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# Mergin sets and arithmetic mean

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Director
Joined: 23 Apr 2010
Posts: 574

Kudos [?]: 93 [0], given: 7

Mergin sets and arithmetic mean [#permalink]

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30 Nov 2011, 07:51
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)

Thanks.

Kudos [?]: 93 [0], given: 7

Manager
Joined: 09 Nov 2011
Posts: 127

Kudos [?]: 64 [0], given: 16

Re: Mergin sets and arithmetic mean [#permalink]

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30 Nov 2011, 07:56
I think the relation would be Avg (S) + Avg(T) < Avg(V)...
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Kudos [?]: 64 [0], given: 16

GMAT Tutor
Joined: 24 Jun 2008
Posts: 1341

Kudos [?]: 1908 [0], given: 6

Re: Mergin sets and arithmetic mean [#permalink]

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30 Nov 2011, 12:04
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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Kudos [?]: 1908 [0], given: 6

Director
Joined: 23 Apr 2010
Posts: 574

Kudos [?]: 93 [0], given: 7

Re: Mergin sets and arithmetic mean [#permalink]

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30 Nov 2011, 15:22

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.

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Re: Mergin sets and arithmetic mean   [#permalink] 30 Nov 2011, 15:22
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