Mean and standard deviations of 10 observations are 20 and 2 respectively. If p `(p ne 0)` is multiplied to each observation and then q `(q ne 0)` is subtracted then new mean and standard deviation becomes half of original value . Then find q

Options

A. -10
B. -20
C. -5
D. 10

Explanation

To solve the problem, we need to find the value of \( q \) given that the mean and standard deviation of 10 observations are transformed by multiplying each observation by \( p \) and then subtracting \( q \). The new mean and standard deviation become half of the original values.

### Step-by-step Solution:

1. **Original Mean and Standard Deviation**:
- Given the original mean \( \bar{x} = 20 \) and standard deviation \( \sigma = 2 \).

2. **Transformation of Observations**:
- Each observation \( x_i \) is transformed to \( p \cdot x_i - q \).

3. **New Mean Calculation**:
- The new mean \( \bar{x}' \) can be calculated as follows:
\[ \bar{x}' = \frac{1}{n} \sum_{i=1}^{n} (p \cdot x_i - q) = p \cdot \bar{x} - q \]
- Substituting the values:
\[ \bar{x}' = p \cdot 20 - q \]

4. **Condition for New Mean**:
- We know that the new mean is half of the original mean:
\[ \bar{x}' = \frac{1}{2} \cdot 20 = 10 \]
- Setting the equations equal gives:
\[ p \cdot 20 - q = 10 \quad \text{(Equation 1)} \]

5. **New Standard Deviation Calculation**:
- The new standard deviation \( \sigma' \) is calculated as:
\[ \sigma' = |p| \cdot \sigma \]
- Substituting the values:
\[ \sigma' = |p| \cdot 2 \]

6. **Condition for New Standard Deviation**:
- We know that the new standard deviation is half of the original standard deviation:
\[ \sigma' = \frac{1}{2} \cdot 2 = 1 \]
- Setting the equations equal gives:
\[ |p| \cdot 2 = 1 \quad \Rightarrow \quad |p| = \frac{1}{2} \quad \text{(Equation 2)} \]

7. **Solving for \( p \)**:
- From Equation 2, we have:
\[ p = \frac{1}{2} \quad \text{or} \quad p = -\frac{1}{2} \]

8. **Substituting \( p \) back into Equation 1**:
- Using \( p = \frac{1}{2} \):
\[ \frac{1}{2} \cdot 20 - q = 10 \quad \Rightarrow \quad 10 - q = 10 \quad \Rightarrow \quad q = 0 \quad \text{(not valid since \( q \neq 0 \))} \]
- Using \( p = -\frac{1}{2} \):
\[ -\frac{1}{2} \cdot 20 - q = 10 \quad \Rightarrow \quad -10 - q = 10 \quad \Rightarrow \quad -q = 20 \quad \Rightarrow \quad q = -20 \]

### Final Answer:
Thus, the value of \( q \) is \( -20 \).