Question

The variance of $20$ observations is $5$. If each observation is multiplied by two, find the standard deviation and variance of the resulting observations.

Answer 1

In the given question, we are provided with some information about a series where we are given the number of terms and their variance. We have to find the standard deviation and variance of the series obtained by making the changes in the original series as suggested in the question.

Complete step by step answer:
The standard deviation is a measure of the amount of variation or dispersion of a set of values. So, we have a series of observations of $20$ observations whose variance is $5$. We know, standard deviation is,
$\sigma = \sqrt {\dfrac{{\sum\limits_{i = 1}^n {{{\left( {{x_i} - \mu } \right)}^2}} }}{n}}$.
Where, $n =$total observations, ${x_i} =$each term and $\mu =$mean of observations.
We also know that variance of observations is the square of the standard deviation. So, let the given series of observations is, ${x_1}$, ${x_2}$, ${x_3}$, ${x_4}$, …., ${x_{20}}$.

Now, on multiplying $2$ to each of the observations, the series becomes, $2{x_1}$, $2{x_2}$, $2{x_3}$, $2{x_4}$, …., $2{x_{20}}$.
Therefore, the mean of the new series becomes, ${\mu _1} = \dfrac{{2{x_1} + 2{x_2} + 2{x_3} + ..... + 2{x_{20}}}}{{20}}$
$\Rightarrow {\mu _1} = \dfrac{{2\left( {{x_1} + {x_2} + {x_3} + .... + {x_{20}}} \right)}}{{20}}$
[Taking $2$ common from the numerator]
$\Rightarrow {\mu _1} = 2\mu$
[$\mu =$ older mean]
Now, using this new mean in the formula of standard deviation, for new series, we get,
${\sigma _1} = \sqrt {\dfrac{{\sum\limits_{i = 1}^{20} {{{\left( {2{x_i} - \left( {2\mu } \right)} \right)}^2}} }}{{20}}}$

Taking, $2$ common from numerator, inside the root, we get,
$\Rightarrow {\sigma _1} = \sqrt {\dfrac{{{{(2)}^2}\sum\limits_{i = 1}^n {{{\left( {{x_i} - \mu } \right)}^2}} }}{n}}$
Now, taking $2$ out of the root, we get,
$\Rightarrow {\sigma _1} = 2\sqrt {\dfrac{{\sum\limits_{i = 1}^n {{{\left( {{x_i} - \mu } \right)}^2}} }}{n}}$
$\Rightarrow {\sigma _1} = 2\sigma$
[$\sigma =$ older standard deviation]
Now, we are given, the variance of the older series is $5$. So, the standard deviation of the older series must be the square root of the variance. So, we get,
$\sigma = \sqrt 5$
Therefore, the standard deviation of the new series is,
${\sigma _1} = 2\sigma = 2\left( {\sqrt 5 } \right) = 2\sqrt 5$
Also, the new variance must be the square of the new standard deviation.

So, we get variance $= {\left( {2\sqrt 5 } \right)^2} = 20$.

Note: From the above problem we can say that the standard deviation remains constant or unchanged if any constant is added or subtracted from all the observations. But the standard deviation gets multiplied or divided by the constant if all the observations are multiplied or divided by the same constant.