Will standard deviation and mean uniquely identify a set?

Question

If I fix the values of standard deviation and mean, is it true that we can get only set that satisfies this condition?

Simply put, is it possible that two sets may have the same standard deviation and mean?

Bunuel · Answer

Below two different sets have the same SD and mean: The SD of {-1, 0, 1} is \sqrt{\frac{2}{3}} and mean is 0; The SD of { -\sqrt{\frac{4}{3}} , 0, 0, \sqrt{\frac{4}{3}} } is \sqrt{\frac{2}{3}} and mean is 0. Please read this: statistics-126672.html#p1035732 Hope it helps.

gmatter0913 · Answer

I guess it is not so.

The following two sets have the same mean and the same deviation.

Does this mean if I say: given a mean, a standard deviation, and fixed number of elements, one can get only one set satisfying those values?

Bunuel · Answer

Please read this: statistics-126672.html#p1035732

gmatter0913 · Answer

The link doesn't have any answer to my question.

My question is: Could we find two sets having the same mean, standard deviation, and the number of elements?

Bunuel · Answer

The link has an answer to another question,  if you are preparing for the GMAT  you just need to understand the concept of it: standard deviation shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Also there is links to the SD chapter of Math Book, to some useful rules and tips on SD along with typical GMAT questions on this subject.

walker · Answer

Yes, you can find two different sets with the same mean, standard deviation, and the number of elements. Here is how you can construct one of such pairs:

set1: {-2, -1, 0, 1, 2}

set2: {-x, -1.1 , 0, 1.1, x} - the same mean and number of elements. Now we should find x for which the set has the same SD as set 1. My guess it's ~1.95

Bunuel · Answer

Sure you can find such sets. Simpler example would be:

{-2, -1, 0, 1, 2} -->  SD=\sqrt{2}  and  mean=0 ;
{- \sqrt{5} , 0, 0, 0,  \sqrt{5} } -->  SD=\sqrt{2}  and  mean=0 .

Both have the same SD, mean and # of elements.

Will standard deviation and mean uniquely identify a set?

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