Question

\(X\) is a random variable whose mean is 3 and standard deviation is 2. A new random variable \(Y\) is defined as:
\[ Y = \frac{X-3}{4} \]
The standard deviation of \(Y\) is

• A. \(0.5\)
• B. \(1\)
• C. \(2\)
• D. \(8\)

Hint:

Standard deviation scales by the absolute value of the multiplying constant; additive shifts do not change it.

Answer 1

The Correct Option is A

Step 1: Recall how a linear transformation affects standard deviation.
If \(Y = aX + b\) for constants \(a\) and \(b\), the standard deviation of \(Y\) is \(|a|\) times the standard deviation of \(X\). Adding a constant \(b\) only shifts every value up or down; it does not change how spread out the values are, so it has no effect on the standard deviation.

Step 2: Identify \(a\) and \(b\) for the given transformation.
We are given \[ Y = \frac{X-3}{4} = \frac{1}{4}X - \frac{3}{4} \]
Comparing with \(Y = aX+b\), we get \(a = \frac{1}{4}\) and \(b = -\frac{3}{4}\).

Step 3: Apply the standard deviation scaling rule.
The standard deviation of \(X\) is given as \(2\).
\[ \text{SD}(Y) = |a| \cdot \text{SD}(X) = \frac{1}{4} \times 2 = 0.5 \]

Step 4: Note why the mean of \(X\) plays no part here.
The mean, 3, only shifts \(X\) inside the transformation before scaling by \(\frac{1}{4}\); a shift changes the mean of \(Y\) but never its spread, so it does not enter the standard deviation calculation.

Step 5: Check why the other options are wrong.
Option (B) 1 and option (C) 2 would come from treating the scale factor as 1 or forgetting to divide by 4. Option (D) 8 comes from multiplying by 4 instead of dividing by 4.

Final Answer:
The standard deviation of \(Y\) is \(0.5\).
\[ \boxed{0.5} \]