Remember that the sum of deviations of all scores from the mean is 0.
i.e. if one score is less than mean, there has to be one (or more) score that is more than mean.
e.g. If mean is 78, all scores can be equal to 78.
If I know there is one score less than 78, then there has to be at least one score more than 78. (If one score is 70, i.e. 8 less than 78, one or more other scores have to make up this deficit of 8 i.e. there might be one score that is 86 or there might be two scores 82 each etc)

So if we know that one score is 78, either the other two will also be 78 or one will be less than 78 and the other will be greater than 78. In either case, median will be 78.
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Hi Ajit,

given 78 is the mean and let the scores be {x,y,z}

Statement 1:

Paul scored 73 in his exam which makes it {73, y,z}

we do know that (x+y+z)/3 is 78 and hence

73+y+z=78*3=234

y+z =234 and we cannot decipher further since we do not knw y or z hence Statement 1 is Insufficient

Statement 2:
Mary scored 78 in her exam.

{ x,78, z}

Since 78 is the mean of the two terms x shld be lesser than 78 and z shld be greater than 78 or x shld be greater than 78 and z shld be lesser than 78. In both the situations 78 would be the middle number if the three terms are arranged in ascending order.Statement 2 is Sufficient

Hence Ans B

I don't think this is necessarily the case. All of them can be 78. Even so, the statement remains sufficient.

Mean = 78
total No of people 3. therefore <78 ~78 >78 could be the three values. [Less than, approx, greater than]

1. 73 is obviously least mark, it doesn't say anything about the middle value.
2. 78 is obtained by one of the person. And we know 78 is the mean => <78,78,>78 or 78,78,78 . In any case, median is 78.

Peter, Paul, and Mary each received a passing score on his/her history midterm. The average (arithmetic mean) of the three scores was 78. What was the median of the three scores?

(1) Peter scored a 73 on his exam. If the other two scores are 73 and 78*3-73-73 (whatever it is) then the median is 73 but if the other two scores are 78 and 83 then the median is 78. Not sufficient.

(2) Mary scored a 78 on her exam. In order the average to be 78, one of the remaining scores (Peter or Paul) must be less than 78 and another must be more than 78 OR all three scores must be 78. In either case, the median is 78. Sufficient.

Answer: B.
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Your explanation is perfect just making some corrections.

Hi Ajit,

given 78 is the mean and let the scores be {x,y,z}

Statement 1:

Paul scored 73 in his exam which makes it {73, y,z}

we do know that (x+y+z)/3 is 78 and hence

73+y+z=78*3=234

y+z =161 and we cannot decipher further since we do not know y or z hence Statement 1 is Insufficient

Statement 2:
Mary scored 78 in her exam.

{ x,78, z}

Since 78 is the mean of the two terms x shld be lesser than 78 and z shld be greater than 78 or x shld be greater than 78 and z shld be lesser than 78. In both the situations 78 would be the middle number if the three terms are arranged in ascending order.Statement 2 is Sufficient
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What this question is basically asking us to find is- whether or not we can find have different possibilities for the mean.

Statement 2 is a bit tricky because even though all three scores could be 78 , ultimately 78 must be the median because you cannot have two values that are smaller than 78 or greater than 78 - a score cannot exceed 100 on the scale.

Target question: What was the median of the three scores?

Since there are 3 values, the median will be the middle-most value (when the values are arranged in ascending order).

We also know that: Total of all values = (median)(# of values)
So, the sum of all 3 scores = (78)(3) = 234

Statement 1: Peter scored a 73 on his exam.
There are several sets of scores that meet this condition. Here are two:
Case a: Peter:73, Paul:74, Mary:87, in which case the median is 74
Case b: Peter:73, Paul:75, Mary:86, in which case the median is 75
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Mary scored a 78 on her exam
NOTE: For scores above 78, I'll use the notation 78+ and for scores below 78, I'll use the notation 78-
If the mean is 78 and Mary scored a 78, then there are only 3 scenarios possible:
scenario 1: Peter:78, Mary:78, Paul:78, in which case the median is 78
scenario 2: Peter:78-, Mary:78, Paul:78+, in which case the median is 78
scenario 3: Peter:78+, Mary:78, Paul:78-, in which case the median is 78

Notice that no other scenarios are possible. For example, consider this scenario:
Peter:78+, Mary:78, Paul:78+
This scenario is impossible, because the sum of all three values must be 234, and we know that 78+78+78=234.
So, it is impossible for (78)+(78+)+(78+) to equal 234

Using similar logic and notation we can show that other scenarios are impossible.
As you can see, statement 2 consistently yields the same answer to the target question.
So, statement 2 is SUFFICIENT

Answer = B

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