Suppose you have two histograms, Histogram A and Histogram B, each displaying the distribution of exam scores for two different classes. Histogram A shows scores tightly clustered around the mean $\mu$, while Histogram B shows scores spread out over a wider range. Which histogram depicts a higher standard deviation $\sigma$?

• A. Both histograms have the same standard deviation $\sigma$
• B. It is impossible to determine from the histograms
• C. Histogram A
• D. Histogram B

Answer: (D)

Verified step by step guidance

1. Understand that the standard deviation measures the amount of variation or dispersion in a data set. A higher standard deviation means the data points are spread out more widely from the mean.
2. Look at Histogram A, which shows scores tightly clustered around the mean. This indicates that most scores are close to the average, suggesting a smaller spread.
3. Look at Histogram B, which shows scores spread out over a wider range. This indicates that the scores vary more widely from the mean, suggesting a larger spread.
4. Since standard deviation quantifies the spread of data, the histogram with scores more spread out (Histogram B) will have a higher standard deviation.
5. Therefore, Histogram B depicts a higher standard deviation because the data points are more dispersed compared to Histogram A.