UPSC CDS 1 2023 Math Paper

Concept:The mode for grouped data is found using the formula: $\text{Mode} = L + \frac{D_1}{D_1 + D_2} \times i$ where $L$ is the lower limit of the modal class, $D_1$ is the difference between the frequency of the modal class and that of the preceding class, $D_2$ is the difference between the frequency of the modal class and that of the following class, and $i$ is the class width.Explanation:The class $60 - 90$ has the highest frequency (7), so it is the modal class.$L = 60$Frequency preceding modal class = 5, so $D_1 = 7 - 5 = 2$Frequency following modal class = 4, so $D_2 = 7 - 4 = 3$Class width $i = 90 - 60 = 30$Substitute into formula: $\text{Mode} = 60 + \frac{2}{2+3} \times 30 = 60 + \frac{2}{5} \times 30 = 60 + 12 = 72$Answer:The mode of the distribution is $72$, which corresponds to option B.