Question

A group of $10$ observations has mean $5$ and S.D.$2\sqrt 6$. Another group of $20$ observations has mean $5$ and S.D. $3\sqrt 2$, then the S.D. of combined group of $30$ observations is
(A) $\sqrt 5$
(B) $2\sqrt 5$
(C) $3\sqrt 5$
(D) None of these

Answer 1

In this question they have given mean and standard deviation of two types of observation. We have to find mean and standard deviation of another observation related to the given observations. By using combined mean and combined standard deviation we will find the required result.

Formula used: The formula of combined mean,
${\overline x _{12}} = \dfrac{{{n_1}.{{\overline x }_1} + {n_2}.{{\overline x }_2}}}{{{n_1} + {n_2}}}$
The formula of combined Standard deviation
${\sigma _{12}} = \sqrt {\dfrac{{{n_1}.\left( {\sigma _1^2 + d_1^2} \right) + {n_2}.\left( {\sigma _2^2 + d_2^2} \right)}}{{{n_1} + {n_2}}}}$
Where,${d_1} = {\overline x _1} - {\overline x _{12}}$, ${d_2} = {\overline x _2} - {\overline x _{12}}$
Putting the given mean and S.D. and number of observations in this formula we will get the required S.D.

Complete step-by-step answer:
It is given that a group of $10$ observations has mean $5$ and S.D. $2\sqrt 6$. Another group of $20$ observations has mean $5$ and S.D. $3\sqrt 2$.
We need to find out the S.D. of a combined group of $30$ observations.
If ${n_1}$ and ${n_2}$ are number of observations, ${\overline x _1}$ and ${\overline x _2}$ are the mean, ${\sigma _1}$ and ${\sigma _2}$ are the S.D. of both the groups then,
Hence the given of a group of $10$ observations,
${n_1} = 10,{\overline x _1} = 5,{\sigma _1} = 2\sqrt 6$
Hence the given of a group of $20$ observations,

${n_2} = 20,{\overline x _2} = 5,{\sigma _2} = 3\sqrt 2$
Combined mean is,
$\Rightarrow {\overline x _{12}} = \dfrac{{{n_1}.{{\overline x }_1} + {n_2}.{{\overline x }_2}}}{{{n_1} + {n_2}}}$
Substituting the given values to find the mean of the $30$ observations,
$\Rightarrow {\overline x _{12}} = \dfrac{{10 \times 5 + 20 \times 5}}{{10 + 20}}$
Simplifying we get,
$\Rightarrow {\overline x _{12}} = \dfrac{{50 + 100}}{{30}}$
$\Rightarrow {\overline x _{12}} = \dfrac{{150}}{{30}} = 5$
Putting the combined mean and mean of the observations in the formula we get,

$${d_1} = {\overline x _1} - {\overline x _{12}} = 5 - 5 = 0 \\\ {d_2} = {\overline x _2} - {\overline x _{12}} = 5 - 5 = 0 \\\$$

Now the formula for Standard deviation of combined group of $30$ observations is
$\Rightarrow {\sigma _{12}} = \sqrt {\dfrac{{{n_1}.\left( {\sigma _1^2 + d_1^2} \right) + {n_2}.\left( {\sigma _2^2 + d_2^2} \right)}}{{{n_1} + {n_2}}}}$
Substituting the given values to find the standard deviation of the $30$ observations,
\Rightarrow {\sigma _{12}} = \sqrt {\dfrac{{10.\left\\{ {{{\left( {2\sqrt 6 } \right)}^2} + 0} \right\\} + 20\left\\{ {{{\left( {3\sqrt 2 } \right)}^2} + 0} \right\\}}}{{10 + 20}}}
Simplifying the terms,
$\Rightarrow {\sigma _{12}} = \sqrt {\dfrac{{10 \times 24 + 20 \times 18}}{{30}}}$
Multiplying the terms we get,
$\Rightarrow {\sigma _{12}} = \sqrt {\dfrac{{240 + 360}}{{30}}}$
Adding the terms we get,
$\Rightarrow {\sigma _{12}} = \sqrt {\dfrac{{600}}{{30}}}$
Dividing the terms,
$\Rightarrow {\sigma _{12}} = \sqrt {20}$
Taking square root,
$\Rightarrow {\sigma _{12}} = 2\sqrt 5$
The S.D. of combined group of $30$ observations is ${\sigma _{12}} = 2\sqrt 5$

So, the correct answer is “Option B”.

Note: Standard deviation:
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
$\sigma = \sqrt {\dfrac{{\sum {{{\left( {{x_i} - \mu } \right)}^2}} }}{N}}$
Where,
$\sigma$=population standard deviation
$N$= the size of the population
${x_i}$=each value from the population
$\mu$= the population mean.