Tag: measures of dispersion

Questions Related to measures of dispersion

To find the median, it is necessary to arrange the data in _______.

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. ascending order

  2. descending order

  3. ascending or descending order

  4. any of them

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Correct Option: D

Calculate standard deviation of the following data.

X $0-10$ $10-20$ $20-30$ $30-40$ $40-50$
f $10$ $8$ $15$ $8$ $4$
business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $-12$

  2. $12$

  3. $12.36$

  4. $152.77$

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Correct Option: C
Explanation:
Marks Mid-values(X) f dxi$=X-25/10$ $fdx _if$ $dx _i^2$
$0-10$ $5$ $10$ $-2$ $-20$ $40$
$10-20$ $15$ $8$ $-1$ $-8$ $8$
$20-30$ $25$ $15$ $0$ $0$ $0$
$30-40$ $35$ $8$ $1$ $8$ $8$
$40-50$ $45$ $4$ $2$ $8$ $16$
Total $45$ $-12$ $72$

$\sqrt{\displaystyle\frac{72}{45}-\frac{144}{45\times 45}}\times 10$
$=12.36$

The standard deviation of $10, 16, 10, 16, 10, 10, 16, 16$ is?

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $4$

  2. $6$

  3. $3$

  4. $0$

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Correct Option: C
Explanation:

Assume $A=13;$ hence $d=A-13$

x d $d^2$
$10$ $-3$ $9$
$16$ $3$ $9$
$10$ $-3$ $9$
$16$ $3$ $9$
$10$ $-3$ $9$
$10$ $-3$ $9$
$16$ $-3$ $9$
$16$ $3$ $9$
$\displaystyle\sum d=0$ $\displaystyle\sum d^2=72$

Use following formula:
$\sqrt{\displaystyle\frac{\displaystyle\sum d^2}{n}-\left[\displaystyle\frac{\displaystyle\sum d}{n}\right]^2}$
$\sqrt{\displaystyle\frac{72}{8}-\left[\displaystyle\frac{0}{8}\right]^2}$
$\sqrt{9}$
$=3$.

Calculate standard deviation of the following data.

X $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$
f $2$ $7$ $10$ $12$ $15$ $11$ $10$ $6$ $3$
business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $3.95$

  2. $1.99$

  3. $14$

  4. None of the above

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Correct Option: B
Explanation:
X f fx X-Mean $f(X-x)^2$
$10$ $2$ $20$ $-4$ $32$
$11$ $7$ $77$ $-3$ $63$
$12$ $10$ $120$ $-2$ $40$
$13$ $12$ $156$ $-11$ $2$
$14$ $15$ $210$ $0$ $0$
$15$ $11$ $165$ $1$ $11$
$16$ $10$ $160$ $2$ $40$
$17$ $6$ $102$ $3$ $54$
$18$ $3$ $54$ $4$ $48$
Total $76$ $fX=1064$ $f(X-X)^2=300$

$^-{X}=\displaystyle\frac{1064}{76}=14$
$\sigma x=\sqrt{\displaystyle\frac{300}{76}}$
$=\sqrt{3.95}$
$=1.99$.

Find the mean and standard deviation of the following observations: X $=2, 5, 7, 8, 13$.

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $7$ & $3.63$

  2. $3.63$ & $7$

  3. $7.63$ & $3$

  4. $3$ & $7.63$

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Correct Option: A
Explanation:

$\displaystyle\frac{2+5+7+8+13}{5}=7$
$\sqrt{\displaystyle\frac{4+25+49+64+169}{5}}-49=3.63$.

Find the present value of Rs. $10,000$ to be required after $5$ years if the interest rate be $9\%$. Given that $(1.09)^5=1.5386$.

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $6,994.42$

  2. $6,949.24$

  3. $6,449.24$

  4. $6,499.42$

Show answer
Correct Option: D
Explanation:

Here, $i=0.09=9\%$
$n=5$
$A _n=10,000$
Required present value $=\displaystyle\frac{A _n}{(1+i)^n}$
$=\displaystyle\frac{10,000}{(1+0.09)^5}$
$=Rs. 6499.42$.

If the standard deviation of the numbers $2,3,a$ and $11$ is $3.5,$ then which of the following is true?

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $3a^2-32a+84=0$

  2. $3a^2-34a+91=0$

  3. $3a^2-23a+44=0$

  4. $3a^2-26a+55=0$

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Correct Option: A
Explanation:
Numbers are $2,3,a$ and $11$
$N=4$
Standard deviation $\sigma=3.5$
Mean of numbers $(\overline{x})=\dfrac{2+3+11+a}{4}=\dfrac{16+a}{4}$

$\Rightarrow$  $\sigma^2=\dfrac{1}{N}\sum(x _i-\overline{x})^2$
$\Rightarrow$  $3.5\times 3.5\times 4\times  16=(8+a)^2+(4+a)^2+(16-3a)^2+(a-28)^2$
$\Rightarrow$  $784=64+a^2+16a+16+a^2+8a+256+9a^2-96a+a^2+784-56a$
$\Rightarrow$  $784=12a^2-128a+1120$
$\Rightarrow$  $196=3a^2-32a+280$

$\Rightarrow$  $3a^2-32a+84=0$

The two observations $A$ & $B$ are given by $100, 101, ........149$ and $200, 201, .......,249$ with $V _{A}$ and $V _{B}$ are variances of $A$ and $B$ than $V _{A}$ is equal to :

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $V _{B}$

  2. $100\ V _{B}$

  3. $50\ V _{B}$

  4. $200\ V _{B}$

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Correct Option: A
Explanation:

$\sigma {x^2} = \frac{{\sum {{d^2}} }}{h}${here deviation are taken  from mean}

Since, A and B have consecutive integers therefore both have same standard deviation and hence variation 
${ V _{ A } }={ V _{ B } }\sum { { d^{ 2 } }\, \, is\, \, same } $

In expected rate of return for constant growth, dividends are expected to grow but with the _____________.

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. constant rate

  2. variable rate

  3. yielding rate

  4. returning yield

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Correct Option: A

If the standard deviation of the values $2,4,6,8$ is $2.33$, then the standard deviation of the values $4,6,8,10$ is

business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

  1. $0$

  2. $2.58$

  3. $4.66$

  4. None of these

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Correct Option: B
Explanation:

Given data values are $4,6,8,10$


Mean of the data is $\dfrac{4+6+8+10}{4}=\dfrac{28}{4}=7$

Therefore standard deviation is $\sqrt{\dfrac{(4-7)^2+(6-7)^2+(8-7)^2+(10-7)^2}{4-1}}=\sqrt{\dfrac{20}{3}}=2.58$