The mean square deviation of a set of observa | vedclass

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(A) Given that the mean square deviation about $c = -2$ is $18$:$\frac{1}{n} \sum (x_i + 2)^2 = 18 \implies \sum (x_i^2 + 4x_i + 4) = 18n \implies \su

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$3$

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(A) Given that the mean square deviation about $c = -2$ is $18$:$\frac{1}{n} \sum (x_i + 2)^2 = 18 \implies \sum (x_i^2 + 4x_i + 4) = 18n \implies \sum x_i^2 + 4 \sum x_i + 4n = 18n \implies \sum x_i^2 + 4 \sum x_i = 14n$  $(1)$Given that the mean square deviation about $c = 2$ is $10$:$\frac{1}{n} \sum (x_i - 2)^2 = 10 \implies \sum (x_i^2 - 4x_i + 4) = 10n \implies \sum x_i^2 - 4 \sum x_i + 4n = 10n \implies \sum x_i^2 - 4 \sum x_i = 6n$  $(2)$Adding $(1)$ and $(2)$:$2 \sum x_i^2 = 20n \implies \frac{\sum x_i^2}{n} = 10$Subtracting $(2)$ from $(1)$:$8 \sum x_i = 8n \implies \frac{\sum x_i}{n} = 1$The variance $\sigma^2$ is given by $\frac{\sum x_i^2}{n} - (\frac{\sum x_i}{n})^2 = 10 - (1)^2 = 9$.Therefore,the standard deviation $\sigma = \sqrt{9} = 3$.

$x_i$	$92$	$93$	$97$	$98$	$102$	$104$	$109$
$f_i$	$3$	$2$	$3$	$2$	$6$	$3$	$3$

The mean square deviation of a set of observations $x_1, x_2, \dots, x_n$ about a point $c$ is defined as $\frac{1}{n} \sum_{i=1}^n (x_i - c)^2$. If the mean square deviations about $-2$ and $2$ are $18$ and $10$ respectively,find the standard deviation of this set of observations.

Similar Questions

If the variance of the numbers $2, 3, 11$ and $x$ is $\frac{49}{4}$,then the values of $x$ are

Find the mean and variance for the following data:

$x_i$ $92$ $93$ $97$ $98$ $102$ $104$ $109$

$f_i$ $3$ $2$ $3$ $2$ $6$ $3$ $3$

Statement-$1$: The variance of the first $n$ even 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 $\frac{n(4n^2 - 1)}{3}$.

The variance of the following data is:

$x_i$ $6$ $10$ $14$ $18$ $24$ $28$ $30$

$f_i$ $2$ $4$ $7$ $12$ $8$ $4$ $3$

The variance of $20$ observations is $5$. If each of the observations is multiplied by $2$,then the variance of the resulting observations is:

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