Tag: measures of dispersion and skewness

Questions Related to measures of dispersion and skewness

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. $6,499.42$

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

The sum of squared deviations of a set of $n$ values from their mean 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. Minimum

  2. Least

  3. Maximum

  4. Zero

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Correct Option: B
Explanation:

The sum of the squared deviations from their their mean is the least value.



If $y=-8x-5$ and SD of $x$ is $3$, then SD of $y$ 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. $8$

  2. $24$

  3. $3$

  4. None of these

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Correct Option: B
Explanation:
Given that std dev of x is $\sigma(x)=3$
Given equation is $y=-8x-5$

applying std dev on both sides we get

$\sigma(y)=\sigma(-8x-5)$

$\implies \sigma(y)=\sigma(-8x)+\sigma(-5)$

$\implies \sigma(y)=\sigma(-8x)+0$ (since, $\sigma(c)=0$)

$\implies \sigma(y)=8\sigma(x)$ (since, $\sigma(aX)=|a|\sigma(X)$)

$\implies \sigma(y)=8\times 3=24$

Therefore, standard deviation of $y$ is $24$.