• Step 1: Use total frequency $N = 100$:
  $$14 + x + 27 + y + 15 = 100 \implies 56 + x + y = 100 \implies x + y = 44$$
• Step 2: Use Mode formula.
  Given Mode = 24, which lies in the class interval 20-30. Hence, the modal class is 20-30:
  Lower limit of modal class ($L$) = 20
  Class width ($h$) = 10
  Frequency of modal class ($f_1$) = 27
  Frequency of pre-modal class ($f_0$) = $x$
  Frequency of post-modal class ($f_2$) = $y$
  
  Mode formula:
  $$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$
  $$24 = 20 + \left(\frac{27 - x}{2(27) - (x + y)}\right) \times 10$$
  $$4 = \left(\frac{27 - x}{54 - 44}\right) \times 10 = \left(\frac{27 - x}{10}\right) \times 10 = 27 - x$$
  $$4 = 27 - x \implies x = 27 - 4 = 23$$
• Step 3: Find $y$:
  $$y = 44 - x = 44 - 23 = 21$$

For a normal distribution, standard empirical relationships link Quartile Deviation (QD), Mean Deviation (MD), and Standard Deviation (SD / $\sigma$):

• $\text{Mean Deviation} = \sqrt{\frac{2}{\pi}}\sigma \approx 0.7979\sigma \approx \frac{4}{5}\sigma \implies 5 \times \text{MD} = 4 \times \text{SD}$ (Option 1 is correct).
• $\text{Quartile Deviation} = 0.6745\sigma \approx \frac{2}{3}\sigma \implies 3 \times \text{QD} = 2 \times \text{SD}$ (Option 2 is correct).
  The overall ratio of dispersion measures is: $$\text{QD} : \text{MD} : \text{SD} = \frac{2}{3}\sigma : \frac{4}{5}\sigma : \sigma = 10 : 12 : 15$$
  Because both Option 1 and Option 2 are mathematically and practically valid for a normal distribution, NTA's official final answer key correctly recognized both '1' and '2' as valid answers.