Question

Variance remains unchanged by the change of
${\text{(A)}}$ Scale
${\text{(B)}}$ Origin
$(C)$ Both
${\text{(D)}}$ None of these

Answer 1

In this question we will first check both the conditions and then see which condition fulfills our given statement.
Finally we conclude the required answer.

Complete step-by-step solution:
In statistics the terms origin and scale are related to the diagrammatic presentation of statistical data.
The term origin means the point of intersection of $X$ axis and $Y$ axis. It is often represented as $(0,0)$
The value plotted at the point of origin is considered as to be zero.
Change of origin means some value has been added or subtracted to the value of the origin.
The origin can be changed to numerical values such as $1, - 1,2, - 2$ etc. Either on $X$ axis or $Y$ axis depending on the data provided.
Change of scale in statistics means that there is some value multiplied to the observations.
Therefore, the change of scale means changing the parameter of the presentation of the statistical data.
Now, when it comes to the variance of a distribution, changing the origin is equivalent to adding some constant values to the data.
Since variance is the measure of the dispersion, the linear change in the values does not affect the distribution and the variance is not affected.
And when we change the scale of the distribution the variance gets changed since the overall values change which affects the variance.

Therefore, the correct answer is origin which is option ${\text{(B)}}$.

Note: Variance is independent of change of origin as the change in origin is uniformly added to all the values and hence the mean also and hence, when ${(x - \bar x)^2}$ is calculated there is no change in the overall answer.