Question

The standard deviation of the data \[6,5,9,13,12,8,10\] is
A. \[\sqrt {\dfrac{{52}}{7}} \]
B. \[\dfrac{{52}}{7}\]
C. \[\sqrt 6 \]
D. 6

Answer 1

Hint: We will first consider the given data set to determine the standard deviation. We will use the formula to calculate the standard deviation, \[\sigma = \sqrt {\dfrac{{\sum {{x_i}^2} }}{n} - {{\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)}^2}} \] where \[n\] is the total number of observations and \[\sum {{x_i}} \] represents the sum of all the observations. Then we will substitute the obtained values in the formula and get the desired result.

Complete step by step answer:

We will first consider the given data set that is \[6,5,9,13,12,8,10\].
The aim is to find the standard deviation.
As we know that the formula for finding the standard deviation is \[\sigma = \sqrt {\dfrac{{\sum {{x_i}^2} }}{n} - {{\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)}^2}} \].
We will first find the value of \[\sum {{x_i}} \] by adding all the given observations.
Thus, we get,
\[
   \Rightarrow \sum {{x_i}} = 6 + 5 + 9 + 13 + 12 + 8 + 10 \\
   \Rightarrow \sum {{x_i}} = 63 \\
 \]
Next, we will square the obtained summation to determine the value of \[{\left( {\sum {{x_i}} } \right)^2}\]
Thus, we get,
\[
   \Rightarrow {\left( {\sum {{x_i}} } \right)^2} = {\left( 6 \right)^2} + {\left( 5 \right)^2} + {\left( 9 \right)^2} + {\left( {13} \right)^2} + {\left( {12} \right)^2} + {\left( 8 \right)^2} + {\left( {10} \right)^2} \\
   \Rightarrow {\left( {\sum {{x_i}} } \right)^2} = 619 \\
 \]
As we have total number of observations as \[n = 7\].
Then we will substitute all the values in the formula of standard deviation,
We have,
\[
   \Rightarrow \sigma = \sqrt {\dfrac{{619}}{7} - {{\left( {\dfrac{{63}}{7}} \right)}^2}} \\
   \Rightarrow \sigma = \sqrt {\dfrac{{4333 - 396}}{{49}}} \\
   \Rightarrow \sigma = \sqrt {\dfrac{{52}}{7}} \\
 \]
Hence, we can conclude that the standard deviation for the given data set is \[\sqrt {\dfrac{{52}}{7}} \].
Thus, option A is correct.

Note: There is no need to find the value of mean in this. Summation means to add all the values. \[n\] is equal to the total number of observations. Do the square of summation properly. We have used the formula \[\sigma = \sqrt {\dfrac{{\sum {{x_i}^2} }}{n} - {{\left( {\dfrac{{\sum {{x_i}} }}{n}} \right)}^2}} \] to find the value of standard deviation. There is no need to set up the data in ascending or descending order. While calculating the square of the summation, do the square of each number and not the square of summation of \[x\] which gives the different answer.