Question

determine whether the statement is true or false. if it is false, rewrite it as a true statement.

a data set can have the same mean, median, and mode.

choose the correct answer below.

a. the statement is true.
b. the statement is false. a data set cannot have the same mean and mode.
c. the statement is false. a data set cannot have the same mean and median.
d. the statement is false. a data set cannot have the same median and mode.

Explanation:

Analyze the statement

The statement is: "A data set can have the same mean, median, and mode."

Evaluate validity with a counterexample

Consider a perfectly symmetrical data set with a single peak, such as:

$$ \{1, 2, 2, 2, 3\} $$

• Mean: \(\frac{1 + 2 + 2 + 2 + 3}{5} = \frac{10}{5} = 2\)
• Median: The middle value of the ordered set is \(2\)
• Mode: The most frequent value is \(2\)

Since all three measures of central tendency are equal to \(2\), the statement is true.

Answer:

• A. The statement is true. (Correct answer)
• B. The statement is false. A data set cannot have the same mean and mode.
• C. The statement is false. A data set cannot have the same mean and median.
• D. The statement is false. A data set cannot have the same median and mode.