If the standard deviation of a variable `X \ is \ sigma` , then standard deviation of variable `(a X+b)/c` is

Options

A. `asigma`
B. `a/csigma`
C. `|a/c|sigma`
D. `(asigma+b)/c`

Explanation

To find the standard deviation of the variable \( Y = \frac{aX + b}{c} \) when the standard deviation of \( X \) is \( \sigma \), we can follow these steps:

### Step 1: Understand the transformation
The variable \( Y \) is defined as:
\[ Y = \frac{aX + b}{c} \]
Here, \( a \) and \( c \) are constants, and \( b \) is an additive constant.

### Step 2: Find the mean of \( Y \)
The mean of \( Y \) can be calculated as:
\[ \bar{Y} = \frac{a\bar{X} + b}{c} \]
where \( \bar{X} \) is the mean of \( X \).

### Step 3: Compute the variance of \( Y \)
The variance of \( Y \) is given by the formula:
\[ \sigma_Y^2 = \text{Var}(Y) = \text{Var}\left(\frac{aX + b}{c}\right) \]
Since the variance is affected by the square of the scaling factor and is not affected by the additive constant, we can simplify this as:
\[ \sigma_Y^2 = \left(\frac{a}{c}\right)^2 \text{Var}(X) \]

### Step 4: Substitute the variance of \( X \)
We know that the variance of \( X \) is \( \sigma^2 \), so we can substitute this into the equation:
\[ \sigma_Y^2 = \left(\frac{a}{c}\right)^2 \sigma^2 \]

### Step 5: Take the square root to find the standard deviation
To find the standard deviation \( \sigma_Y \), we take the square root of the variance:
\[ \sigma_Y = \sqrt{\sigma_Y^2} = \sqrt{\left(\frac{a}{c}\right)^2 \sigma^2} \]
This simplifies to:
\[ \sigma_Y = \left|\frac{a}{c}\right| \sigma \]

### Final Answer
Thus, the standard deviation of the variable \( Y = \frac{aX + b}{c} \) is:
\[ \sigma_Y = \left|\frac{a}{c}\right| \sigma \]