Question

What is the standard deviation of$5,5,10,10,10$?
A. $2.44$
B. $1.44$
C. $5$
D. $0$

Answer 1

Hint:
we know the formula of the standard deviation is given by the formula
$\sigma = \sqrt {\dfrac{{\sum {{{(x - \overline x )}^2}} }}{n}} $
Here $\sigma $represents the standard deviation. And $\overline x $Represents the mean of the given five numbers. And $n$ is the size or the number of items in the sample.

Complete step by step solution:
First of all we are given the question that we need to find the standard deviation of the given numbers which are $5, 5, 10, 10, 10$. First of all we need to understand what the standard deviation means. As the name itself suggests it is the deviation which means how spread the numbers are.
It is denoted by the Greek letter $\sigma $ or we can also define it as the square root of the variance. For calculating the standard deviation we need to find first the mean which we know is the ratio of the sum of all the numbers and the number of numbers.
So mean$ = \dfrac{{5 + 5 + 10 + 10 + 10}}{5} = \dfrac{{40}}{5} = 8$
So $\overline x = 8$
So it is the mean of the given numbers.
Now we need to calculate the mean of the given numbers.
$\sum {{{(x - \overline x )}^2}} $$ = {({x_1} - \overline x )^2} + {({x_2} - \overline x )^2} + {({x_3} - \overline x )^2} + {({x_4} - \overline x )^2} + {({x_5} - \overline x )^2}$
$
   = {(5 - 8)^2} + {(5 - 8)^2} + {(10 - 8)^2} + {(10 - 8)^2} + {(10 - 8)^2} \\
   = 9 + 9 + 4 + 4 + 4 \\
   = 30 \\
 $
Putting the value of the above values in the formula we get that
$\sigma = \sqrt {\dfrac{{\sum {{{(x - \overline x )}^2}} }}{n}} $
$\sigma = \sqrt {\dfrac{{30}}{5}} = \sqrt 6 $$ = 2.44$

Note:
Here we know that mean$ = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
So here $5,10$ are two and three times. So we get that frequency of $5$ is $2$ and the frequency of $10$ is $3$
Therefore mean$ = \dfrac{{5(2) + 10(3)}}{5} = 8$
We can find the mean by this formula. Hence the formula is the same but written in different types. So we must know each type of the formula that must be used in order to get the results of mean.