Question

Let $x_1$,$x_2$,…………..$x_100$ are 100 observation such that \[\sum {{x_i}} = 0\], \[\sum\limits_{1 \leqslant i \leqslant j \leqslant 100} {\left| {{x_i}{x_j}} \right|} = 80000\] and mean deviation from their mean is 5, then their standard deviation, is:
A.10
B.30
C.40
D.50

Answer 1

Hint: In this question since the mean for 100 observations is given so we first find the mean deviation from the mean and then by using standard deviation formula we will find the standard deviation.

Complete step-by-step answer:
Given
\[\sum {{x_i}} = 0\]
\[\sum\limits_{1 \leqslant i \leqslant j \leqslant 100} {\left| {{x_i}{x_j}} \right|} = 80000\]
\[{x_1},{x_2}, \ldots \ldots \ldots \ldots ..{x_{100}}\] are the 100 observations
We know mean for a set of data is the ratio of the sum of the observations by the total number of observations made, hence we can write
\[\bar x = \dfrac{{\sum {{x_i}} }}{{100}}\]
Now since \[\sum {{x_i}} = 0\] hence we get
\[\Rightarrow Mean = \bar x = \dfrac{{\sum {{x_i}} }}{{100}} = 0\]
Now the mean deviation from the mean is given 5, so the mean deviation for the mean can be written as
\[\dfrac{{\sum {\left| {{x_i} - \bar x} \right|} }}{{100}} = 5\]
This can be further written as
\[\sum {\left| {{x_i} - \bar x} \right|} = 500\]
Now we take square on the both side of the equation, hence we get
\[
\Rightarrow {\left( {\sum {\left| {{x_i} - \bar x} \right|} } \right)^2} = {500^2} \\
  {\left( {\sum {\left| {{x_i}} \right|} } \right)^2} = {500^2} \;
 \]
Now we can further write the obtained equation as
\[\sum {{{\left| {{x_i}} \right|}^2}} + 2\sum {{x_i}{x_j}} = {500^2}\]
Now since \[\sum\limits_{1 \leqslant i \leqslant j \leqslant 100} {\left| {{x_i}{x_j}} \right|} = 80000\] as already given hence we can further write the equation as
\[\sum {{{\left| {{x_i}} \right|}^2}} + 2\left( {8000} \right) = {500^2}\]
By solving this
\[
\Rightarrow \sum {{{\left| {{x_i}} \right|}^2}} = {500^2} - 2\left( {8000} \right) \\
\Rightarrow \sum {{{\left| {{x_i}} \right|}^2}} = {500^2} - 16000 \\
   = {500^2} - {400^2} \\
   = \left( {500 + 400} \right)\left( {500 - 400} \right) \\
   = \left( {900} \right)\left( {100} \right) \\
   = 90000 \;
 \]
We know the standard deviation is given by the formula
\[
\Rightarrow {\sigma ^2} = \dfrac{{\sum {{{\left( {{x_i} - \bar x} \right)}^2}} }}{{100}} \\
   = \dfrac{{\sum {{{\left( {{x_i}} \right)}^2}} }}{{100}} - - (ii) \;
 \]
Now since\[\sum {{{\left| {{x_i}} \right|}^2}} = 90000\], hence by substituting the value in equation (ii) we can write
\[
\Rightarrow {\sigma ^2} = \dfrac{{\sum {{{\left( {{x_i}} \right)}^2}} }}{{100}} \\
   = \dfrac{{90000}}{{100}} \\
   = 900 \;
 \]
Hence by solving this we get
\[
\Rightarrow {\sigma ^2} = 900 \\
   = 30 \;
 \]
Therefore standard deviation for 100 observations \[ = 30\]
So, the correct answer is “Option B”.

Note: Standard deviation for the mean deviation is given by the formula \[{\sigma ^2} = \dfrac{{\sum {{{\left( {{x_i} - \bar x} \right)}^2}} }}{{100}}\], where \[\bar x\]is the mean for the observations. Students must note that mean tells us where the highest part of the curve for data should go whereas standard deviation tells how wide the curve is going to be.