Tag: measures of dispersion and skewness

Questions Related to measures of dispersion and skewness

Suppose for $40$ observations, the variance is $50$. If all the observations are increased by $20$, the variance of these increased observation will be

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. $20$

  2. $50$

  3. $30$

  4. None of these

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Correct Option: B
Explanation:

The variance for $40$ observations is $50$
Variance is independent of the number of observations. Therefore, variance for $60$ observations is $50$

The variance of $5$ numbers is $10$. If each number is divided by $2$, then the variance of new numbers 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. $5.5$

  2. $2.5$

  3. $5$

  4. None of these

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Correct Option: B
Explanation:

Given that variance of $5$ number is $10$


Therefore, $Var(X)=10$

Given that each number is divided by $2$

Therefore, variance of new numbers is $Var(\dfrac X2)$

$Var(\dfrac X2)=\dfrac 14Var(X)=\dfrac 1 4(10)=2.5$

Hence, variance of new numbers is $2.5$

Find $Var(2X+3)$

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. $5 Var(X)+3$

  2. $4 Var(X)+3$

  3. $4 Var(X)$

  4. None of these

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Correct Option: C
Explanation:

$Var(2X+3)=Var(2X)+Var(3)$


$\implies Var(2X+3)=Var(2X)+0$ (Since, $Var(c)=0$)

$\implies Var(2X+3)=2^2Var(X)$ (Since, $Var(aX)=a^2Var(X)$)

$\implies Var(2X+3)=4Var(X)$

If a, b are constants then, $Var(a+bX)$ 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. $Var(a)+Var(X)$

  2. $Var(a)-Var(X)$

  3. $b^2Var(X)$

  4. None of these

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Correct Option: C
Explanation:

$Var(a+bX)=Var(a)+Var(bX)$


$\implies Var(a+bX)=0+Var(bX)$ (Since, $Var(c)=0$)

$\implies Var(a+bX)=b^2Var(X)$ (Since, $Var(aX)=a^2Var(X)$)

If the standard deviation of a population is $9$, the population variance 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. $9$

  2. $3$

  3. $21$

  4. $81$

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Correct Option: D
Explanation:

We know that variance is square of standard deviation.


Given that standard deviation is $9$

Therefore, variance $=9^2=81$

If $X, Y $ are independent then $SD(X-Y)$ 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. $SD(X)-SD(Y)$

  2. $SD(X)+SD(Y)$

  3. $\sqrt{SD(X)+SD(Y)}$

  4. None of these

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Correct Option: B
Explanation:

$SD(X-Y)=SD(X)+SD(-Y)$


$\implies SD(X-Y)=SD(X)+SD(Y)$ (since, $SD(aX)=|a|SD(X)$)

The variance of $20$ observations is $5$. If each observation is multiplied by $2$, then what is the new variance of the resulting observations?

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. $5$

  2. $10$

  3. $20$

  4. $40$

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Correct Option: C
Explanation:

If each observation is multiplied by $2$, then mean$(\mu )$ will get doubled.

Now, variance $=\sigma ^{ 2 }=\dfrac { (X-\mu )^{ 2 } }{ n } $
Now here both $X$ and $\mu $ will be doubled, so the variance will become four times.
Thus, the new variance will be $4$ times the older variance which is $4\times 5=20$.
Hence, C is correct.

The formula Of $X^2$ distribution 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. ${X^2} = \dfrac {{S}^{2} _{1}}{{S}^{2} _{2}}$

  2. ${X^2} = \dfrac {E}{{\Sigma}(O - E)^2}$

  3. ${X^2} = {\Sigma} \dfrac {E}{(E - O)^2}$

  4. ${X^2} = {\Sigma} \dfrac {(O - E)^2}{E}$

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Correct Option: D
Explanation:

The correct formula for Chi square distribution is given in option D, where O means the observed number in the table and E means the corresponding expected number.

For a large sample, the sampling distribution of $X^2$ may form________.

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. a continuous curve

  2. severely skewed curve

  3. the symmetrical curve

  4. either (B) or (C)

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Correct Option: A
Explanation:

For a large sample, the sampling distribution of Chi square distribution may form a continuous curve. Chi square distribution helps in measuring how well the observed distribution of data fits with the distribution that is expected.

${X}^2$ test 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. ${\sum _{i = 1}^{n}} {A{x}^1} = A{x}^1 + A{x}^2 + ... + A{x}^n$

  2. $V = (r - 1) (e - 1)$

  3. $\dfrac {{\Sigma}(O - E)^2}{E}$

  4. $ r = \dfrac {{\Sigma} _{xy}}{\sqrt{{\Sigma}{{x}^2}{{y}^2}}}$

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Correct Option: C
Explanation:

The correct formula for the estimation of Chi-Square test is mentioned in Option C, where O represents the frequency observed while E represents the frequency expected.