In the context of probability and data distributions, which type of distribution is most likely to have a $\mathrm{mean}$ and $\mathrm{median}$ that are not close in value?

• A. A $\mathrm{normal}$ distribution
• B. A $\mathrm{uniform}$ distribution
• C. A perfectly symmetric distribution
• D. A highly skewed distribution

Answer: (D)

Verified step by step guidance

1. Understand the concepts of mean and median: The mean is the average of all data points, while the median is the middle value when data points are ordered.
2. Recall that in symmetric distributions (like normal, uniform, or perfectly symmetric distributions), the mean and median are typically very close or equal because the data is evenly distributed around the center.
3. Recognize that in skewed distributions, the data is not symmetric; it has a longer tail on one side, which pulls the mean toward the tail more than the median.
4. Therefore, in a highly skewed distribution, the mean and median differ significantly because the mean is affected by extreme values (outliers) in the tail, while the median is more robust and less influenced by these extremes.
5. Conclude that the distribution type most likely to have a mean and median that are not close in value is a highly skewed distribution.