b)

52. All measures of dispersion are independent of change of ____, but are dependent on

change of ____.

a) scale, origin

b) origin, scale

c) time, scale


Range:

53. Which measure of dispersion represents the maximum possible difference between

any two observations?

a) SD

b) mean deviation

c) range

d) quartile deviation

54. The quickest measure of dispersion is ___.

a) standard deviation

b) mean deviation about mean

c) range

d) inter-quartile range

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55. If each item is reduced by 5, the Range is:

a) Increased by 5

b) decreased by 5

c) neither a) nor b)

d) either a) or b)

Quartile Deviation

56. Quartile Deviation is also known as:

a) quartile range

b) inter-quartile range

c) semi-inter-quartile range

d) mean deviation

57. Which measure of dispersion is based on only the central 50% of the observations?

a) range

b) quartile deviation

c) standard deviation

d) mean deviation

58. Quartile Deviation is:

a) difference between the upper and the lower quartiles

b) sum of upper and the lower percentiles

c) half the difference between the upper and the lower quartiles


59. For a open-end classes in the frequency distribution, which of the following measures

of dispersion can be calculated?

a) Range

b) Standard Deviation

c) Mean Deviation

d) Quartile Deviation

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60. Coefficient of Quartile Deviation is defined as:

a) the ratio of QD to Mode, multiplied by 100

b) the ratio of QD to Median

c) the ratio of QD to Median, multiplied by 100

d) the ratio of QD to Range, multiplied by 100

61. The appropriate measure of dispersion from a frequency distribution with open-end

class is ______.

a) mean deviation


c) SD

d) range

Mean Deviation:

62. Mean Deviation is also known as:

a) median deviation

b) median absolute deviation

c) absolute deviation

d) mean absolute deviation

63. Mean Deviation has the minimum value when deviations are taken from:

a) arithmetic mean

b) geometric mean

c) median

d) mode

64. Mean Deviation about an arbitrary origin A is defined as:

a) half difference between the upper and lower quartiles

b) the mean of absolute deviations from A

c) the difference between the largest and the smallest items


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Standard Deviation:

65. When Variance(x) = 0, what can you say about the set of observations?

a) all the values are low in magnitude

b) all the values are very high in magnitude

c) all the values are not equal

d) all the values are equal

66. Which measure of dispersion is based on all the observations?

a) standard deviation


c) mean deviation

d) both a) and c) above

67. Standard Deviation is also known as:

a) Root Mean Deviation

b) Root Mean Square Deviation

c) Root Square Deviation

d) Mean Square Deviation

68. The square of standard deviation is known as:

a) deviation

b) variation

c) variance

d) co variance

69. Standard Deviation is:

a) negative square root of variance

b) positive square root of variance

c) either a) or b)

d) both a) and b)

70. The standard deviation of two values a and b is equal to:

a) sum of the two values

b) half their difference

c) their difference

d) square root of their difference

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71. Coefficient of Variation is defined as:

a) the ratio of standard deviation to median

b) the ratio of standard deviation to arithmetic mean

c) the ratio of SD to AM, multiplied by 100


72. Root-Mean-Square Deviation is minimum when deviations are taken about _____.

a) Arithmetic Mean

b) Median

c) Mode

d) Harmonic Mean

73. The most useful measure of dispersion is:

a) Range

b) Quartile Deviation

c) Mean Deviation

d) Standard Deviation

74. For any two numbers, standard deviation is:

a) square of their range

b) double of the range

c) half of the range

d) thrice the range

75. If each item is reduced by 22. the Standard Deviation is:

a) increased by 22

b) decreased by 22

c) remains unchanged


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76. If all the variables are increased or decreased by the same amount the standard

deviation:

a) decreases or increases respectively

b) increases or decreases respectively

c) remains unchanged

d) can’t be determined

77. Which of the following measures of dispersion has some mathematical properties?

a) quartile deviation

b) mean deviation

c) standard deviation

d) range

78. Which of the following measures of dispersion would you use to find a pooled

measure of dispersion after combining several groups?

a) mean deviation about median


c) mean deviation about mode

d) standard deviation

79. Which of the following statements is true?

a) Mean is an absolute measure and standard deviation is based upon it.

Therefore, standard deviation is a relative measure.

b) For a symmetrical distribution, semi-inter quartile range is one fourth of the

range.

c) Whole frequency table is needed for the calculation of quartile deviation.

d) None of the above is true.

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80. Which of the following statement(s) is TRUE?

a) When all observations are same, any measure of dispersion would be 1.0.

b) Mean Absolute Deviation from median and mean are 5 and 3 respectively.

c) The unit of Variance is that of the observations.

d) None of the above is true.

THEORY ANSWERS:

46 c 56 c 66 d 76 c

47 c 57 b 67 b 77 c

48 c 58 c 68 c 78 d

49 a 59 d 69 b 79 d

50 b 60 c 70 b 80 d

51 a 61 b 71 c

52 b 62 d 72 a

53 c 63 c 73 d

54 c 64 b 74 c

55 c 65 d 75 c

: 341 :

Documents

J. K. SHAH CLASSES Measures of Dispersion€¦ · Different central tendency and different dispersion Types of Measures of Dispersion There are two types of measures of dispersion,