Question

The standard deviation for the scores $1$ , $2$, $3$, $4$, $5$, $6$ and $7$ is $2$. Then, the standard deviation of $12$, $23$,$34$, $45$,$56$, $67$ and $78$ is :
A. $2$
B. $4$
C. $22$
D. $11$

Answer 1

Here we will be using the formula of calculating the mean and standard deviation for a particular series of numbers. The formula as given below:
${\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{{{\text{number of terms}}}}$ and
${\text{S}}{\text{.D}} = \sqrt {\dfrac{{{{\left( {{x_1} - \mu } \right)}^2} + {{\left( {{x_2} - \mu } \right)}^2} + {{\left( {{x_3} - \mu } \right)}^2}..... + {{\left( {{x_n} - \mu } \right)}^2}}}{{{\text{number of terms}}}}}$, where
${x_1}$ is the first term of the series,
${x_2}$ known as the second term, and
${x_n}$ denotes the $nth$.
$\mu$ denotes the mean of that series.

Complete step-by-step solution:
Step 1: For calculating the standard deviation of the given series $12$,$23$, $34$, $45$, $56$,
$67$ and $78$, at first we will be calculating the mean of that series as shown below:
${\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{{{\text{number of terms}}}}$
By substituting the values of the sum of terms and number of terms which is $7$, we get:
$\Rightarrow {\text{Mean}} = \dfrac{{12 + 23 + 34 + 45 + 56 + 67 + 78}}{7}$
By doing the addition in the RHS side of the above expression we get:
$\Rightarrow {\text{Mean}} = \dfrac{{315}}{7}$
By doing the final division in the above expression we get:
$\Rightarrow {\text{Mean}} = 45$
Step 2: By using the formula of standard deviation we get:
$\Rightarrow {\text{S}}{\text{.D}} = \sqrt {\dfrac{{{{\left( {12 - 45} \right)}^2} + {{\left( {23 - 45} \right)}^2} + {{\left( {34 - 45} \right)}^2} + {{\left( {45 - 45} \right)}^2} + {{\left( {56 - 45} \right)}^2} + {{\left( {67 - 45} \right)}^2} + {{\left( {78 - 45} \right)}^2}}}{7}}$ , where
${x_1} = 12$,
${x_2} = 23$,
${x_3} = 34$,
${x_4} = 45$,
${x_5} = 56$,${x_6} = 67$,
${x_7} = 78$ and $\mu ({\text{mean}}) = 45$.
Solving the brackets by doing addition and subtraction we get:
$\Rightarrow {\text{S}}{\text{.D}} = \sqrt {\dfrac{{{{\left( { - 33} \right)}^2} + {{\left( { - 22} \right)}^2} + {{\left( { - 11} \right)}^2} + {{\left( 0 \right)}^2} + {{\left( {11} \right)}^2} + {{\left( {22} \right)}^2} + {{\left( {33} \right)}^2}}}{7}}$
By solving the powers of the particular terms in the above expression we get:
$\Rightarrow {\text{S}}{\text{.D}} = \sqrt {\dfrac{{1089 + 484 + 121 + 0 + 121 + 484 + 1089}}{7}}$
By doing the addition in the numerator of the RHS side of the above expression, we get:
$\Rightarrow {\text{S}}{\text{.D}} = \sqrt {\dfrac{{3388}}{7}}$
By dividing the RHS side of the above expression we get:
$\Rightarrow {\text{S}}{\text{.D}} = \sqrt {484}$
Finally, by solving the root of the RHS side of the above expression we get:
$\Rightarrow {\text{S}}{\text{.D}} = 22$

Option C is correct.

Note: Students should remember the formulas for calculating mean, median, mode, and standard deviation. The symbol of mean and standard deviation is $\mu$ and $\sigma$ respectively. Also, you should remember that for calculating the standard deviation for any series of numbers first we need to calculate the mean of that series.