Given four histograms representing different data sets, each with the same mean $\mu$ but varying spreads, which histogram would correspond to the largest standard deviation $\sigma$?

• A. The histogram with data values spread out most widely from the mean $\mu$
• B. The histogram with a perfectly symmetrical bell shape and no outliers
• C. The histogram with data values most tightly clustered around the mean $\mu$
• D. The histogram with all data values equal to the mean $\mu$

Answer: (A)

Verified step by step guidance

1. Recall that the standard deviation measures the average distance of data points from the mean, quantifying the spread or dispersion of the data set.
2. Identify that if all histograms have the same mean, the key difference affecting standard deviation is how spread out the data values are around that mean.
3. Understand that a histogram with data values spread out most widely from the mean will have larger deviations for many points, increasing the overall standard deviation.
4. Recognize that a perfectly symmetrical bell-shaped histogram with no outliers typically represents a normal distribution with moderate spread, so its standard deviation is moderate.
5. Note that a histogram with data values tightly clustered around the mean or all equal to the mean will have very small or zero deviations, resulting in the smallest standard deviation.