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28 May 2020, 05:06
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Hello guys,
I have a query regarding averages so I will try to ask the question by providing some examples to make myself clear.
Let's say that we have 2 sets A and B and we want to find the average of both those sets.
set A (1,3,8) set B (1,2,3)
Average of A=4
Average of B= 2
Average of both sets (4+2)/2=3
Alternatively (1+3+8+1+2+3)/6=3
As we can see in both cases the average is the same
However there are cases in which the above 2 approaches have 2 different results
set A ( 1,5,7,9) set B(13,14)
Average of set A= 5.5
Average of set B= 13.5
Average of set A and B= 19/2= 9.5
Alternatively (1+5+7+9+13+14)/6=8.16
Could someone shed some light on the issue that I have?
I have a query regarding averages so I will try to ask the question by providing some examples to make myself clear.
Let's say that we have 2 sets A and B and we want to find the average of both those sets.
set A (1,3,8) set B (1,2,3)
Average of A=4
Average of B= 2
Average of both sets (4+2)/2=3
Alternatively (1+3+8+1+2+3)/6=3
As we can see in both cases the average is the same
However there are cases in which the above 2 approaches have 2 different results
set A ( 1,5,7,9) set B(13,14)
Average of set A= 5.5
Average of set B= 13.5
Average of set A and B= 19/2= 9.5
Alternatively (1+5+7+9+13+14)/6=8.16
Could someone shed some light on the issue that I have?
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Updated on: 15 Jul 2020, 10:28
Originally posted by BrushMyQuant on 29 May 2020, 03:49.
Last edited by BrushMyQuant on 15 Jul 2020, 10:28, edited 1 time in total.
Last edited by BrushMyQuant on 15 Jul 2020, 10:28, edited 1 time in total.
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Average of two sets = Sum of all the numbers in both the sets/Total Number of numbers in both the sets
Example 1:
In this case you have the same number of elements in both the sets (3 each)
So when you are trying to do (Average of A + Average of B)/2 then it is still equal to average of both the sets because
((Average of A)*3 + (Average of B)*3) / 6 will give you the same result.
So, if you have same number of elements in two or more sets and if you want to take the average of all the sets together then you can still take the average of individual sets and then take average of all the sets to get average of all the sets together.
Example 2:
In this case you have different number of elements in both the sets (First has 4 and other has 2)
If you want to use the averages calculated for individual sets then you can do the following:
(4 *(Average of A) + 2*(Average of B)) / 6
But if you notice then 4*Average of A is same as Sum of all elements of A. SO, you can directly do some of all the elements in both the sets / total number of elements in both the sets.
When we find averages of two or more sets then you can also think about it as weighted average, instead of thinking of it as just average. Here weight is nothing but the total number of elements.
Hope it helps!
Example 1:
In this case you have the same number of elements in both the sets (3 each)
So when you are trying to do (Average of A + Average of B)/2 then it is still equal to average of both the sets because
((Average of A)*3 + (Average of B)*3) / 6 will give you the same result.
So, if you have same number of elements in two or more sets and if you want to take the average of all the sets together then you can still take the average of individual sets and then take average of all the sets to get average of all the sets together.
Example 2:
In this case you have different number of elements in both the sets (First has 4 and other has 2)
If you want to use the averages calculated for individual sets then you can do the following:
(4 *(Average of A) + 2*(Average of B)) / 6
But if you notice then 4*Average of A is same as Sum of all elements of A. SO, you can directly do some of all the elements in both the sets / total number of elements in both the sets.
When we find averages of two or more sets then you can also think about it as weighted average, instead of thinking of it as just average. Here weight is nothing but the total number of elements.
Hope it helps!
29 May 2020, 04:10
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If you're combining two sets, and finding the average of the combined set, then you're always calculating what is known as a "weighted average". The average is "weighted" by the sizes of the sets (by the ratio of their sizes, technically). So if one set is twice as big as the other, the average you calculate will be "twice as close" to the bigger set's average -- the bigger set is twice as important as the smaller one.
So if you wanted to find the average income for people in the USA and in Canada, then since the USA has about ten times as many people as Canada, the USA's average is going to be far more important than Canada's average. You could not just find the USA average, and the Canada average, and then average those two numbers. You'd need to do a weighted average calculation. Weighted averages are too big a topic to get into in a short post, but one way to get the right answer is to give the USA average a weight of 10 (i.e. multiply it by 10) and the Canada average a weight of 1, and then divide by 10 + 1 = 11. So if average income in the US is $31,000 and in Canada is $20,000, then the overall average will be
[(10)(31,000) + (1)(20,000)] / 11 = 30,000
If you look at how far the answer here is from the individual averages, you'll notice it's 10 times as far from Canada's average than from the USA average. In non-grammatical speak, you could say the answer is "10 times as close" to the USA average (that's not correct English though). That's identical to the ratio of the sizes of the two populations, and weighted averages always work out that way. That observation is the basis for a different weighted average method that is sometimes called "alligation", which you could look into, though that method takes a bit of practice. If you can learn it though, it's the best weighted average method for GMAT purposes.
So if you wanted to find the average income for people in the USA and in Canada, then since the USA has about ten times as many people as Canada, the USA's average is going to be far more important than Canada's average. You could not just find the USA average, and the Canada average, and then average those two numbers. You'd need to do a weighted average calculation. Weighted averages are too big a topic to get into in a short post, but one way to get the right answer is to give the USA average a weight of 10 (i.e. multiply it by 10) and the Canada average a weight of 1, and then divide by 10 + 1 = 11. So if average income in the US is $31,000 and in Canada is $20,000, then the overall average will be
[(10)(31,000) + (1)(20,000)] / 11 = 30,000
If you look at how far the answer here is from the individual averages, you'll notice it's 10 times as far from Canada's average than from the USA average. In non-grammatical speak, you could say the answer is "10 times as close" to the USA average (that's not correct English though). That's identical to the ratio of the sizes of the two populations, and weighted averages always work out that way. That observation is the basis for a different weighted average method that is sometimes called "alligation", which you could look into, though that method takes a bit of practice. If you can learn it though, it's the best weighted average method for GMAT purposes.
