If set S consists of an odd number of even integers that have a normal

Question

Source: Princeton Review 1012 P239 #8.

If set S consists of an odd number of even integers that have a normal distribution, what is the standard deviation of set S?

1) the mean of set S is 4.
2) the median of set S is 4.

The official answer is C. The logic is that when the mean and the median are equal, we have a consecutive set. Since the set is only even integers, this implies a consecutive even set with a mean of 4. The book claims that this means the standard deviation is 2. The standard deviation is 2 if you have a set consisting of three integers. However, if you have more than 3 integers, such as 5 (also an odd number) the standard deviation is different. To test the different standard deviations, calculate them with {2,4,6} and then again with {0,2,4,6,8}. My take is that the answer is E - we don't have enough information to solve it until we know the number of figures in the set.

Any insight is appreciated.

IanStewart · Answer

If that's what the book says, just throw it away. I don't think a single thing in that explanation is actually correct:

* even the question itself is flawed: finite sets can't have a 'normal distribution' (normal distributions are infinite distributions); at best they can have 'approximately normal' distributions. If that's the case, though, the set won't be equally spaced, as the explanation goes on to conclude. Further, normal distributions are simply never tested on the GMAT, so the question is irrelevant to test takers.

* just because your mean and median are equal is not enough to let you conclude that your set is equally spaced - that's nonsense. You can't even make the weaker conclusion that your set is 'symmetric' just from the fact that your mean and median are equal. For example, the set {-20,2,4,8,26} has a mean and median of 4, but is not equally spaced, nor is it even symmetric. I'd add that it is very useful to know, on many GMAT questions, that when you have an equally spaced set, your mean and median are always equal, but the converse of that fact is false - when your mean and median are equal, your set does not need to be equally spaced.

* The standard deviation of an equally spaced set is not straightforward to calculate. The standard deviation, for example, of the set {2, 4, 6} is certainly not equal to 2. To find the standard deviation, you find the distances from each element to the mean - those distances are 2, 0 and 2 - square them and average to get 8/3, then take the square root. So the standard deviation of {2, 4, 6} is just a bit less than  \sqrt{2.7}  and is certainly less than 2.

* The number of elements matters. The set {2, 4, 6} has a smaller standard deviation than the set {0, 2, 4, 6, 8}, because we can make this latter set by adding elements to the first set which are far from the mean.

garimavyas · Answer

where is it mentioned that the even integers are consecutive, a set like {2,4,4,4,6} has a mean of 4 and a median also of 4. a set like {2,4,6} also has a mean of 4 and median of 4, but the standard deviation in the two cases is different.

C can not be the answer . stupid question

C can not be the answer . stupid question

amit2k9 · Answer

the set can be either (4,4,4) or (2,4,6) or so.
STD deviation is 0 or approx 2 (considering the differences only).
Hence a clean E here.

manishgeorge · Answer

Does normal distribution as said in the question mean that the numbers in the set are in some sequence?

if we look at it that way

statemene 1: mean is 4

sum of all the elements could be 12, 16, 20

2 4 6

0 2 4 6 8 etc ...

second option

median is 4: there for there are equal number of digits before are after 4

possible values are,

246
02468

now, combining 1 and 2. the only result that has a sum of 20, mean 4, median 4 and in normal distribution is 02468

here is the dictionary definition for normal distribution:

Statistics) Statistics a continuous distribution of a random variable with its mean, median, and mode equal,

If set S consists of an odd number of even integers that have a normal

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If set S consists of an odd number of even integers that have a normal

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