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

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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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.
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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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Thanks & Regards,
Anaira Mitch

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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