1. Let the original observations be $x_1, x_2, \dots, x_{10}$.

   Given: $\bar{x} = 20, \qquad \sigma = 2$

2. Each observation is transformed as: $y = px - q$

   For such a linear transformation:

   • New mean: $\bar{y} = p\bar{x} - q$
   • New standard deviation: $\sigma_y = |p|\sigma$

3. It is given that the new mean and new s.d. are half of their original values.

   So, $\bar{y} = \frac{20}{2} = 10$ and $\sigma_y = \frac{2}{2} = 1$

4. Use the standard deviation condition: $|p|\cdot 2 = 1$ $|p| = \frac{1}{2}$

   Hence, $p = \frac{1}{2} \quad \text{or} \quad p = -\frac{1}{2}$

5. Now use the mean condition: $p(20) - q = 10$ $20p - q = 10$ $q = 20p - 10$

6. Check both possible values of $p$:

   • If $p = \frac{1}{2}$, $q = 20\left(\frac{1}{2}\right) - 10 = 10 - 10 = 0$ But given $q \ne 0$, so this is not allowed.

   • If $p = -\frac{1}{2}$, $q = 20\left(-\frac{1}{2}\right) - 10 = -10 - 10 = -20$

7. Therefore, $q = -20$

8. Option check:

   • A: $10$ ❌
   • B: $-20$ ✅
   • C: $-10$ ❌
   • D: $-5$ ❌

So the correct answer is Option B.