Question

Assertion :If $x_i=(2i-1);i=1,2,\mathrm{3...}$. Then, the sum of the deviations of $x_1,x_2,.....x_n$ from $x=n$ is zero Reason: The algebraic sum of the deviations of a set of observations about their mean is zero.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

We know that
"Algebraic sum of deviations of set of observations above their mean is 0"
Assertion:
According to given definition of $x_i=2i-1;i=1,2,\mathrm{3...}n$, we have data as
$1,3,5,....2n-1$
Sum of these observations ${\sum }_{{i=1}^n}{x}_i$
$=1+3+5+....+2n-1$
$=\frac{n}{2}(2+(n-1\left)2\right)$
$=n^2$
Mean $=\frac{\sum x_i}{n}$
$\overline{x}=\frac{n^2}{n}=n$
Now, deviation of observations about mean are $1-n,2-n,3-n....,(2n-1-n)$
Sum of deviations about mean $=\sum (x_i-\overline{x})$
$=1-n+2-n+.....+(2n-1-n)$
$=(1+2+3+....+2n-1)-n^2$
$=0$
Hence,assertion is correct and reason is the correct explanation for assertion.