Which of the following statements must be true ? I. The range of a data set must be smaller than its standard deviation.
II. The standard deviation of a data set must be smaller than its mean.
III. The median of a data set must be smaller than its mode.

Options

A. I only
B. I and II
C. II only
D. none are true

Explanation

To determine which of the statements must be true, we will analyze each statement one by one, providing examples to prove or disprove them.

### Statement I: The range of a data set must be smaller than its standard deviation.

1. **Example Dataset**: Let's consider the dataset: 1, 5, 10.
2. **Calculate the Mean**:
\[
\text{Mean} = \frac{1 + 5 + 10}{3} = \frac{16}{3} \approx 5.33
...