Question

What is the standard deviation of $7, 9, 11, 13, 15$?
A. $2.4$
B. $2.5$
C. $2.7$
D. $2.8$

Answer 1

Mean of the $5$ numbers are given as $\overline x = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{5}$
And the standard deviation is given by the formula $\sum\limits_{i = 1}^5 {\dfrac{{\left| {{x_i} - \overline x } \right|}}{5}}$

Complete step by step solution:
Here in the question we are required to find the standard deviation of the five numbers given which are $7, 9, 11, 13, 15$. Now we know that standard deviation in statistics is the measure of the amount of the variation or dispersion in the set of values. A low standard deviation is close to the mean of the set of the numbers while the high standard deviation has a very large difference in the mean of the numbers contained in the set.
Here we should know the formula of the mean and the standard deviation of the set of the values given which are as follows:
Mean of the $5$ numbers are given as $\overline x = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}}}{5}$
And the standard deviation is given by the formula $\sum\limits_{i = 1}^5 {\dfrac{{\left| {{x_i} - \overline x } \right|}}{5}}$
And here ${x_1},{x_2},{x_3},{x_4},{x_5}$ are the five numbers whose mean and the standard deviation is found.
So here we are given the five numbers as $7,9,11,13,15$ so the mean will be
$\overline x = \dfrac{{79 + 11 + 13 + 15}}{5} = \dfrac{{55}}{5} = 11$
So we get that mean$= 11$
Now we need to find the deviation
So let us draw the table which will be

${x_i}$| Deviation from mean $\left| {\overline x - {x_i}} \right|$
---|---
$7$| $\left| {11 - 7} \right| = 4$
$9$| $\left| {11 - 9} \right| = 2$
$11$| $\left| {11 - 11} \right| = 0$
$13$| $\left| {11 - 13} \right| = 2$
$15$| $\left| {11 - 15} \right| = 4$

So we can find sum of the values of $\left| {\overline x - {x_i}} \right|$ which will be $4 + 2 + 0 + 2 + 4 = 12$
Now we know that standard deviation is given by
$SD =$ $\sum\limits_{i = 1}^5 {\dfrac{{\left| {{x_i} - \overline x } \right|}}{5}}$
$= \dfrac{{12}}{5} = 2.4$
Here we must know that the sign of mod means like $\left| { - 2} \right| = 2$ and $\left| 2 \right| = 2$ so the modulus function changes the negative function to the positive function. Hence here we must be aware of what the modulus function means.
Hence we get that $SD = 2.4$

Hence option A is correct.

Note:
As we know about standard deviation we must also know about the variance. Variance is the square of the standard deviation which can be written as
$SD = \sqrt {{\text{variance}}} =$ $\sqrt {\sum\limits_{i = 1}^n {\dfrac{{\left| {{x_i} - \overline x } \right|}}{n}} }$