To determine which statement is true given that the spread of data set X is greater than that of data set Y, and both data sets are normally distributed, let's analyze each option:
A. The mean of data set X is greater than the mean of data set Y.
- The spread (variability) of a data set does not provide information about the central tendency (mean) of the data. Therefore, this statement is not necessarily true.
B. The median of data set X is less than the median of data set Y.
- Similarly, the spread of a data set does not give us information about the median. The spread indicates how wide the data values are, not their central location. Therefore, this statement is not necessarily true.
C. The standard deviation of data set X is greater than the standard deviation of data set Y.
- The spread of a data set is commonly measured by the standard deviation, which quantifies the amount of variation or dispersion in the data set. Since we know that the spread of data set X is greater than that of data set Y, it follows that the standard deviation of X must be greater than that of Y. This statement is true.
D. The range of data set X is less than the range of data set Y.
- The range is another measure of spread, and if the overall spread of data set X is greater than that of data set Y, the range of X will not be less than the range of Y. Therefore, this statement is not true.
E. The mode of data set X is greater than the mode of data set Y.
- Like the mean and median, the mode is a measure of central tendency and does not provide information about the spread. Therefore, this statement is not necessarily true.
Given these analyses, the correct and true statement is:
○ C. The standard deviation of data set X is greater than the standard deviation of data set Y.
