A group of 7 students took a test. In the test, one student scored 100% and 2 students scored 0%. If the median score of

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A group of 7 students took a test. In the test, one student scored 100% and 2 students scored 0%. If the median score of the group is 20%, what is the value of the average (arithmetic mean) score of the group of students?

(1) If the students who scored either 0% or 100% are not considered, the median score of the group improves to 25%.

(2) If the students who scored either 0% or 100% are not considered, the range of the scores of the group is 10%.

Answer: C

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Since we have an odd number of scores, the median must be one of the test scores. So we know two of the scores are zero, one is 20, and one is 100. We don't know three of the scores. Writing all seven scores, using unknowns, in increasing order (some of the middle scores might equal each other) we have:

0, 0, a, 20, b, c, 100

Using Statement 1, removing the 0's and 100's, we have a, 20, b, c, in increasing order. The median of this list is now the average of 20 and b, and if the median of this list is 25, then b = 30. But we need more information about a and c to find the average.

Using Statement 2, we know that a and c are 10 apart. That restricts the values quite a lot -- since a can't be larger than 20, we can see that c can't be larger than 30, for example. But we still have many possible values for a, b and c, and cannot find the average.

Using both Statements, the set must look like this, from Statement 1:

0, 0, a, 20, 30, c, 100

in increasing order. If a were less than 20, then c would be less than 30, but then the median described in Statement 1 would no longer be 25, so it's impossible that a is less than 20. Simliarly, a cannot be greater than 20, since then the median of the entire list is no longer 20. So a must be exactly 20, and our list can only be:

0, 0, 20, 20, 30, 30, 100

Now we know our values, so we can compute any statistic at all, and the answer is C.
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