Suppose a graph shows a normal distribution of data with a mean of $\mu =5$ and the data points are most densely clustered between $3.5$ and $6.5$. What is the standard deviation $\sigma$ of the data?

• A. $2.0$
• B. $2.5$
• C. $0.5$
• D. $1.5$

Answer: (D)

Verified step by step guidance

1. Recognize that the data is normally distributed with a mean (\mu) of 5, and the data points are most densely clustered between 3.5 and 6.5.
2. Understand that in a normal distribution, about 68% of the data lies within one standard deviation (\sigma) from the mean, i.e., between \( \mu - \sigma \) and \( \mu + \sigma \).
3. Set up the equation using the given interval: \( \mu - \sigma = 3.5 \) and \( \mu + \sigma = 6.5 \).
4. Substitute the mean value into the equations: \( 5 - \sigma = 3.5 \) and \( 5 + \sigma = 6.5 \).
5. Solve either equation for \( \sigma \) to find the standard deviation: \( \sigma = 5 - 3.5 \) or \( \sigma = 6.5 - 5 \).