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Mergin sets and arithmetic mean

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Director
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Mergin sets and arithmetic mean [#permalink] New post 30 Nov 2011, 06:51
Say that we have two sets S and T that we merge to form a single set V. What is the relation between the sum of averages of S and T and an average of V if we know that the sets contain only positive numbers?

I think that it is: AVG(S) + AVG(T) > AVG(V)

Please double check me.

Thanks.
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Re: Mergin sets and arithmetic mean [#permalink] New post 30 Nov 2011, 06:56
I think the relation would be Avg (S) + Avg(T) < Avg(V)...
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Re: Mergin sets and arithmetic mean [#permalink] New post 30 Nov 2011, 11:04
nonameee wrote:
Say that we have two sets S and T that we merge to form a single set V. What is the relation between the sum of averages of S and T and an average of V if we know that the sets contain only positive numbers?

I think that it is: AVG(S) + AVG(T) > AVG(V)


If both averages are positive, then yes, your inequality will be true. But we can say a lot more. The situation you're describing is that of a conventional weighted average. We have two groups, each with their own average. When we combine the two groups, the combined average must be *in the middle* of the two group averages. So if you know, say, that AVG(S) < AVG(T), then you could be certain that:

AVG(S) < AVG(V) < AVG(T)

The larger set T is relative to set S (in ratio terms), the closer the average of V will be to the average of T.

You encounter this situation in word problems very frequently. If, say, at a company, women earn on average $44/hour, and men earn on average $36/hour, then the average wage for all employees certainly must be somewhere between $36 and $44. The higher the proportion of employees who are women, the closer the average will be to $44.

Finally in this example, it certainly is true that the combined average is less than the sum of 36 and 44, but that doesn't really give you especially useful information.
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Re: Mergin sets and arithmetic mean [#permalink] New post 30 Nov 2011, 14:22
Ian, thanks for your reply.

The reason I asked this question is because it appeared in one of the GmatClub tests (Test24):

Quote:
21. If sets S and T are merged into a single set, will the mean of this set be smaller than the sum of means of sets S and T ?

(1) S and T are one-element sets
(2) Neither set S nor set T contains negative numbers


So, I wanted to study the properties of averages when combining sets containing various numbers (positive, negative, same numbers etc.). If we allow numbers to be negative, I found no straightforward rule as the one I mentioned in my first post.
Re: Mergin sets and arithmetic mean   [#permalink] 30 Nov 2011, 14:22
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