Statement 1: The variance of the first n odd | vedclass

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(D) The first $n$ odd natural numbers are $1, 3, 5, \dots, (2n-1)$.The mean $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} (2i-1) = \frac{1}{n} \cdot n^2 = n$.

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Statement $1$ is true,Statement $2$ is true; Statement $2$ is a correct explanation for Statement $1$.

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(D) The first $n$ odd natural numbers are $1, 3, 5, \dots, (2n-1)$.The mean $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} (2i-1) = \frac{1}{n} \cdot n^2 = n$.The sum of squares is $\sum_{i=1}^{n} (2i-1)^2 = \sum (4i^2 - 4i + 1) = 4 \frac{n(n+1)(2n+1)}{6} - 4 \frac{n(n+1)}{2} + n = \frac{n(4n^2-1)}{3}$.Variance $\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2 = \frac{4n^2-1}{3} - n^2 = \frac{4n^2-1-3n^2}{3} = \frac{n^2-1}{3}$.Thus,Statement $1$ is true and Statement $2$ is true. Statement $2$ provides the correct formulas used to derive Statement $1$.

$x_i$	$4$	$8$	$11$	$17$	$20$	$24$	$32$
$f_i$	$3$	$5$	$9$	$5$	$4$	$3$	$1$

\text{Particulars}	\text{Firm } $A$ \text{ and Firm } $B$
\text{No. of wage earners}	$A: 586, B: 648$
\text{Mean of monthly wages}	$A: Rs. 5253, B: Rs. 5253$
\text{Variance of wages}	$A: 100, B: 121$

Statement $1$: The variance of the first $n$ odd natural numbers is $\frac{n^2 - 1}{3}$.
Statement $2$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2 - 1)}{3}$.

Similar Questions

Find the variance and standard deviation for the following data:

$x_i$ $4$ $8$ $11$ $17$ $20$ $24$ $32$

$f_i$ $3$ $5$ $9$ $5$ $4$ $3$ $1$

Calculate the variance if $\Sigma x_i^2 = 18000$ and $\Sigma x_i = 960$,for $60$ observations.

The mean and the standard deviation of a data set of $8$ items are $25$ and $5$ respectively. If two items $15$ and $25$ are added to this data,then the variance of the new data is:

Let $x_{i} (1 \leq i \leq 10)$ be ten observations of a random variable $X$. If $\sum_{i=1}^{10} (x_{i} - p) = 3$ and $\sum_{i=1}^{10} (x_{i} - p)^{2} = 9$,where $0 \neq p \in R$,then the standard deviation of these observations is:

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Statement $1$: The variance of the first $n$ odd natural numbers is $\frac{n^2 - 1}{3}$.Statement $2$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2 - 1)}{3}$.