Measures of Dispersion Test

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Measures of Dispersion Test - 18

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Question 1
1 / -0

If mean of a series is 40 and variance 1486, then coefficient of variation is

A
$$0.9021$$

B
$$0.9637$$

C
$$0.8864$$

D
$$0.9853$$

Solution

If mean of the given dist. be $$\bar{x}$$ and S.D be $$\sigma $$
then given $$\bar{x} = 40, \sigma^2 = 1486$$
$$\therefore$$ Coefficient of variation $$=\cfrac{\sigma}{\bar{x}}=\cfrac{\sqrt{1486}}{40}=.9637$$
Question 2
1 / -0

Mean deviation of the observations 70, 42, 63,34, 44, 54, 55, 46, 38, 48 from median is

A
$$7.8$$

B
$$8.6$$

C
$$7.6$$

D
$$8.8$$

Solution

Arranging the given data in ascending order-
$$34, 38, 42, 44, 46, 48, 54, 55, 63, 70$$
$$\because n = 10$$
Therefore,
Median $$\left( M \right) = \cfrac{{\left( \cfrac{n}{2} \right)}^{th} \text{ term } + {\left( \cfrac{n}{2} + 1 \right)}^{th} \text{ term}}{2}$$
$$\Rightarrow M = \cfrac{46 + 48}{2} = 47$$
Therefore,
$${x}_{i}$$ $$\left| {x}_{i} - M \right|$$
$$34$$ 13
$$38$$ 9
$$42$$ 5
$$44$$ 3
$$46$$ 1
$$48$$ 1
$$54$$ 7
$$55$$ 8
$$63$$ 16
$$70$$ 23

$$\sum{\left| {x}_{i} - M \right|} = 86$$
Therefore,
Mean deviation $$= \cfrac{\sum{\left| {x}_{i} - M \right|}}{n} \\ = \cfrac{86}{10} = 8.6$$
Hence the mean deviation of the observations is $$8.6$$.
Question 3
1 / -0

The S.D. of the following frequency distribution is
Class 0-10 10-20 20-30 30-40
$$\displaystyle f_{i}$$ 1 3 4 2

A
$$7.8$$

B
$$9$$

C
$$8.1$$

D
$$0.9$$

Solution

Class $$x_i$$ $$f_i$$ $$u_i=\dfrac{x_i-a}{h}$$ $$f_iu_i$$ $$f_iu_i^2$$
0-10 5 1 -2 -2 4
10-20 15 3 -1 -3 3
20-30 25 4 0 0 0
30-40 35 2 1 2 2
$$N=10$$ $$\sum f_iu_i=-3$$ $$\sum f_iu_i^2=9$$
$$\displaystyle \sigma =h\sqrt{\frac{ \Sigma f_{1}u_{1}^{2}}{N}-\left ( \frac{\Sigma f_{1}u_{1}}{N} \right )^{2}}$$

$$\displaystyle =10\sqrt{\frac{9}{10}-\left ( \frac{-3}{10} \right )^{2}}=9$$
Question 4
1 / -0

The sum of the squares of deviation of 10 observations from their mean 50 is 250, then coefficient of varition is

A
10%

B
40%

C
50%

D
None of these

Solution

Given $$\displaystyle \Sigma \left ( x_{i}-\overline{x} \right )^{2}=250$$,$$n=10,\overline{x}=50$$

Now, $$\sigma=\sqrt{\dfrac{1}{n}\Sigma \left ( x_{i}-\overline{x} \right )^{2}}$$

$$= \sqrt{\dfrac{1}{10}\times 250}=5$$
Hence coefficient of variation $$\displaystyle =\dfrac{\sigma }{\overline{x}}\times 100=\dfrac{5}{50}\times 100=10$$%
Question 5
1 / -0

If the coefficient of variation and standard deviation of a distribution are 50% and 20 respectively, the its mean is

A
40

B
30

C
20

D
None of these

Solution

We know if a distribution having mean $$\bar{x}$$ and standard deviation $$\sigma$$
then coefficient of variation $$=\cfrac{\sigma}{\bar{x}}\times 100$$
$$\therefore \cfrac{20}{\bar{x}}\times 100=50\Rightarrow \bar{x} = 40$$
Hence required mean is $$=40$$
Question 6
1 / -0

The coefficient of range of the following distribution $$10, 14, 11, 9, 8, 12, 6$$

A
$$0.4$$

B
$$2.5$$

C
$$8$$

D
$$0.9$$

Solution

Here greatest term in the given observation is, $$x_m =14$$
and least term is, $$x_l = 6$$
Hence coefficient of range is $$=\cfrac{x_m-x_l}{x_m+x_l}=\cfrac{8}{20}=0.4$$
Question 7
1 / -0

The coefficient of mean deviation from median of observations $$40, 62, 54, 90, 68, 76$$ is

A
$$2.16$$

B
$$0.2$$

C
$$5$$

D
None of these

Solution

Arrange the given observations in ascending order
$$40,54,62,68,76,90$$
Here, number of terms $$n=6 (even) $$
$$\displaystyle \therefore $$ Median (M) $$\displaystyle =\frac{\left ( \frac{n}{2} \right )th\:term+\left ( \frac{n}{2}+1 \right )th\:term}{2}=\frac{62+68}{2}=65$$

$$\Sigma \left | x_{i}-M \right |=25+11+3+3+11+25=78$$
Mean deviation from median $$\displaystyle =\frac{\Sigma \left | x_{i}-M \right |}{n}=\frac{78}{6}=13 $$
$$\therefore $$ Coefficient of M.D.=$$\displaystyle =\frac{M.D.}{median}=\frac{13}{65}=0.2$$
Question 8
1 / -0

The S.D. of the following freq. dist.
Class 0-10 10-20 20-30 30-40
$$f_i$$ 1 3 4 2

A
$$7.8$$

B
$$9$$

C
$$8.1$$

D
$$0.9$$

Solution

Class $$x_i$$ $$f_i$$ $$u_i=\dfrac{x_i-A}{h}$$ $$u_i^2$$ $$f_iu_i$$ $$f_iu_i^2$$
0-10 5 1 -2 4 -2 4
10-20 15 3 -1 1 -3 3
20-30 25 4 0 0 0 0
30-40 35 2 1 1 2 2
Total 10 -3 9
$$S.D.$$$$=h\sqrt { \cfrac { \sum { { f }_{ i }{ u }_{ i }^{ 2 } } }{ \sum { { f }_{ i } } } -{ \left( \cfrac { \sum { f } _{ i }{ u }_{ i } }{ \sum { { f }_{ i } } } \right) }^{ 2 } } $$
$$S.D.=10\sqrt { \cfrac { 9 }{ 10 } -{ \left( \cfrac { -3 }{ 10 } \right) }^{ 2 } } =9$$
Question 9
1 / -0

The mean of a dist. is $$4$$. if its coefficient of variation is $$58\%$$. Then the S.D. of the dist. is

A
$$2.23$$

B
$$3.23$$

C
$$2.32$$

D
None of these

Solution

We know if a distribution having mean $$\bar{x}$$ and standard deviation $$\sigma$$
then coefficient of variation $$=\cfrac{\sigma}{\bar{x}}\times 100$$
$$\therefore \cfrac{\sigma}{4}\times 100=58\Rightarrow \sigma = \cfrac{58}{25}=2.32$$
Hence required standard deviation is $$=2.32$$
Question 10
1 / -0

The mean deviation from mean of observations 5, 10, 15, 20, ......85 is

A
$$43.71$$

B
$$21.17$$

C
$$38.7$$

D
None of these

Solution

Given series is $$5,10,15,....,85$$, which is an AP.
Last term $$85=a+(n-1)d$$
$$\Rightarrow n=17$$
Sum of $$n$$ terms of AP $$=\dfrac{17}{2}(5+85)$$
$$\sum x_i=765$$

$$\bar x=\dfrac{\sum x_i}{n}$$

$$\bar x=\dfrac{765}{17}=45$$

So, deviations from mean are $$40,35,25,20,15,10,5,0,5,10,15,20,25,30,35,40$$
Sum of deviations $$=360$$

Mean deviation from mean $$=\dfrac{360}{17}=21.17$$

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$${x}_{i}$$	$$\left\| {x}_{i} - M \right\|$$
$$34$$	13
$$38$$	9
$$42$$	5
$$44$$	3
$$46$$	1
$$48$$	1
$$54$$	7
$$55$$	8
$$63$$	16
$$70$$	23
	$$\sum{\left\| {x}_{i} - M \right\|} = 86$$

Class	$$x_i$$	$$f_i$$	$$u_i=\dfrac{x_i-a}{h}$$	$$f_iu_i$$	$$f_iu_i^2$$
0-10	5	1	-2	-2	4
10-20	15	3	-1	-3	3
20-30	25	4	0	0	0
30-40	35	2	1	2	2
		$$N=10$$		$$\sum f_iu_i=-3$$	$$\sum f_iu_i^2=9$$

Measures of Dispersion Test - 18

Result

Measures of Dispersion Test - 18

Score

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Rank

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SHARING IS CARING

Question 1

$$0.9021$$

$$0.9637$$

$$0.8864$$

$$0.9853$$

Solution

Question 2

$$7.8$$

$$8.6$$

$$7.6$$

$$8.8$$

Solution

Question 3

$$7.8$$

$$9$$

$$8.1$$

$$0.9$$

Solution

Question 4

10%

40%

50%

None of these

Solution

Question 5

40

30

20

None of these

Solution

Question 6

$$0.4$$

$$2.5$$

$$8$$

$$0.9$$

Solution

Question 7

$$2.16$$

$$0.2$$

$$5$$

None of these

Solution

Question 8

$$7.8$$

$$9$$

$$8.1$$

$$0.9$$

Solution

Question 9

$$2.23$$

$$3.23$$

$$2.32$$

None of these

Solution

Question 10

$$43.71$$

$$21.17$$

$$38.7$$

None of these

Solution

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