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(D) The given observations are $a, b, c, d, e$ with mean $m$ and standard deviation $S$.Standard deviation is defined as $S = \sqrt{\frac{\sum (x_i -

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$S$

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(D) The given observations are $a, b, c, d, e$ with mean $m$ and standard deviation $S$.Standard deviation is defined as $S = \sqrt{\frac{\sum (x_i - m)^2}{n}}$.When a constant $k$ is added to each observation,the new observations are $x_i' = x_i + k$.The new mean $m'$ is $\frac{\sum (x_i + k)}{n} = \frac{\sum x_i}{n} + k = m + k$.The new standard deviation $S'$ is $\sqrt{\frac{\sum (x_i' - m')^2}{n}}$.Substituting the values,$S' = \sqrt{\frac{\sum ((x_i + k) - (m + k))^2}{n}} = \sqrt{\frac{\sum (x_i - m)^2}{n}}$.Thus,$S' = S$.Adding a constant to each observation does not change the standard deviation.

Class interval	$0-10$	$10-20$	$20-30$	$30-40$
Frequency	$1$	$3$	$4$	$2$

Let $a, b, c, d$ and $e$ be the observations with mean $m$ and standard deviation $S$. The standard deviation of the observations $a+k, b+k, c+k, d+k$ and $e+k$ is

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