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 new standard deviation after multiplying each of the n observations by a constant \( c \). Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding Standard Deviation**:
The standard deviation \( S \) of \( n \) observations \( a_1, a_2, a_3, \ldots, a_n \) 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.

2. **New Observations**:
When each observation is multiplied by a constant \( c \), the new observations become:
\[ c \cdot a_1, c \cdot a_2, c \cdot a_3, \ldots, c \cdot a_n \]

3. **Calculating the New Mean**:
The new mean \( \bar{x}' \) of the new observations is:
\[ \bar{x}' = \frac{c \cdot a_1 + c \cdot a_2 + c \cdot a_3 + \ldots + c \cdot a_n}{n} = c \cdot \bar{x} \]
This shows that the new mean is \( c \) times the old mean.

4. **New Standard Deviation Calculation**:
The new standard deviation \( S' \) can be calculated using the new observations:
\[ S' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (c \cdot a_i - \bar{x}')^2} \]
Substituting \( \bar{x}' = c \cdot \bar{x} \):
\[ S' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (c \cdot a_i - c \cdot \bar{x})^2} \]

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

6. **Conclusion**:
Therefore, the new standard deviation \( S' \) after multiplying each observation by \( c \) is:
\[ S' = c \cdot S \]

### Final Answer:
The standard deviation of the resulting numbers is \( c \cdot S \).

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