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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