teal wrote:

What is the median of a set {2,2,2,2} .....a set with all elements with the same value?

Median of a set that has a single element will be the same as the element that is if set = {2} then median = 2 as well.

Please confirm.

Generally:

The median of a set with odd # of terms is just a middle term (when ordered in ascending/descending order).

The median of a set with even # of terms is the average of two middle terms (when ordered in ascending/descending order).So, the median of {2,2,2,2} is (2+2)/2=2 and median of {2} is 2. Basically the median of a set with all equal numbers is this number itself (in this case it does not matter whether a set has an odd or even # of elements).

Hope it's clear.

As for the question:

If the mean of set S does not exceed mean of any subset of set S , which of the following must be true about set S?I. Set S contains only one element

II. All elements in set S are equal

III. The median of set S equals the mean of set 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

"The mean of set S does not exceed mean of

any subset of set S" --> set S can be:

A. \(S=\{x\}\) - S contains only one element (eg {7});

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

Why is that? Because if set S contains two (or more) different elements, then we can always consider the subset with smallest number and the mean of this subset (mean of subset=smallest number) will be less than mean of entire set (mean of full set>smallest number).

Example: S={3, 5} --> mean of S=4. Pick subset with smallest number s'={3} --> mean of s'=3 --> 3<4.

Now let's consider the statements:

I. Set S contains only one element - not always true, we can have scenario B too (\(S=\{x, x, ...\}\));

II. All elements in set S are equal - true for both A and B scenarios, hence always true;

III. The median of set S equals the mean of set S - - true for both A and B scenarios, hence always true.

So statements II and III are always true.

Answer: D.

This question is also discussed here:

ps-challenge-93565.htmlHope it helps.

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