Statistical Methods & Experimental Designs

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.

When equal distances are traversed at different speeds ($s_1$ and $s_2$), the average speed is given by the Harmonic Mean (HM) of the speeds:

Given:
$s_1 = 45\text{ km/hr}$
$s_2 = 55\text{ km/hr}$

Calculation:

Note: Using the arithmetic mean $\frac{45 + 55}{2} = 50\text{ km/hr}$ is a classic conceptual mistake because the rider spends more time travelling at the slower speed.

Calculations:

• Total sum of all 21 numbers $= 21 \times 9 = 189$.
• If the number 19 is removed, the sum of the remaining 20 numbers is:
  $$\text{New Sum} = 189 - 19 = 170$$
• The mean of the remaining 20 numbers is:
  $$\text{New Mean} = \frac{170}{20} = 8.5$$

Mathematical derivation of harmonic mean:
Let $x_1 = \frac{a}{1-ab}$ and $x_2 = \frac{a}{1+ab}$.

• Incorrect: The calculation yields $a$, not $a/b$.
• Correct: The harmonic mean $H$ of two quantities $x_1$ and $x_2$ is given by:
  $$H = \frac{2}{\frac{1}{x_1} + \frac{1}{x_2}}$$
  Reciprocals are:
  $$\frac{1}{x_1} = \frac{1-ab}{a}, \quad \frac{1}{x_2} = \frac{1+ab}{a}$$
  Adding the reciprocals:
  $$\frac{1}{x_1} + \frac{1}{x_2} = \frac{(1-ab) + (1+ab)}{a} = \frac{2}{a}$$