The question asks which of the statements I, II, and III are true about the mode. I. The mode is not unique. II. The mode of a set of data may not exist. III. It is not affected by outliers.

1. Problem Description

The question asks which of the statements I, II, and III are true about the mode.

I. The mode is not unique.

II. The mode of a set of data may not exist.

III. It is not affected by outliers.

2. Solution Steps

The mode is the value that appears most frequently in a data set.

I. The mode is not unique: This is true. A data set can have multiple modes (bimodal, trimodal, etc.) if several values have the same highest frequency.

II. The mode of a set of data may not exist: This is true. If all values in a data set occur with the same frequency, then there is no mode. For instance, the dataset {1, 2, 3, 4, 5} does not have a mode.

III. It is not affected by outliers: This is false. Consider the dataset {1, 2, 3, 4, 5}. The mode doesn't exist. Now consider {1, 2, 3, 4, 100}. The outlier does not affect the mode since there is still no mode. But consider {1,1,1,2,3,4,5}. Mode is

1. Now consider {1,1,1,2,3,4,100}. The mode is still

1. However, consider the dataset {1, 2, 2, 2, 3, 4, 100}. Mode is

2. If the dataset were {1, 2, 2, 2, 3, 4}, the mode would also be

2. Outliers can alter the mode if they increase the frequency of a number.

Therefore, statements I and II are true, and statement III is false. Since none of the answer choices match this, the best choice is D.

3. Final Answer

D. None of the above.