Whenever a data value is less than the $\mathrm{mean}$, _______.

• A. it is equal to the $\mathrm{mean}$
• B. it is located to the left of the $\mathrm{mean}$ on a number line
• C. it is located to the right of the $\mathrm{mean}$ on a number line
• D. it is always the smallest value in the data set

Answer: (B)

Verified step by step guidance

1. Understand the definition of the mean: The mean is the average of all data values, calculated by summing all values and dividing by the number of values. Mathematically, the mean \( \bar{x} \) is given by \( \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \).
2. Visualize the number line: Imagine placing all data values on a number line, where smaller values are to the left and larger values are to the right.
3. Interpret what it means for a data value to be less than the mean: If a data value \( x \) satisfies \( x < \bar{x} \), then it must be positioned to the left of the mean on the number line because smaller numbers are on the left side.
4. Eliminate incorrect options: The data value cannot be equal to the mean if it is less than the mean, nor can it be to the right of the mean. Also, being less than the mean does not guarantee it is the smallest value in the data set.
5. Conclude that any data value less than the mean is located to the left of the mean on the number line.