Measures of Dispersion Test

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Measures of Dispersion Test - 22

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Question 1
1 / -0

What are the objectives of measure of dispersion?

A
Helpful in use of further statistical analysis as in regression, correlation etc.

B
Reliability of measure of central tendency

C
Control of variability

D
All of the above

Solution

Objectives of measure of dispersion:
(i)Reliability of measure of central tendency
(ii)Control of variability
(iii)Helpful in use of further statistical analysis as in regression, correlation etc
Question 2
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Which of the following are measures of central tendency.

A
Percentile, Quartile, Median

B
Median,Mode, Percentile

C
Percentile, Quartile, Mode

D
Mean,Mode, Median

Solution

Mean, Mode, Median are the measures of central tendency. Thus the correct option is C.
Question 3
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The runs scored by Virat in 7 different matches are given as:
$$89,91,54,66,13,97,06$$
Calculate the mean deviation about the median for this data.

A
$$31.26$$

B
$$30.06$$

C
$$29.142$$

D
$$27.85$$

Solution

Arranging data in ascending order,
$$06,13,54,66,89,91,97$$
Here, number of observations $$(n)=7(odd)$$
$$\therefore$$ $$Median=\left(\dfrac{7+1}{2}\right)^{th}observation=4^{th}observation.$$
$$\therefore$$ $$Median(M) =66$$

$$x$$ $$|x-M|$$
$$06$$ $$60$$
$$13$$ $$53$$
$$54$$ $$12$$
$$66$$ $$0$$
$$89$$ $$23$$
$$91$$ $$25$$
$$97$$ $$31$$
$$\sum |x-M|=204$$
$$\Rightarrow$$ Mean deviation about median $$=\dfrac{\sum|x-M|}{7}=\dfrac{204}{7}=29.142$$
Question 4
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The formula of standard error of sample mean is ________.

A
$$\dfrac{\sqrt{PQ}}{\sqrt{n}}$$

B
$$\dfrac{\sigma}{\sqrt{n}}$$

C
$$\dfrac{\sqrt{N-n}}{\sqrt{N-1}}$$

D
$$\dfrac{\sqrt{N}}{\sigma}$$

Solution

The standard deviation of the sample mean is equivalent to the standard deviation of the error in the sample mean with respect to the true mean. Therefore, the standard error of the mean can also be understood as the standard deviation of the error in the sample mean with respect to the true mean.
Question 5
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Find the mean deviation about the mean as well as the coefficient of Mean Deviation about mean of the following set data:$$ 4, 7, 14, 11, 9$$.

A
$$2.8$$ and $$0.311$$

B
$$2.1$$ and $$0.211$$

C
$$24.8$$ and $$0.411$$

D
$$21.3$$ and$$0.566$$

Solution

Mean of the following set of data can be calculated by $$(\bar{x})=\dfrac{4+7+14+11+9}{5}=\dfrac{45}{5}=9$$

$$x_i$$ $$\bar{x}-x_i$$ $$|\bar{x}-x_i|$$
$$4$$ $$9-4=5$$ $$|5|=5$$
$$7$$ $$9-7=2$$ $$|2|=2$$
$$14$$ $$14-9=-5$$ $$|-5|=5$$
$$11$$ $$11-9=-2$$ $$|-2|=2$$
$$9$$ $$9-9=0$$ $$ |0|=0$$
Mean Deviation can be calcuated by (M.D) $$=\dfrac{\sum |\bar{x}-x_i|}{n}=\dfrac{5+2+5+2+0}{5}=\dfrac{14}{5}=2.8$$

Coefficient of Mean Deviation about mean can be calculated by $$=\dfrac{M.D}{Mean \enspace (\bar{x})}=\dfrac{2.8}{9}=0.311$$
Question 6
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Which of the following is correct about measure of dispersion?

A
It does not measure direction of the variation.

B
Dispersion measures the extent to which the items vary from some central value

C
Measures only degree of variation

D
All of the above

Solution

Dispersion measures the extent to which the items vary from some central value. It measures only degree of variation not measure direction of variation.
Question 7
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The mean deviation about median for the following data is 4.4.calculate value of $$x$$
$$2,3,x,10,17$$.

A
$$3$$

B
$$5$$

C
$$7$$

D
$$17$$

Solution

Median $$=$$ $$\dfrac{N+1}{2}$$ $$=$$ 3rd term

$$\Rightarrow x$$ is median

Mean Deviation $$=$$ $$\dfrac{1}{N} \sum fi |Xi-M|$$

$$4.4 = \dfrac{|2-x| + |3-x| + |x-10| + |x-17|}{5}$$

Now as $$x > 2$$

$$|x-2|$$ $$=$$ $$x-2$$
$$22 =$$ $$x-2 + |3-x+10-x+17-x|$$
$$22 =$$ $$x-3x-2+30$$
$$2x=6$$
$$x=3$$
Question 8
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If mean deviation about Mean of a particular data consisting $$10$$ observations is$$7$$, then what will be value of mean deviation when each is multiplied by $$5$$?

A
$$35$$

B
$$45$$

C
$$55$$

D
$$65$$

Solution

Suppose original numbers were : x1 , x2, x3, ……, xn
Thus, $$(x1+x2+x3+…….+xn)/n = 10$$ (by the very definition of mean
After adding 1 to each number, the sum would be : 5$$\times$$ $$(x1+x2+x3+…….+xn) + n$$

Thus new mean would be: $$(5\times(x1+x2+x3+…….+xn) + n)/n = 5\times10 +1 = 51$$
Question 9
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The study of standard error will help_____________.

A
to test if the value of sample differs significantly from their corresponding value of the population

B
to test significant difference between two sample means and proportions

C
to calculate point estimate of the population

D
all of the above

Solution

Calculation of standard error helps in determining the difference between two sample means and proportion or how much the value of sample differs from the corresponding value of population.
Question 10
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The standard error estimate is used to measure____________.

A
the dispersion from the arithmetic mean

B
the deviation from the average

C
both (A) and (B)

D
none of the above

Solution

Standard error refers to the standard deviation of its sampling distribution from its population mean. Standard error is a statistical term that measures the accuracy with which sample represents the population.

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$$x$$	$$\|x-M\|$$
$$06$$	$$60$$
$$13$$	$$53$$
$$54$$	$$12$$
$$66$$	$$0$$
$$89$$	$$23$$
$$91$$	$$25$$
$$97$$	$$31$$
	$$\sum \|x-M\|=204$$

$$x_i$$	$$\bar{x}-x_i$$	$$\|\bar{x}-x_i\|$$
$$4$$	$$9-4=5$$	$$\|5\|=5$$
$$7$$	$$9-7=2$$	$$\|2\|=2$$
$$14$$	$$14-9=-5$$	$$\|-5\|=5$$
$$11$$	$$11-9=-2$$	$$\|-2\|=2$$
$$9$$	$$9-9=0$$	$$ \|0\|=0$$

Measures of Dispersion Test - 22

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Measures of Dispersion Test - 22

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SHARING IS CARING

Question 1

Helpful in use of further statistical analysis as in regression, correlation etc.

Reliability of measure of central tendency

Control of variability

All of the above

Solution

Question 2

Percentile, Quartile, Median

Median,Mode, Percentile

Percentile, Quartile, Mode

Mean,Mode, Median

Solution

Question 3

$$31.26$$

$$30.06$$

$$29.142$$

$$27.85$$

Solution

Question 4

$$\dfrac{\sqrt{PQ}}{\sqrt{n}}$$

$$\dfrac{\sigma}{\sqrt{n}}$$

$$\dfrac{\sqrt{N-n}}{\sqrt{N-1}}$$

$$\dfrac{\sqrt{N}}{\sigma}$$

Solution

Question 5

$$2.8$$ and $$0.311$$

$$2.1$$ and $$0.211$$

$$24.8$$ and $$0.411$$

$$21.3$$ and$$0.566$$

Solution

Question 6

It does not measure direction of the variation.

Dispersion measures the extent to which the items vary from some central value

Measures only degree of variation

All of the above

Solution

Question 7

$$3$$

$$5$$

$$7$$

$$17$$

Solution

Question 8

$$35$$

$$45$$

$$55$$

$$65$$

Solution

Question 9

to test if the value of sample differs significantly from their corresponding value of the population

to test significant difference between two sample means and proportions

to calculate point estimate of the population

all of the above

Solution

Question 10

the dispersion from the arithmetic mean

the deviation from the average

both (A) and (B)

none of the above

Solution

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