Solveeit Logo
Login

Question

Question: For a series, the information available for us is \(n = 10\), \(\sum x = 60\), and \(\sum {{x^2} = 1...

For a series, the information available for us is n=10n = 10, ∑x=60\sum x = 60, and ∑x2=1000\sum {{x^2} = 1000} . The standard deviation is?
(A) 88
(B) 6464
(C) 2424
(D) 128128

Explanation

Solution

In the given question, we are provided with some information about a series where n represents the number of terms and xi′s{x_i}'s represent the observations. So, the summation of all the xi′s{x_i}'s is given to us as well as the summation of the squares of all the xi′s{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 ∑x=60\sum x = 60.
Also, we are given the sum of squares of observations as ∑x2=1000\sum {{x^2} = 1000} and the number of terms in the series as 1010.
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 σ=∑x2n−(∑xn)2\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,
⇒σ=100010−(6010)2\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,
⇒σ=100−(6)2\Rightarrow \sigma = 100 - {\left( 6 \right)^2}
We know that the square of 66 is 3636. Hence, we get,
⇒σ=100−36\Rightarrow \sigma = 100 - 36
Doing the calculation, we get,
⇒σ=64\Rightarrow \sigma = 64
So, the standard deviation of the given series with n=10n = 10, ∑x=60\sum x = 60, and ∑x2=1000\sum {{x^2} = 1000} is 6464.

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.