Which of the following best describes the relationship among the $\mathrm{mean}$, $\mathrm{median}$, and $\mathrm{mode}$ in a perfectly symmetric (normal) distribution?

• A. The $\mathrm{mean}$ is always greater than the $\mathrm{median}$ and $\mathrm{mode}$.
• B. The $\mathrm{mean}$, $\mathrm{median}$, and $\mathrm{mode}$ are all equal.
• C. The $\mathrm{mean}$ is always less than the $\mathrm{median}$ and $\mathrm{mode}$.
• D. The $\mathrm{mode}$ is always less than the $\mathrm{mean}$ and $\mathrm{median}$.

Answer: (B)

Verified step by step guidance

1. Understand the characteristics of a perfectly symmetric (normal) distribution: it is symmetric about its center, meaning the left and right sides are mirror images.
2. Recall the definitions of mean, median, and mode: the mean is the average of all data points, the median is the middle value when data is ordered, and the mode is the most frequently occurring value.
3. In a perfectly symmetric distribution, the data is evenly spread around the center, so the mean, median, and mode all coincide at the center point.
4. Therefore, the mean, median, and mode have the same value in a perfectly symmetric (normal) distribution.
5. This implies that none of the inequalities (mean > median and mode, mean < median and mode, mode < mean and median) hold true; instead, they are all equal.