Questions & Answers

Question

Answers

Answer

Verified

91.5k+ views

The variance will be calculated by,

${\sigma ^2} = \dfrac{{{{\sum {\left( {x - \bar x} \right)} }^2}}}{{n - 1}}$

$\sigma $ , will be the variance

$x$ , will be the first term

$\bar x$ , will be the next term

$n$ , will be the number of terms

Standard deviation,

$Standard{\text{ deviation = }}\sqrt {Variance} $

Mean deviation,

$M.D = \dfrac{{\sum x }}{N}$

So we have the series given as \[x = 160,160,161,162,163,163,164,164,170\]

So the, $\sum x = 160 + 160 + 161 + 162 + 163 + 163 + 164 + 164 + 170$

So on adding it, we will get the series as

\[\sum x = 1,467\]

Here, we have $N = 9$

Therefore, $M.D = \dfrac{{\sum x }}{N}$

On substituting the values, we get

$M.D = \dfrac{{\sum {1,467} }}{9}$

On solving the above division, we get

$M.D = 163$

Hence, the mean deviation will be equal to $163$ .

Now we will calculate the variance,

${\sigma ^2} = \dfrac{{{{\sum {\left( {x - \bar x} \right)} }^2}}}{{n - 1}}$

Now on substituting the values, we get

\[{\sigma ^2} = \dfrac{{{{\left( {160 - 160} \right)}^2} + {{\left( {161 - 160} \right)}^2} + {{\left( {162 - 161} \right)}^2} + {{\left( {163 - 162} \right)}^2} + {{\left( {163 - 163} \right)}^2} + {{\left( {164 - 163} \right)}^2} + {{\left( {164 - 164} \right)}^2} + {{\left( {170 - 164} \right)}^2}}}{{9 - 1}}\]

Now on solving it, we will get

$\Rightarrow$ ${\sigma ^2} = \dfrac{{{0^2} + {1^2} + {1^2} + {1^2} + {0^2} + {0^2} + {6^2}}}{8}$

Now on adding, we get

$\Rightarrow$ ${\sigma ^2} = \dfrac{{0 + 1 + 1 + 1 + 0 + 0 + 36}}{8}$

And on solving we get

$\Rightarrow$ ${\sigma ^2} = \dfrac{{39}}{8}$

So S.D will be calculated by the formula $Standard{\text{ deviation = }}\sqrt {Variance} $

Substituting the values, we get

$Standard{\text{ deviation = }}\sqrt {\sqrt {\dfrac{{39}}{8}} } $

Hence, the above will be the standard deviation.

Students Also Read