Which characteristic must be known about a data set before the empirical rule can be applied?

• A. The data set must have a mean of $0$.
• B. The data set must contain only positive values.
• C. The data set must have equal mean and median.
• D. The data set must be approximately normally distributed.

Answer: (D)

Verified step by step guidance

1. Understand that the empirical rule, also known as the 68-95-99.7 rule, applies specifically to data sets that follow a normal distribution or are approximately normally distributed.
2. Recall that the empirical rule describes the percentage of data within 1, 2, and 3 standard deviations from the mean in a normal distribution: about 68%, 95%, and 99.7%, respectively.
3. Recognize that the mean value does not have to be zero for the empirical rule to apply; the rule is about the shape of the distribution, not the specific mean value.
4. Note that the data set does not need to contain only positive values; normal distributions can include negative values as well.
5. Understand that having equal mean and median is a characteristic of symmetric distributions, which includes normal distributions, but the key requirement is that the data be approximately normally distributed for the empirical rule to hold.