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Can we add standard deviations ?

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by wilderness » Tue Jul 15, 2008 1:56 pm
Hi,

Set A and B are 2 sets of numbers.

A has a standard deviation 3 and a mean 5.

B has the same mean but standard deviation 4.

Can we find the standard deviation of the set A U B ?

If no, then is it possible with the following additional information

1. Both sets have the same number of members each (lets say 4)
2. Both sets have distinct members (no number is common to both sets).

Thanks.
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by VP_Tatiana » Thu Jul 17, 2008 11:59 am
We cannot find the SD of A U B with the original information. We determine this by thinking about an extreme counter example. If A has a million members, and B has one member, the SD of the union will be nearly equal to the SD of A. However, if B is the much larger set, then the SD of the union will be much closer to B.

By telling us that the sets contain equal members, we know we can weight the variances equally when calculating the combined SD. sqrt(variance A) = 3, and sqrt(variance B) = 4. The SD of A U B would just be sqrt ((Va + Vb)/2).

Statement 2 does not give us any useful information. Duplicate entries become irrelevant as we do the variance calculation, and don't affect the calculation in a different way from unique entries.

So, statement 1 alone is sufficient information.

Best wishes,

Tatiana
Tatiana Becker | GMAT Instructor | Veritas Prep
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by wilderness » Thu Jul 17, 2008 12:08 pm
Thanks VP_Tatiana
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