Class 11: Statistics – Very Short Answer Questions

$\displaystyle \textbf{Question 1. }\text{Write the variance of first }n\text{ natural numbers.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The first }n\text{ natural numbers are }1,2,3,\ldots,n.$
$\displaystyle \text{Mean }(\overline{x})=\frac{1+2+\cdots+n}{n}=\frac{n+1}{2}$
$\displaystyle \text{Variance }(\sigma^2)=\frac{\sum x^2}{n}-\overline{x}^{\,2}$
$\displaystyle =\frac{\frac{n(n+1)(2n+1)}{6}}{n}-\left(\frac{n+1}{2}\right)^2$
$\displaystyle =\frac{(n+1)(2n+1)}{6}-\frac{(n+1)^2}{4}$
$\displaystyle =\frac{(n+1)\left[4n+2-3n-3\right]}{12}$
$\displaystyle =\frac{(n+1)(n-1)}{12}$
$\displaystyle =\frac{n^2-1}{12}$
$\displaystyle \therefore \text{Variance of first }n\text{ natural numbers }=\frac{n^2-1}{12}$
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$\displaystyle \textbf{Question 2. }\text{If the sum of the squares of deviations for }10\text{ observations taken from}$
$\displaystyle \text{their mean is }2.5,\text{ then }\text{write the value of standard deviation.}$
$\displaystyle \text{Answer:}$
$\displaystyle \sigma=\sqrt{\frac{\sum (x-\overline{x})^2}{n}}$
$\displaystyle =\sqrt{\frac{2.5}{10}}$
$\displaystyle =\sqrt{0.25}$
$\displaystyle =0.5$
$\displaystyle \therefore \text{Standard deviation}=0.5$
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$\displaystyle \textbf{Question 3. }\text{If }x_1,x_2,\ldots,x_n\text{ are }n\text{ values of a variable }X\text{ and }y_1,y_2,\ldots,y_n\text{ are }n\text{ values}$
$\displaystyle \text{of variable }Y\text{ such that }y_i=ax_i+b,\ i=1,2,\ldots,n,\text{ then write Var}(Y)\text{ in terms}$
$\displaystyle \text{of Var}(X).$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Var}(aX+b)=a^2\text{Var}(X)$
$\displaystyle \therefore \text{Var}(Y)=a^2\text{Var}(X)$
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$\displaystyle \textbf{Question 4. }\text{If }X\text{ and }Y\text{ are two variates connected by the relation }Y=\frac{aX+b}{c}\text{ and}$
$\displaystyle \text{Var}(X)=\sigma^2,\text{ then write the expression for the standard deviation of }Y.$
$\displaystyle \text{Answer:}$
$\displaystyle Y=\frac{a}{c}X+\frac{b}{c}$
$\displaystyle \text{Var}(Y)=\left(\frac{a}{c}\right)^2\text{Var}(X)$
$\displaystyle =\left(\frac{a}{c}\right)^2\sigma^2$
$\displaystyle \text{S.D.}(Y)=\sqrt{\text{Var}(Y)}$
$\displaystyle =\sqrt{\left(\frac{a}{c}\right)^2\sigma^2}$
$\displaystyle =\left|\frac{a}{c}\right|\sigma$
$\displaystyle \therefore \text{Standard deviation of }Y=\left|\frac{a}{c}\right|\sigma$
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$\displaystyle \textbf{Question 5. }\text{In a series of }20\text{ observations, }10\text{ observations are each equal to }$
$\displaystyle k\text{ and each of the remaining }\text{half is equal to }-k.\text{ If the standard deviation of the}$
$\displaystyle \text{observations is }2,\text{ then write the value }\text{of }k.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Mean}=\frac{10k+10(-k)}{20}=0$
$\displaystyle \sigma=\sqrt{\frac{10(k-0)^2+10(-k-0)^2}{20}}$
$\displaystyle =\sqrt{\frac{10k^2+10k^2}{20}}$
$\displaystyle =\sqrt{k^2}=|k|$
$\displaystyle \text{Given, }\sigma=2$
$\displaystyle |k|=2$
$\displaystyle \therefore k=2$
$\\$

$\displaystyle \textbf{Question 6. }\text{If each observation of a raw data whose standard deviation is }\sigma\text{ is multiplied by }$
$\displaystyle a,\text{ then } \text{write the S.D. of the new set of observations.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{If each observation is multiplied by }a,\text{ then standard deviation is multiplied by }|a|.$
$\displaystyle \therefore \text{New S.D.}=|a|\sigma$
$\\$

$\displaystyle \textbf{Question 7. }\text{If a variable }X\text{ takes values }0,1,2,\ldots,n\text{ with frequencies }$
$\displaystyle {^nC_0},{^nC_1},{^nC_2},\ldots,{^nC_n},\text{ then write} \text{variance }X.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Here frequencies are binomial coefficients }{^nC_r}.$
$\displaystyle \therefore X\text{ follows a binomial distribution with }p=\frac{1}{2},\quad q=\frac{1}{2}$
$\displaystyle \text{Variance}=npq$
$\displaystyle =n\times\frac{1}{2}\times\frac{1}{2}$
$\displaystyle =\frac{n}{4}$
$\displaystyle \therefore \text{Variance of }X=\frac{n}{4}$
$\\$