Step 1: Work with variance instead of standard deviation directly.
Variance and standard deviation are related by $\text{Var}(X) = [\text{SD}(X)]^2$. It is often easier to first find the variance of $Y$ using the scaling property of variance, then take a square root at the end to recover the standard deviation.
Step 2: Find the variance of $X$.
$\text{SD}(X) = 2$, so $\text{Var}(X) = 2^2 = 4$.
Step 3: Apply the variance scaling rule for a linear transformation.
For $Y = aX + b$, variance scales as $\text{Var}(Y) = a^2 \text{Var}(X)$, since variance is a squared quantity and any constant shift $b$ drops out completely.
Here $Y = \frac{X-3}{4} = \frac{1}{4}X - \frac{3}{4}$, so $a = \frac{1}{4}$.
\[ \text{Var}(Y) = \left(\frac{1}{4}\right)^2 \times 4 = \frac{1}{16} \times 4 = \frac{1}{4} \]
Step 4: Take the square root to get the standard deviation.
\[ \text{SD}(Y) = \sqrt{\text{Var}(Y)} = \sqrt{\frac{1}{4}} = \frac{1}{2} = 0.5 \]
Step 5: Match with the given options.
This value, $0.5$, corresponds to option (A). The mean value $3$ never enters this calculation, because it only shifts the variable and cancels out completely in a variance calculation.
Final Answer:
The standard deviation of $Y$ is $0.5$.
\[ \boxed{0.5} \]