Measures of Dispersion Test

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Measures of Dispersion Test - 20

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Question 1
1 / -0

The mean of a finite set X of numbers is $$14$$, the median of this set of numbers is $$12$$, and the standard deviation is $$1.8$$. A new set Y is formed by multiplying each member of the set S by $$3$$. Determine the correct statements w.r.t. set Y:
I. The mean of the numbers is set Y is $$42$$
II. The median of the numbers in set Y is $$36$$
III. The standard deviation of the numbers is set Y is $$5.4$$

A
I only

B
II only

C
I and II only

D
I and III only

E
I, II, and III

Solution

For set $$x$$
Mean $$=14$$
Median$$=12$$
S.D.$$=\sqrt { \sigma } $$
$$=1.8$$.

For set $$y$$
Mean $$=14\times 3= 42$$.
Median $$=12\times 3=36$$.
SD$$=1.8\times 3= 5.4$$.

Hence all three are correct.
Question 2
1 / -0

Find the mean for the following data using step deviation method.

A
$$43$$

B
$$44$$

C
$$45$$

D
$$46$$

Solution

Answer:- By shortcut Method
Class interval width (i)= 24-12 = 12

X F $$d=\cfrac{x-A}{i} $$ Fd
12=A 1 0 0
24 2 1 2
36 3 2 6
48 4 3 12
60 5 4 20
$$\Sigma f = 15$$ $$\Sigma fd = 40$$
Mean = $$A+\cfrac{\Sigma fd}{\Sigma f} \times i=12+\left(\cfrac{40}{15} \times 12\right)={12+32}=44$$
B)44
Question 3
1 / -0

Which of the following can be used as measures of dispersions?

A
Range

B
Percentiles

C
Standard Deviation

D
All

Solution

Following terms used as measures of dispersion are:-

a) Range
b) Percentile
c) Standard eviation
d) Mean deviation.
Question 4
1 / -0

Directions For Questions

Refer to the data given in the table below:
$$X:$$ $$3$$ $$5$$ $$12$$ $$3$$ $$2$$

...view full instructions

Find the coefficient of variation.

A
$$72.66$$

B
$$81.24$$

C
$$264$$

D
$$330$$

E
None

Solution

$$ \quad X\\ \quad 03\\ \quad 05\\ \quad 12\\ \quad 03\\ \quad 02\\ \_ \_ \_ \_ \_ \\ \quad 25$$ $$x-\overline { x } \\ -2\\ \quad 00\\ \quad 07\\ -2\\ -3 $$ $${ \quad \quad (x-\overline { x } ) }^{ 2 }\\ \quad \quad \quad 04\\ \quad \quad \quad 00\\ \quad \quad \quad 49\\ \quad \quad \quad 04\\ \quad \quad \quad 09\\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \sum { { (x-\overline { x } ) }^{ 2 }=66 } $$

Variance $$= \cfrac { \sum { { (x-\overline { x } ) }^{ 2 } } }{ n-1 } \\ =\cfrac { 66 }{ 4 } \\ =16.5$$.

$$\sigma =S.D.=\sqrt { 16.5 } $$.
Coefficient of variation $$= \cfrac { \sigma }{ x } \\ =\cfrac { \sqrt { 16.5 } }{ 5 } \\ =0.8124$$.

% of coefficient of variation $$= 0.8124*100\\ =81.24$$.
Question 5
1 / -0

The mean of a distribution is $$14$$ and standard deviation is $$5$$. What is the value of the coefficient of variation?

A
$$57.7\%$$

B
$$45.7\%$$

C
$$35.7\%$$

D
None of these

Solution

Coefficient of variation is given by $$CV = \dfrac{SD}{Mean}\times 100 $$
$$\Rightarrow \dfrac{5}{14}\times 100 = 35.7\%$$

Question 6

1 / -0

Find the mean for the following data using step deviation method.

$$15.833$$

$$16.833$$

$$17.833$$

$$18.833$$

Solution

Answer:- By shortcut Method

Class interval width (i) = $$5-0 = 5$$

X	C. I. mid point	F	$$d=\cfrac{X-A}{i}$$	Fd
0-5	2.5	4	-1	-4
5-10	7.5=A	8	0	0
10-15	12.5	12	1	12
15-20	17.5	16	2	32
20-25	22.5	20	3	60
		$$\Sigma F = 60$$		$$\Sigma Fd=100$$

Mean = $$A+\cfrac{\Sigma fd}{\Sigma f} \times i=7.5+\left(\cfrac{100}{60} \times 5\right)={7.5+8.33}=15.83$$

A) $$15.83$$

Question 7
1 / -0

A random variable $$X$$ has the probability distribution given below. Its variance is
X 1 2 3 4 5
P(X=x) k 2k 3k 2k k

A
$$\dfrac{16}{3}$$

B
$$\dfrac{4}{3}$$

C
$$\dfrac{5}{3}$$

D
$$\dfrac{10}{3}$$

Solution

We know that
$$\sum P(x) = 1$$
$$k+2k+3k+2k+k=1$$
$$9k=1$$
$$k=\frac{1}{9}$$
Now, var $$\sigma^2 = \sum xi^2P(xi)-(\sum\,xiP(xi))^2$$
$$=1(k)+2^2(2k)+3^2(3k)+4^2(2k)+5^2(k)-[1(k)+2(2k)+3(3k)+4(2k)+5(k)]^2$$
$$=93k-(27k)^2$$
$$=93(\cfrac{1}{9})-[27(\frac{1}{9})]^2$$
$$=\cfrac{31}{3}-3^2$$
$$=\cfrac{31-27}{3}$$
$$=\cfrac{4}{3}$$
Question 8
1 / -0

If the standard deviation of a set of scores is $$1.2$$ and their mean is $$10$$, then the coefficient of variation of the scores is

A
$$12$$

B
$$0.12$$

C
$$20$$

D
$$120$$

Solution

Given : standard deviation$$(\sigma)=1.2,$$ mean$$(\overline {X})=10$$.
Coefficient of variation(C.V.) $$=\dfrac{\sigma}{\overline {X}}\times 100=\dfrac{1.2}{10}\times 100=12$$
$$\therefore$$ C.V. $$=12$$
Hence, option $$A$$ is correct.
Question 9
1 / -0

Standard Deviation of n observations $$a_1, a_2, a_3 .....a_n$$ is $$\sigma$$ Then the standard deviation of the observations $$\lambda a_1, \lambda a_2, ..., \lambda a_n$$ is

A
$$\lambda \sigma$$

B
$$-\lambda \sigma$$

C
$$|\lambda| \sigma$$

D
$${\lambda}^n \sigma$$

Solution

Let the mean of original data is $$\mu$$
Then mean of observation if all the terms are multiplied by the constant $$\lambda$$ will be $$\lambda \mu$$
If $$\sigma$$ be the standard deviation of original observation , then
$$\sigma=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n (x_i-\mu)^2 }$$
Now let the standard deviation of new observation is $$\sigma'$$, then
$$\sigma'=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n (\lambda x_i-\lambda\mu)^2 }$$
$$=\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n \lambda^2( x_i-\mu)^2 }$$
$$=|\lambda|\sqrt{\dfrac{1}{n}\displaystyle \sum_{i=1}^n( x_i-\mu)^2 }=|\lambda |\sigma$$
Note that: standard deviation can never be negative
Question 10
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Which one of the following is a measure of dispersion?

A
Mean

B
Median

C
Mode

D
Standard deviation

Solution

Dispersion is the extent to which the distribution is stretched or squeezed.
The most common measures of dispersion are variance, standard deviation, mean deviation, range etc.

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X	F	$$d=\cfrac{x-A}{i} $$	Fd
12=A	1	0	0
24	2	1	2
36	3	2	6
48	4	3	12
60	5	4	20
	$$\Sigma f = 15$$		$$\Sigma fd = 40$$

X	1	2	3	4	5
P(X=x)	k	2k	3k	2k	k

Measures of Dispersion Test - 20

Result

Measures of Dispersion Test - 20

Score

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Rank

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SHARING IS CARING

Question 1

I only

II only

I and II only

I and III only

I, II, and III

Solution

Question 2

$$43$$

$$44$$

$$45$$

$$46$$

Solution

Question 3

Range

Percentiles

Standard Deviation

All

Solution

Question 4

$$72.66$$

$$81.24$$

$$264$$

$$330$$

None

Solution

Question 5

$$57.7\%$$

$$45.7\%$$

$$35.7\%$$

None of these

Solution

Question 6

$$15.833$$

$$16.833$$

$$17.833$$

$$18.833$$

Solution

Question 7

$$\dfrac{16}{3}$$

$$\dfrac{4}{3}$$

$$\dfrac{5}{3}$$

$$\dfrac{10}{3}$$

Solution

Question 8

$$12$$

$$0.12$$

$$20$$

$$120$$

Solution

Question 9

$$\lambda \sigma$$

$$-\lambda \sigma$$

$$|\lambda| \sigma$$

$${\lambda}^n \sigma$$

Solution

Question 10

Mean

Median

Mode

Standard deviation

Solution

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