Question

For a series, the information available for us is $n = 10$, $\sum x = 60$, and $\sum {{x^2} = 1000} $. The standard deviation is?
(A) $8$
(B) $64$
(C) $24$
(D) $128$

Answer 1

Hint: In the given question, we are provided with some information about a series where n represents the number of terms and \[{x_i}'s\] represent the observations. So, the summation of all the \[{x_i}'s\] is given to us as well as the summation of the squares of all the \[{x_i}'s\] is provided in the question itself. We have to find the standard deviation of the series making use of the available information.

Complete step by step answer:
So, we have a series with the sum of the terms or observations given to us as $\sum x = 60$.
Also, we are given the sum of squares of observations as $\sum {{x^2} = 1000} $ and the number of terms in the series as $10$.
Now, we have to calculate the standard deviation of the observations of the given series.
Standard deviation is the amount of variation or dispersion in a given data set.
So, we know the formula to calculate the standard deviation of a data set is $\sigma = \dfrac{{{{\sum x }^2}}}{n} - {\left( {\dfrac{{\sum x }}{n}} \right)^2}$.
Hence, the standard deviation for the given series can be calculated as,
$ \Rightarrow \sigma = \dfrac{{1000}}{{10}} - {\left( {\dfrac{{60}}{{10}}} \right)^2}$
Now, we have to simplify the expression in order to get to the right answer.
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow \sigma = 100 - {\left( 6 \right)^2}$
We know that the square of $6$ is $36$. Hence, we get,
$ \Rightarrow \sigma = 100 - 36$
Doing the calculation, we get,
$ \Rightarrow \sigma = 64$
So, the standard deviation of the given series with $n = 10$, $\sum x = 60$, and $\sum {{x^2} = 1000} $ is $64$.

So, the correct answer is “Option B”.

Note: The given question revolves around the concepts of statistics. So, we should know the formulae for calculating the standard deviation of a given data set. One should take care while carrying out the calculations in order to be sure of the answer.