UPSC CDS 1 2023 Math Paper

Concept:Median for grouped data uses the formula $M = l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h$, where $l$ = lower limit of median class, $n$ = total frequency, $cf$ = cumulative frequency before median class, $f$ = frequency of median class, $h$ = class size.Explanation:First, compute cumulative frequencies (cf) for the given data:Total observations $n = 10$, so $\frac{n}{2} = 5$.The class where cumulative frequency just exceeds 5 is $50-60$.Thus, median class: $l = 50$, $cf = 4$ (cumulative frequency of previous class), $f = 3$, $h = 10$.Plug into formula: $M = 50 + \left(\frac{5 - 4}{3}\right) \times 10 = 50 + \frac{1}{3} \times 10 = 50 + \frac{10}{3} = \frac{150 + 10}{3} = \frac{160}{3}$.The question asks for $3M$: $3M = 3 \times \frac{160}{3} = 160$.Answer:$3M = 160$, which corresponds to option C.