Question

the standard deviation of 20 observations is root 5.if each observation is multiplied by 2. find the standard deviation and variance of the resulting observation.​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{Standard\;deviation\;of\;20\;items\;is\;\sqrt{5}}$

$\underline{\textsf{To find:}}$

$\textsf{Standard deviation and variance when each observation}$

$\textsf{is multiplied by 2}$

$\underline{\textsf{Solution:}}$

$\textsf{Concept:}$

$\boxed{\begin{minipage}{9cm} \\\textsf{If each data is multiplied by a constant k, the new}\\\\\textsf{standard deviation is}\;\mathsf{k}\;\textsf{times the old standard deviation} \end{minipage}}$

$\textsf{When each observation is multiplied by 2}$

$\textsf{New standard deviation}$

$\mathsf{=2{\times}}\textsf{old standard deviation}$

$\mathsf{=2{\times}\sqrt{5}}$

$\mathsf{=2\sqrt{5}}$

$\boxed{\mathsf{Variance=(Standard\;deviation)^2}}$

$\mathsf{New\;variance=(New\;standard\;deviation)^2}$

$\mathsf{New\;variance=(2\sqrt{5})^2}$

$\mathsf{New\;variance=4{\times}5}$

$\mathsf{New\;variance=20}$

$\underline{\textsf{Answer:}}$

$\mathsf{New\;standard\;deviation=2\sqrt{5}}$

$\mathsf{New\;variance=20}$

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Answer 2

Given :- The standard deviation of 20 observations is root 5.if each observation is multiplied by 2. find the standard deviation and variance of the resulting observation. ?

Solution :-

we know that,

• when each observation is multiplied by a , standard deviation of observations is also multiply by a .

So,

→ standard deviation = √5

→ each observation is multiplied by = 2.

then,

→ New standard deviation = 2 * √5 = 2√5

now, we know that,

• variance = (standard deviation)² .

therefore,

→ New variance = (new standard deviation)² = (2√5)² = 4 * 5 = 20 .

Hence, the standard deviation and variance of the resulting observation is 2√5 and 20 respectively .

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