M16-23

Official Solution:


If the average (arithmetic mean) of dataset \(S\) is not greater than the average (arithmetic mean) of any non-empty subset of \(S\), which of the following statements must be true?

I. \(S\) contains only one element.

II. All elements in \(S\) are equal.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).


A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III


Consider a dataset \(S\). If the average of \(S\) is not greater than the average of ANY subset of \(S\), then the following scenarios are possible:

A. \(S=\{x\}\), where \(S\) contains only one element (e.g. {7});

B. \(S=\{x, x, ...\}\), where \(S\) contains more than one element and all elements are equal (e.g. {7,7,7,7}).

This is because if the dataset \(S\) contains two or more different elements, then we can always consider the subset with the smallest number, and the mean of this subset (mean of the subset = smallest number) will be less than the mean of the entire dataset (mean of the full dataset > smallest number).

For example, if \(S=\{3, 5\}\), then the mean of \(S=4\). If we pick the subset with the smallest number, \(s'=\{3\}\), then the mean of \(s'=3\). Thus, \(3 < 4\).

Now let's consider the statements:

I. \(S\) contains only one element. This statement is not always true since scenario B is also possible (\(S=\{x, x, ...\}\)).

II. All elements in \(S\) are equal. This statement is true for both scenarios A and B, hence always true.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\). This statement is true for both scenarios A and B, hence always true.

Therefore, statements II and III are always true.


Answer: D