Let s be the standard deviation of n observations. Each of the n observations is multiplied by a constant c. Then the standard deviation of the resulting number is

Options

A. s
B. cs
C. `s sqrtc`
D. None of these

Answer

To solve the problem, we need to determine the standard deviation of a set of observations after each observation has been multiplied by a constant \( c \). Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding Standard Deviation**:
The standard deviation \( S \) of a set of \( n \) observations \( a_1, a_2, \ldots, a_n \) is given by the formula:
\[ S = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (a_i - \bar{x})^2} \]
where \( \bar{x} \) is the mean of the observations, calculated as:
\[ \bar{x} = \frac{a_1 + a_2 + \ldots + a_n}{n} \]

2. **New Observations**:
If each observation \( a_i \) is multiplied by a constant \( c \), the new observations become \( c a_1, c a_2, \ldots, c a_n \).

3. **Calculating the New Mean**:
The mean of the new observations \( \bar{x}' \) is:
\[ \bar{x}' = \frac{c a_1 + c a_2 + \ldots + c a_n}{n} = c \cdot \frac{a_1 + a_2 + \ldots + a_n}{n} = c \cdot \bar{x} \]

4. **Calculating the New Standard Deviation**:
The standard deviation of the new observations \( S' \) is given by:
\[ S' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (c a_i - \bar{x}')^2} \]
Substituting \( \bar{x}' = c \bar{x} \):
\[ S' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (c a_i - c \bar{x})^2} \]

5. **Factoring out the Constant**:
We can factor out \( c \) from the expression:
\[ S' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} c^2 (a_i - \bar{x})^2} \]
This simplifies to:
\[ S' = \sqrt{c^2 \cdot \frac{1}{n} \sum_{i=1}^{n} (a_i - \bar{x})^2} = c \cdot \sqrt{\frac{1}{n} \sum_{i=1}^{n} (a_i - \bar{x})^2} \]

6. **Final Result**:
Since \( \sqrt{\frac{1}{n} \sum_{i=1}^{n} (a_i - \bar{x})^2} \) is the original standard deviation \( S \), we have:
\[ S' = c \cdot S \]

### Conclusion:
Thus, the standard deviation of the resulting observations after multiplying each observation by a constant \( c \) is:
\[ S' = cS \]