Question

Find the Standard Deviation of the following data: $5,9,8,12,6,10,6,8$
A. $2.14$
B. $2.16$
C. $2.15$
D. $2.17$

Answer 1

To find the standard deviation of the given data, we will first find the absolute mean, then variance, and finally the standard deviation.
The absolute mean of data $\overline x = \dfrac{{\left( {{\text{Sum of observations}}} \right)}}{{\left( {{\text{Total number of observations}}} \right)}}$ .
The variance of data $\delta = \dfrac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}$ , where $n$ is the total number of observations.
Standard Deviation is the square root of the variance, i.e., $\sigma = \sqrt {\dfrac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}}$ .

Complete step-by-step answer:
Given data: $5,9,8,12,6,10,6,8$ .
To find the standard deviation of the given data.
Total number of entities in the given data $= 8$
We will first find the absolute mean of the data
Mean $= \dfrac{{\left( {{\text{Sum of observations}}} \right)}}{{\left( {{\text{Total number of observations}}} \right)}}$
Putting all the values, we get,
$= \dfrac{{5 + 9 + 8 + 12 + 6 + 10 + 6 + 8}}{8}$
$= \dfrac{{64}}{8} = 8$
So, mean of the given data $\overline x = 8$ .
Now, we will find the values of $\left( {{x_i} - \overline x } \right)$ , where ${x_i}$ are the individual entities of the given data.

${x_i}$	$\left( {{x_i} - \overline x } \right)$
$5$	$5 - 8 = - 3$
$9$	$9 - 8 = 1$
$8$	$8 - 8 = 0$
$12$	$12 - 8 = 4$
$6$	$6 - 8 = - 2$
$10$	$10 - 8 = 2$
$6$	$6 - 8 = - 2$
$8$	$8 - 8 = 0$

Now, squaring all $\left( {{x_i} - \overline x } \right)$ , we get,

${x_i}$	$\left( {{x_i} - \overline x } \right)$	$\left( {{x_i} - \overline x } \right)$
$5$	$5 - 8 = - 3$	${\left( { - 3} \right)^2} = 9$
$9$	$9 - 8 = 1$	${\left( 1 \right)^2} = 1$
$8$	$8 - 8 = 0$	${\left( 0 \right)^2} = 0$
$12$	$12 - 8 = 4$	${\left( 4 \right)^2} = 16$
$6$	$6 - 8 = - 2$	${\left( { - 2} \right)^2} = 4$
$10$	$10 - 8 = 2$	${\left( 2 \right)^2} = 4$
$6$	$6 - 8 = - 2$	${\left( { - 2} \right)^2} = 4$
$8$	$8 - 8 = 0$	${\left( 0 \right)^2} = 0$

Now, values of $\left( {{x_i} - \overline x } \right)$ , find the variance of the given data by putting the values in the formula of variance,
Variance $\delta = \dfrac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}$
Putting all the values, we get,
$= \dfrac{{9 + 1 + 0 + 16 + 4 + 4 + 4 + 0}}{8}$
$= \dfrac{{38}}{8} = 4.75$
So, the variance of the given data is $4.75$ .
Now, the standard deviation is the square root of the variance.
Hence, Standard Deviation $\sigma = \sqrt {\dfrac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}}$
Putting the values, we get,
$= \sqrt {4.75} = 2.17$
Hence, the standard deviation of the given data is $2.17$ .
So, the correct answer is “Option D”.

Note: Here, Mean simply means to add all the given terms and divide the sum by the total number of terms.Do remember in case of summation while finding variance, take the square of each term and then add and not the square of the summation.No need to arrange the given data in ascending or descending order.