Statistical Measures of Dispersion

Last Updated on: 2nd December 2024, 03:27 pm

Statistical Measures of Dispersion – MCQ

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1. Concept of standard deviation was introduced by:
(a) Karl Pearson
(b) R.A. Fisher
(c) W.S. Gossett
(d) S.K Peterson.

The Concept of standard deviation was introduced by Karl Pearson. So, option (a) is correct

2. What is the best method of measuring dispersion?
(a) S.D.
(b) Q.D.
(c) Regression
(d) A.M.
Standard deviation is the best method of measuring dispersion.

3. For a normal distribution
(a) Q.D.> M.D.
(b) Q.D.< M.D.
(c) Q.D. = M.D.

For a normal distribution Quartile deviation < Mean deviation.

4. For an approximately normal distribution
(a) M.D. = $\displaystyle \frac{2}{5}$ S.D.

(b) M.D. = $\displaystyle \frac{3}{5}$ S.D.

(d) 1

For a normal distribution. Mean deviation (MD) = 4/5 standard deviation (SD)

5. Positional measure of dispersion is
(a) SD
(b) A.M.
(c) QD
(d) None of the above.

Quartile Deviation (QD) is positional measure of dispersion.

6. The square of the S.D. is known as
(a) Co. of SD
(b) H.M.
(c) $\displaystyle \sigma$
(d) Variance.

The square of standard deviation (S.D). is known as Variance.

7. Which is the relative measurement of dispersion?
(a) M.D.
(b) S.D.
(c) C.V.
(d) Range.

Co-efficient of variation is relative measurement of dispersion.

8. Which is absolute measurement of dispersion?
(a) M.D.
(b) Variance
(c) C.V.
(d) Range

Mean deviation is absolute measurement of dispersion.

9. Standard Deviation is always________________Mean Deviation.
(a) Less than
(b) More than
(c) Equal to
(d) None.

Standard Deviation is always more than Mean Deviation.

10. Standard Deviation is the______________of variance.
(a) Square
(b) Qube
(c) Square-root
(d) Qubic-root.

Standard Deviation is the square-root of variance.

11. When C.V. is 50, and s is 10, then $\displaystyle \overline{x}$ =____________.
(a) 50
(b) 35
(c) 20
(d) 45

C.V. = $\displaystyle \frac{\sigma }{x}$ x 100
$\displaystyle \therefore \overline{x}\frac{\sigma }{{C.V}}$ x 100 = $\displaystyle \frac{{10}}{{50}}\times 100$ = 20

12. When Q1 = 30, Q3 = 50, the co-efficient Q.D. is___________ .
(a) 10%
(b) 25%
(c) 30%
(d) 35%.

Quartile deviation =Q₃-Q₁/2

= $\displaystyle \frac{{50-30}}{2}$

= $\displaystyle \frac{{20}}{2}$

=10

13. If C.V. is 20, C is 50, s is____________.
(a) 30
(b) 10
(c) 25
(d) 40.

C.V.= $\displaystyle \frac{\sigma }{X}\times 100$

$\displaystyle \therefore \sigma =\frac{{C.V.x\overline{x}}}{{100}}$

$\displaystyle =\frac{{20\times 50}}{{100}}$

=10

14. If Variance is 81 and Mean is 50, CV is_____________ .
(a) 12
(b) 25
(c) 18
(d) 24.
$\displaystyle {{\sigma }^{2}}$ =81
$\displaystyle \sigma =9$
$\displaystyle \overline{x}=50$
C.V.= $\displaystyle \frac{9}{{50}}\times 100$
=18

15. The range of 15, 12, 10, 9, 17, 20 is
(a) 15
(b) 20
(c) 11
(d) 24
Range = $\displaystyle \infty -S$
$\displaystyle \infty$ = 20 S = 9
$\displaystyle \therefore$ Range = 20 – 9 = 11

16. If in series co-efficient of variation = 20, arithmetic mean = 40, the value of standard deviation will be
(a) 12
(b) 5
(c) 8
(d) 7
C.V.= $\displaystyle \frac{\sigma }{{\overline{x}}}\times 100$
$\displaystyle \therefore \sigma =\frac{{C.V.\times \overline{x}}}{{100}}$
$\displaystyle =\frac{{20\times 40}}{{100}}$
=8

17. If maximum and minimum values of a series are 50 and 0 respectively, the co-efficient of range will be
(a) 2
(b) +1
(c) -1
(d) 3
$\displaystyle \infty$ = 50 S = 0
$\displaystyle \infty$ – S = 50 – 0 = 50
Co-efficient of range = $\displaystyle \frac{{\infty -S}}{{\infty +S}}$
$\displaystyle =\frac{{50}}{{50+0}}$
$\displaystyle =\frac{{50}}{{50}}$
=+1

18. Quartile Deviation for the data 1, 3, 4, 5, 6, 6, 10 is
(a) 0.5
(b) 2.1
(c) 1.4
(d) 1.5
N = 7
Q₁= $\displaystyle =\frac{{N+1}}{4}=\frac{{7+1}}{4}=2$

Q₃= $\displaystyle \frac{{3\left( {N+1} \right)}}{4}=\frac{{3\left( {7+1} \right)}}{4}=\frac{{3\times 8}}{4}=6$

Q₁=3 , Q₃=6

19. Calculate the standard deviation of 6, 8, 10, 12, 14
(a) 5
(b) 4.84
(c) 2.83
(d) 2.73
$\displaystyle \sum{{}}$ x = 6 + 8 + 10 + 12 + 14 = 50
$\displaystyle \sum{{}}$ x² = 36 + 64 + 100 + 144 + 196 = 540

$\displaystyle \sigma ={{\sqrt{{\frac{{\sum{{{{x}^{2}}}}}}{N}-{{{\left( {\frac{{\sum{x}}}{N}} \right)}}^{2}}}}}^{{N=5}}}$

$\displaystyle ={{\sqrt{{\frac{{540}}{5}-\left( {\frac{{50}}{5}} \right)}}}^{2}}$

$\displaystyle =\sqrt{{\frac{{540}}{5}-\frac{{2500}}{{25}}}}$

$\displaystyle =\sqrt{{108-100}}$

$\displaystyle =\sqrt{8}$

=2.83

21. If the variance of a data is 64, then what is the standard deviation
(a) 12
(b) 19
(c) 8
(d) 24

Standard deviation = √64= 8

22. When QD is 60, then variance shall be
(a) 280
(b) 81
(c) 8100
(d) 9500

Q.D. = $\displaystyle \frac{2}{3}\text{ }\sigma$
$\displaystyle \therefore$ Q.D. = 60
$\displaystyle \therefore \sigma$ = $\displaystyle \frac{2}{3}$ x Q.D. = $\displaystyle \frac{3}{2}$ x 60 = 90
$\displaystyle \therefore$ variance = $\displaystyle \sigma$ ² = (90)² = 8100

23. What is the formula to calculate coefficient of range
(a) H – L

(b) H + L

(d) $\displaystyle \frac{{H+L}}{{H-L}}$

Formula for co-efficient of range
$\displaystyle \frac{{H-L}}{{H+L}}$
H = Highest value
L = Lowest value

24. If Quartile deviation is 7.4, Q₃ = 36, then Q₁ is ………
(a) 28.6
(b) 21.2
(c) 15.75
(d) 26.02

Q.D. = Q₃-Q₁/2
Q.D. = 7.4 Q₃ = 36
7.4 = $\displaystyle \frac{{36-{{Q}_{1}}}}{2}$
Or, 36 – Q₁ = 14.8
Or, Q₁ = 36 – 14.8 = 21.2

25. What is the value of mean deviation about mean for the following numbers? 5, 8, 3, 4
(a) 5.5
(b) 7.51
(c) 1.5
(d) 2.5

We know for individual series
Mean deviation about mean
$\displaystyle =\frac{{\sum{{x-\overline{x}}}}}{n}$
Numbers are 5,8,3,4,
$\displaystyle \therefore \overline{x}=\frac{{5+8+3+4}}{4}=\frac{{20}}{4}=5$
$\displaystyle \therefore \sum{{\left| {x-\overline{x}} \right|}}$ = (5 – 5) + (8 – 5) + (3 – 5) + (4 – 5)
= 0 + 3 + 2 + 1= 6
$\displaystyle \therefore$ M.D.= $\displaystyle \frac{6}{4}$ = 1.5

26. Find value of Q₃ if co-efficient of Quartile deviation = 0.39 and Q₁ = 23.26:
(a) 53.24
(b) 29.02
(c) 36.07
(d) 37.02

Co-efficient of Quartile deviation
Q₃ – Q₁/Q₃ + Q₁
$\displaystyle \therefore$ .39 = Q₃ – 23.26/Q₃ + 23.26
Or, .39 (Q₃ + 23.26) = Q₃ – 23.26
Or, .39. Q3 + 9.0714 = Q₃ – 23.26
Or, Q₃ – .39 Q₃ = 9.0714 + 23.26
Or, .61 Q₃ = 32.3314
Or, Q₃ = 32.3314 = 53.002

27. If Q₃ is 52 and Q₁ is 12, coefficient of quartile deviation shall be:
(a) 0.625
(b) 0.575
(c) 0.425
(d) 0.875

Co-efficient of quartile deviation = Q₃ – Q₁/Q₃ + Q₁
Q₃ = 52 , Q₁= 12
$\displaystyle \therefore$ Co-efficient of quartile deviation = $\displaystyle \frac{{52-12}}{{52+12}}$
= 40 = .625

28. What is the coefficient of range for the following wages of 8 workers?
Rs.80, Rs.65, Rs.90, Rs.60, Rs.75, Rs.70, Rs.72, Rs.85
(a) 0.50
(b) 0.10
(c) 30
(d) 20

Highest value = 90
Lowest value = 60
$\displaystyle \therefore$ Co-efficient of range = $\displaystyle \frac{{H-L}}{{H+L}}$
= $\displaystyle \frac{{90-60}}{{90+60}}$ = $\displaystyle \frac{{30}}{{150}}$ = $\displaystyle \frac{1}{5}$
= .20

29. The limits of range are
(a) 5 $\displaystyle \sigma$ to 9 $\displaystyle \sigma$
(b) $\displaystyle \sigma$ to 2 $\displaystyle \sigma$
(c) 3 $\displaystyle \sigma$ to 6 $\displaystyle \sigma$
(d) 4 $\displaystyle \sigma$ to 6 $\displaystyle \sigma$

The limits of range are 4 $\displaystyle \sigma$ to 6 $\displaystyle \sigma$ .

30. What is the formula to calculate variance?
(a) $\displaystyle \frac{\sigma }{X}$

(b) $\displaystyle \frac{\sigma }{X}$ x 100

31. What is the formula to calculate coefficient of $\displaystyle \sigma$ (SD)
(a) $\displaystyle \frac{\sigma }{X}$

(b) $\displaystyle \frac{\sigma }{X}$ x 100

$\displaystyle \frac{\sigma }{X}$ is the formula to calculate coefficient of $\displaystyle \sigma$ (SD)

32. If variance is 144, then standard deviation is_______________.
(a) 24
(b) 20
(c) 16
(d) 12

Variance = $\displaystyle {{{\sigma }^{2}}}$ = 144
$\displaystyle \sigma$ = 12
$\displaystyle \sigma$ Standard deviation ( $\displaystyle \sigma$ ) = 12

33. If N =10, $\displaystyle \sum{X}=60,\text{ }\sum{{{{X}^{2}}}}$ = 1,000, then standard deviation is
(a) 8
(b) 25
(c) 36
(d) 80

$\displaystyle \sum{{x=60\text{ }\sum{{{{x}^{2}}=1000\text{ N=10}}}}}$

$\displaystyle \sigma =\sqrt{{\frac{{\sum{{{{x}^{2}}}}}}{N}-\left( {\frac{{\sum{x}}}{N}} \right)}}$

$\displaystyle =\sqrt{{\frac{{1000}}{{10}}-{{{\left( {\frac{{60}}{{10}}} \right)}}^{2}}}}$

$\displaystyle =\sqrt{{100-\frac{{3600}}{{100}}}}=\sqrt{{100-36}}$

$\displaystyle =\sqrt{{164}}=8$

34. What is the amount of CV is $\displaystyle \sigma$ = 10, N = 100, $\displaystyle \overline{X}$ 20
(a) 5%
(b) 20%
(c) 50%
(d) 30%

Co-efficient of variation
= $\displaystyle \frac{{Standard\text{ }deviation}}{{A.M.}}\times 100$
= $\displaystyle \frac{{10}}{{20}}$ x 100 = 50%

35. The quartiles of a variable are 45, 52 and 65 respectively. Its quartile deviation is
(a) 10
(b) 20
(c) 45
(d) 30

Quartile of a variable are 45, 52, 65
$\displaystyle \therefore$ Q₃ = 45, Q₃ = 52, Q₃ = 65
Quartile deviation = Q₃ – Q₃/2
= $\displaystyle \frac{{65-45}}{2}=\frac{{20}}{2}=10$