Which of the following best describes the $\mathrm{mean}$ in a data set?

• A. It is always the most frequently occurring value in the data set.
• B. It is the difference between the largest and smallest values in the data set.
• C. It is the value that divides the data set into two equal halves.
• D. It represents the arithmetic center of the data by averaging all values, that is, $\mathrm{mean}=\frac{\sum x_i}{n}$.

Answer: (D)

Verified step by step guidance

1. Understand the definition of the mean: The mean is a measure of central tendency that represents the average value of a data set.
2. Recall that the mean is calculated by summing all the values in the data set and then dividing by the number of values. This can be expressed as: \(\text{Mean} = \frac{\sum_{i=1}^n x_i}{n}\), where \(x_i\) are the data points and \(n\) is the total number of data points.
3. Compare the mean to other measures of central tendency mentioned: the mode (most frequently occurring value), the range (difference between largest and smallest values), and the median (value dividing the data into two equal halves).
4. Recognize that the mean is not necessarily the most frequent value (mode), nor the middle value (median), nor the range, but rather the arithmetic average of all values.
5. Conclude that the best description of the mean is that it represents the arithmetic center of the data by averaging all values.