Question

select the correct answer.

the spread of data set \\(x\\) is greater than the spread of data set \\(y\\), and the data sets are normally distributed. which statement is true?

• the mean of data set \\(x\\) is greater than the mean of data set \\(y\\).
• the median of data set \\(x\\) is less than the median of data set \\(y\\).
• the standard deviation of data set \\(x\\) is greater than the standard deviation of data set \\(y\\).
• the range of data set \\(x\\) is less than the range of data set \\(y\\).
• the mode of data set \\(x\\) is greater than the mode of data set \\(y\\).

Explanation:

Analyze the relationship between spread and standard deviation

$$ \text{Spread of } X > \text{Spread of } Y \implies \sigma_X > \sigma_Y $$

Evaluate the given options

• Mean, median, and mode measure central tendency, not spread.
• Range is a measure of spread, so the range of \(X\) must be greater than (not less than) the range of \(Y\).
• Standard deviation is a direct measure of spread, meaning a larger spread corresponds to a larger standard deviation.

Answer:

• The mean of data set X is greater than the mean of data set Y.
• The median of data set X is less than the median of data set Y.
• The standard deviation of data set X is greater than the standard deviation of data set Y. (Correct answer)
• The range of data set X is less than the range of data set Y.
• The mode of data set X is greater than the mode of data set Y.