Measures of Dispersion Test

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Measures of Dispersion Test - 21

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Weekly Quiz Competition

Question 1

1 / -0

Find the variance of the following distribution.

Class interval	$$20-24$$	$$25-29$$	$$30-34$$	$$35-39$$	$$40-44$$	$$45-49$$
Frequency	$$15$$	$$25$$	$$28$$	$$12$$	$$12$$	$$8$$

$$416$$

$$46$$

$$16$$

None of these

Solution

$$Class$$	$$Mid-point$$ $$(x)$$	$$Frequency$$ $$(f)$$	$$xf$$	$$(x-\overline{x})$$	$$(x-\overline{x})^2$$	$$(x-\overline{x})^2f$$
$$20-24$$	$$22$$	$$15$$	$$330$$	$$-10.25$$	$$105.0625$$	$$1575.9375$$
$$25-29$$	$$27$$	$$25$$	$$675$$	$$-5.25$$	$$27.5625$$	$$689.0625$$
$$30-34$$	$$32$$	$$28$$	$$896$$	$$-0.25$$	$$0.0625$$	$$1.75$$
$$35-39$$	$$37$$	$$12$$	$$444$$	$$4.75$$	$$22.5625$$	$$270.75$$
$$40-44$$	$$42$$	$$12$$	$$504$$	$$9.75$$	$$95.0625$$	$$1140.75$$
$$45-49$$	$$47$$	$$8$$	$$376$$	$$14.75$$	$$217.5625$$	$$1740.5$$
		$$\sum f=100$$	$$\sum xf=3225$$			$$\sum(x-\overline{x})^2f=5418.75$$

$$\Rightarrow$$ $$Mean(\overline{x})=\dfrac{\sum xf}{\sum f}=\dfrac{3225}{100}=32.25$$

$$\Rightarrow$$ $$Variance(\sigma^2)=\dfrac{\sum(x-\overline{x})^2f}{\sum f}=\dfrac{5418.75}{100}=54.18$$

Question 2
1 / -0

If the mean of the numbers $$a,b,8,5,10$$ is $$6$$ and their variance is $$68$$, then $$ab$$ is equal to

A
$$6$$

B
$$7$$

C
$$12$$

D
$$14$$

E
$$25$$

Solution

Given,
$$\cfrac { a+b+8+5+10 }{ 5 } =6$$
$$\Rightarrow a+b+c+23=30$$
$$\Rightarrow a+b=7....(i)$$
and $$\quad \cfrac { \sum { { \left( { x }_{ 1 }-\overline { x } \right) }^{ 2 } } }{ n } ={ \sigma }^{ 2 }\quad $$
$$\cfrac { { (a-6) }^{ 2 }+{ (b-6) }^{ 2 }+4+1+16 }{ 5 } =\cfrac { 34 }{ 5 } \quad $$
$$\Rightarrow { (a-6) }^{ 2 }+{ (b-6) }^{ 2 }+21=34\quad $$
$$\Rightarrow { a }^{ 2 }+{ b }^{ 2 }-84+93=34\quad \left[ \because a+b=7 \right] $$
$$\quad \Rightarrow { a }^{ 2 }+{ b }^{ 2 }=25...(i)\quad $$
From Eq.(i)
$$\Rightarrow { (a+b) }^{ 2 }={ 7 }^{ 2 }$$
$$\Rightarrow { a }^{ 2 }+{ b }^{ 2 }+2ab=49$$
$$\Rightarrow 25+2ab=49$$ (From Eq.(ii))
$$\Rightarrow 2ab=24\Rightarrow ab=12\quad $$
Question 3
1 / -0

If the median of the data 6,7,x-2,x,18,21 written in ascending order is 16, then the variance of that data is

A
$$30\dfrac{1}{5}$$

B
$$31\dfrac{1}{3}$$

C
$$32\dfrac{1}{2}$$

D
$$33\dfrac{1}{3}$$

Solution

$$M=\dfrac{x-2+x}{2}=16$$

Variance = $$\dfrac{1}{n}\sum (x_i-\overline{x})^{2}$$
Required variance=$$31\dfrac{1}{3}$$
Question 4
1 / -0

If $$n=10, \bar{x}=12$$ and $$\sum x^2=1530$$, then calculate the coefficient of variation.

A
$$20$$

B
$$25$$

C
$$30$$

D
$$35$$

Solution

$$\sigma=\sqrt{\dfrac{\sum x^2}{n}-\left(\dfrac{\sum x}{n}\right)^2}$$

$$=\sqrt{\dfrac{1530}{10}-(12)^2}$$

$$=\sqrt{153-144}$$
$$=\sqrt{9}$$
$$=3$$

Coefficient of variation $$=\dfrac{\sigma}{\overline{x}}\times 100$$

$$=\dfrac{3}{12}\times 100$$

$$=\dfrac{1}{4}\times 100$$
$$=25$$

Question 5

1 / -0

Calculate the standard deviation of the following data.

x	$$3$$	$$8$$	$$13$$	$$18$$	$$23$$
f	$$7$$	$$10$$	$$15$$	$$10$$	$$8$$

$$6.32$$.

$$2.32$$.

$$3.32$$.

None of these

Solution

$$x$$	$$f$$	$$xf$$	$$(x-\overline{x})$$	$$(x-\overline{x})^2$$	$$(x-\overline{x})^2f$$
$$3$$	$$7$$	$$21$$	$$-10.2$$	$$104.04$$	$$728.28$$
$$8$$	$$10$$	$$80$$	$$-5.2$$	$$27.04$$	$$270.4$$
$$13$$	$$15$$	$$195$$	$$-0.2$$	$$0.04$$	$$0.6$$
$$18$$	$$10$$	$$180$$	$$4.8$$	$$23.04$$	$$230.4$$
$$23$$	$$8$$	$$184$$	$$9.8$$	$$96.04$$	$$768.32$$
	$$\sum f=50$$	$$\sum xf=660$$			$$\sum(x-\overline{x})^2=1998$$

$$\Rightarrow$$ $$\overline{x}=\dfrac{\sum xf}{\sum f}=\dfrac{660}{50}=13.2$$

Now, calculate standard deviation :

$$S=\sqrt{\dfrac{(x-\overline{x})^2}{\sum f}}=\sqrt{\dfrac{1998}{50}}=\sqrt{39.96}=6.32$$

Question 6

1 / -0

Calculate the standard deviation of the following data.
$$10, 20, 15, 8, 3, 4$$

$$5.97$$

$$59.7$$

$$4.97$$

None of these

Solution

$$\Rightarrow$$ The given numbers are $$3,4,8,10,15,20$$.

$$\Rightarrow$$ $$\overline{X}=\dfrac{3+4+8+10+15+20}{6}=10$$

$$X$$	$$X-\overline{X}$$	$$(X-\overline{X})^2$$
$$3$$	$$-7$$	$$49$$
$$4$$	$$-6$$	$$36$$
$$8$$	$$-2$$	$$4$$
$$10$$	$$0$$	$$0$$
$$15$$	$$5$$	$$25$$
$$20$$	$$10$$	$$100$$
		$$\sum (X-\overline{X})^2=214$$

$$\Rightarrow$$ $$S=\sqrt{\dfrac{(X-\overline{X})^2}{N}}$$

$$=\sqrt{\dfrac{214}{6}}$$

$$=\sqrt{35.66}$$

$$=5.97$$

Question 7
1 / -0

Let X denote the number of scores which exceed 4 in 1 toss of a symmetrical die. Consider the following statements :
1. The arithmetic mean of X is 6.
2. The standard deviation of X is 2.
Which of the above statements is/are correct ?

A
1 only

B
2 only

C
Both 1 and 2

D
Neither 1 nor 2

Solution

The only possible favorable outcomes are $$5$$ and $$6$$. Hence, the arithmetic mean of $$X$$ is 5.5.
The standard deviation of $$X=\dfrac{1}{2}((6-5.5)^2+(5-55)^2)=0.25$$
Question 8
1 / -0

The arithmetic mean of the observations 10,8,5,a,b is 6 and their variance 6.8. Then ab=

A
6

B
4

C
3

D
12

Solution

$$\dfrac{10+8+5+a+b}{5}=6,$$

$$\dfrac{1}{n}\sum \left ( x_i-\overline{x} \right )^{2}=6.8$$
solving above two equations we get values of a and b.
Question 9
1 / -0

Directions For Questions

As median divides an arranged sense into two equal parts, in similar way quartile divides an arranged series in $$4$$ equal part. For ungrouped frequency distribution formula of finding $$i^{th}$$ quartile $$=Q_i=\left\{i\cdot \left(\displaystyle\frac{N+1}{4}\right)\right\}^{th}$$ term, $$i=1, 2, 3$$.
Quartile deviation: half of difference between upper quartile & lower quartile.
$$\Rightarrow$$ Quartile deviation(Q.D.)$$=\displaystyle\frac{1}{2}(Q_3-Q_1)$$
$$\Rightarrow$$ Coefficient of quartile $$=\displaystyle\frac{Q_3-Q_1}{Q_3+Q_1}$$.

...view full instructions

Coefficient of quartile deviation of numbers $$6, 8, 9, 10, 11, 12, 14$$ is?

A
$$1/5$$

B
$$1/10$$

C
$$2/5$$

D
$$1/4$$

Solution

Arranging given numbers in ascending order,
$$6,8,9,10,11,12,14$$
Number of observation $$(N)=7$$
$$Q_i=\left\{i\left(\dfrac{N+1}{4}\right)\right\}^{th}term$$

$$Q_1=\left\{1.\left(\dfrac{7+1}{4}\right)\right\}^{th}term=2^{nd}term$$
$$2^{nd}term=8$$
$$\therefore$$ $$Q_1=8$$

$$Q_3=\left\{3.\left(\dfrac{7+1}{4}\right)\right\}^{th}term=6^{th}term$$
$$6^{th}term=12$$
$$\therefore$$ $$Q_3=12$$

$$\Rightarrow$$ Coefficient of quartile $$=\dfrac{Q_3-Q_1}{Q_3+Q_1}$$

$$=\dfrac{12-8}{12+8}$$

$$=\dfrac{4}{20}$$

$$=\dfrac{1}{5}$$
Question 10
1 / -0

Which of the following are measures of dispersion?

A
Standard Deviation,Median,Range

B
Standard Deviation,Mode,Range

C
Standard Deviation,Variance,Range

D
Mean,Mode,Median

Solution

Standard Deviation, Variance, and Range are measures of dispersion but the Mean, Mode, and Median are the measure of central tendency.

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Measures of Dispersion Test - 21

Result

Measures of Dispersion Test - 21

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Question 1

$$416$$

$$46$$

$$16$$

None of these

Solution

Question 2

$$6$$

$$7$$

$$12$$

$$14$$

$$25$$

Solution

Question 3

$$30\dfrac{1}{5}$$

$$31\dfrac{1}{3}$$

$$32\dfrac{1}{2}$$

$$33\dfrac{1}{3}$$

Solution

Question 4

$$20$$

$$25$$

$$30$$

$$35$$

Solution

Question 5

$$6.32$$.

$$2.32$$.

$$3.32$$.

None of these

Solution

Question 6

$$5.97$$

$$59.7$$

$$4.97$$

None of these

Solution

Question 7

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Solution

Question 8

6

4

3

12

Solution

Question 9

$$1/5$$

$$1/10$$

$$2/5$$

$$1/4$$

Solution

Question 10

Standard Deviation,Median,Range

Standard Deviation,Mode,Range

Standard Deviation,Variance,Range

Mean,Mode,Median

Solution

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