Suppose you are shown four histograms, each representing a different data set with the same mean $\mu$. Which data set is most likely to have the smallest standard deviation $\sigma$?

• A. The data set with values spread out evenly across a wide range.
• B. The data set where all values are tightly clustered around the mean $\mu$.
• C. The data set with a uniform distribution from the minimum to the maximum value.
• D. The data set with two distinct peaks far from the mean $\mu$.

Answer: (B)

Verified step by step guidance

1. Understand that the standard deviation measures the average distance of data points from the mean, reflecting the spread or variability of the data.
2. Recognize that if all data values are tightly clustered around the mean, the distances from the mean are small, leading to a smaller standard deviation.
3. Contrast this with data sets where values are spread out evenly across a wide range or have multiple peaks far from the mean, which increase the average distance from the mean and thus increase the standard deviation.
4. Recall that a uniform distribution spreads values evenly across a range, so the data points are not clustered, resulting in a larger standard deviation compared to tightly clustered data.
5. Conclude that among the given options, the data set with values tightly clustered around the mean will have the smallest standard deviation because the variability is minimal.