UPSC NDA 2 2017 Mathematics Solved Paper

Concept:Variance measures dispersion around the mean, while the coefficient of variation (CV) is a unit-free ratio.Explanation:Statement 1:Variance is unaffected by change of origin but is affected by change of scale.A change of origin adds a constant to all $x_i$; the mean also shifts by the same constant, so deviations $x_i - \mu$ remain unchanged. Hence variance stays the same.A change of scale multiplies all values by a constant $c$; the mean also multiplies by $c$, so squared deviations become $c^2$ times the original. Therefore variance changes by factor $c^2$.Thus Statement 1 is false because it claims variance is unaffected by both origin and scale.Statement 2:The coefficient of variation is $\displaystyle CV = \frac{\sigma}{\mu} \times 100\%$.Both $\sigma$ (standard deviation) and $\mu$ (mean) have the same unit as the observations, so their ratio is dimensionless.Thus CV is independent of the unit of observation.Hence Statement 2 is true.Answer:Only statement 2 is correct. The correct option is B: 2 only.