Question

Find the standard deviation of $9,16,23,30,37,44,51$ is
${\text{A}}{\text{.}}$$7$
${\text{B}}{\text{.}}$$9$
\[{\text{C}}{\text{.}}\]$12$
${\text{D}}{\text{.}}$$14$
${\text{E}}{\text{.}}$$16$

Answer 1

Hint: From the question, we have to find the standard deviation for the given data and choose the correct answer. On finding the solution we have to apply the standard deviation formula on the given data. Then, we will get the required result and choose the correct answer.

Formula used: Standard deviation is the measure of depression of a set of data from its mean. It measures the absolute variability of a distribution; the higher the dispersion or variability, the greater is the standard deviation and greater will be the magnitude of the deviation of the value from their mean.
Formula for standard deviation:
${{\sigma }} = \sqrt {\dfrac{{\text{1}}}{{\text{N}}}\sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\left( {{{\text{x}}_{\text{i}}} - {{\mu }}} \right)}^2}} } $
Where, ${{\sigma }} = $ population standard deviation
\[{\text{N}} = \] The size of the population
${{\text{x}}_{\text{i}}} = $ Each value from the population
 ${{\mu }} = $ The population mean
We know that, \[{\text{Mean}} = \dfrac{{{\text{Total of all items}}}}{{{\text{No}}{\text{. of items}}}}\]

Complete step-by-step solution:
From the given data, we are going to find the standard deviation. First, we find the above terms are mentioned in the formula.
Here, the size (given number of items) ${\text{N}} = 7$.
First, we have to find the mean (average) of the set of numbers by using the formula given below:
\[{\text{Mean}} = \dfrac{{{\text{Total of all items}}}}{{{\text{No}}{\text{. of items}}}}\]
$\Rightarrow$${{\mu }} = \dfrac{{9 + 16 + 23 + 30 + 37 + 44 + 51}}{7}$
On simplification, we get
$\Rightarrow$${{\mu }} = \dfrac{{210}}{7}$
$\therefore {{\mu }} = 30$
Now, we have to substitute the above values in the standard deviation formula. Then, we get
$\Rightarrow$${{\sigma }} = \sqrt {\dfrac{{\text{1}}}{{\text{N}}}\sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\left( {{{\text{x}}_{\text{i}}} - {{\mu }}} \right)}^2}} } $
$\Rightarrow$${{\sigma }} = \sqrt {\dfrac{{\text{1}}}{7}\sum\limits_{{\text{i}} = 1}^7 {{{\left( {{{\text{x}}_{\text{i}}} - 30} \right)}^2}} } $
Now, we have to expand the summation series of the square root.
$\Rightarrow$\[{{\sigma }} = \sqrt {\dfrac{{\text{1}}}{7}\left( {{{\left( {{{\text{x}}_1} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_2} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_3} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_4} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_5} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_6} - {\text{30}}} \right)}^2} + {{\left( {{{\text{x}}_7} - {\text{30}}} \right)}^2}} \right)} \]
Simplifying we get,
$\Rightarrow$\[{{\sigma }} = \sqrt {\dfrac{{{{\left( {9 - {\text{30}}} \right)}^2} + {{\left( {16 - {\text{30}}} \right)}^2} + {{\left( {23 - {\text{30}}} \right)}^2} + {{\left( {30 - {\text{30}}} \right)}^2} + {{\left( {37 - {\text{30}}} \right)}^2} + {{\left( {44 - {\text{30}}} \right)}^2} + {{\left( {51 - {\text{30}}} \right)}^2}}}{7}} \]
On simplification, we get
$\Rightarrow$\[{{\sigma }} = \sqrt {\dfrac{{{{\left( { - 21} \right)}^2} + {{\left( { - 14} \right)}^2} + {{\left( { - 7} \right)}^2} + {{\left( 0 \right)}^2} + {{\left( 7 \right)}^2} + {{\left( {14} \right)}^2} + {{\left( {21} \right)}^2}}}{7}} \]
Squaring the all term,
$\Rightarrow$\[{{\sigma }} = \sqrt {\dfrac{{441 + 196 + 49 + 0 + 49 + 196 + 441}}{7}} \]
Adding the terms,
$\Rightarrow$${{\sigma }} = \sqrt {\dfrac{{1372}}{7}} $
Dividing the terms we get,
$\Rightarrow$${{\sigma }} = \sqrt {196} $
$\therefore {{\sigma }} = 14$
Therefore, the standard deviation of a given set of numbers is $14$ .

$\therefore $ The correct answer is option ${\text{D}}$.

Note: In financial terms, standard deviation is used to measure risks involved in an investment instrument. Standard deviation provides investors a mathematical basis for decisions to be made regarding their investment in the financial market, standard deviation is a common term used in deals involving stocks, mutual funds, EFTs and others.
The given problem is easy to solve. The students should concentrate on the standard deviation formula and on the simple calculations. Particularly, on expanding the square terms and squaring those terms.