Two histograms (A and B) each summarize 100 observations and have the same mean. Histogram A is tightly clustered around the mean, while Histogram B is more spread out with more observations far from the mean. Which histogram depicts a higher standard deviation?

• A. They must have the same standard deviation because the means are the same
• B. Histogram A
• C. Cannot be determined without knowing the exact bin widths
• D. Histogram B

Answer: (D)

Verified step by step guidance

1. Recall that the standard deviation measures the spread or dispersion of data points around the mean. It quantifies how much the observations deviate from the average value.
2. Understand that two datasets can have the same mean but different spreads. The mean alone does not determine the variability of the data.
3. Histogram A is described as tightly clustered around the mean, meaning most observations are close to the average value, indicating low variability.
4. Histogram B is described as more spread out with more observations far from the mean, indicating higher variability in the data.
5. Since standard deviation increases as data points are more spread out from the mean, Histogram B, which has more spread, will have a higher standard deviation than Histogram A.