Question

In a series of \[2{\rm{n}}\] observations, half of them equal to \[{\rm{a}}\] and remaining half equal \[ - {\rm{a}}\]. If the standard deviation of the observations is 2, then\[\left| {\rm{a}} \right|\] equals:
A) \[\dfrac{1}{{\rm{n}}}\]
B) \[\sqrt 2 \]
C) \[2\]
D) \[\dfrac{{\sqrt 2 }}{{\rm{n}}}\]

Answer 1

Hint:
Here we have to use the basic formula of the standard deviation to find out the value of \[\left| {\rm{a}} \right|\]. Firstly we will use the formula of the standard deviation to find its value in terms of the variable \[{\rm{a}}\]. Then we have to equate the value of the standard deviation to the given value in the question i.e. 2 then we will get the value of \[\left| {\rm{a}} \right|\].

Complete step by step solution:
It is given that the series has \[2{\rm{n}}\] number of observations.
It is also given that half of the observation is equal to \[{\rm{a}}\] and the remaining half equal to \[ - {\rm{a}}\].
So the mean of the total observation is \[{\rm{\bar x = 0}}\]
Now we have to use the formula of the standard deviation and find its value in terms of a variable \[{\rm{a}}\].
Standard deviation \[{\rm{ = }}\sum {\dfrac{{{{\left| {{{\rm{x}}_{\rm{i}}} - {\rm{\bar x}}} \right|}^2}}}{{2{\rm{n}}}}} \]and we know that\[{\rm{\bar x = 0}}\]. So, we get
Standard deviation \[{\rm{ = }}\sum {\dfrac{{{{\left| {{{\rm{x}}_{\rm{i}}} - {\rm{0}}} \right|}^2}}}{{2{\rm{n}}}}} = \sum {\dfrac{{{{\left| {{\rm{a}} - {\rm{0}}} \right|}^2}}}{{2{\rm{n}}}}} = \dfrac{{2{\rm{n}}{{\rm{a}}^2}}}{{2{\rm{n}}}} = {{\rm{a}}^2}\]
It is given that the value of the standard deviation equals 2. Then we have to equate the equation of the standard deviation with the value 2, we get
Standard deviation \[ = {{\rm{a}}^2} = 2\]
\[ \Rightarrow \left| {\rm{a}} \right| = \sqrt 2 \]
Therefore, the value of \[\left| {\rm{a}} \right|\] is\[\sqrt 2 \].

Hence, option B is the correct option.

Note:
Standard deviation is defined as the measure of the variation of the numbers within the given set of numbers. Standard deviation is used to check the range of deviation of the data from the mean of the data. Wider the range of the data from the mean in a set of numbers then higher will be the standard deviation and lower the range of the data in a set of numbers then lower will be the standard deviation of the data.
We should also know the meaning of
Mean is equal to the ratio of the sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
Mode is the most common or most repeating numbers in the given data.
Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.