Meaning of calculating the mean

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Calculating the mean of a set of numbers typically involves summing them and dividing by the count, yielding a straightforward average. The discussion explores an alternative method of calculating the mean by averaging pairs sequentially, which leads to different results. This method is criticized for lacking clarity and not providing a meaningful representation of the data. It is noted that while various operations can be performed on numbers, they may not yield valid or useful averages. Ultimately, the standard mean calculation remains the most reliable approach for summarizing a dataset.
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Hi,

Given the numbers: 1, 2, 3, 4, 5; to calculate the mean [as we all know] we would sum them up and divide, in this case, by 5. This gives 3. This is not the same as calculating the mean of each pair, which would be performed as follows:
1 + 2 = 3 / 2 = 1.5
1.5 + 3 = 4.5 / 2 = 2.25
2.25 + 4 = 6.25 / 2 = 3.125
3.125 + 5 = 8.125 / 2 = 4.0625

My question is, what does the above (calculating the mean of each pair) show us?
 
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King said:
Hi,

Given the numbers: 1, 2, 3, 4, 5; to calculate the mean [as we all know] we would sum them up and divide, in this case, by 5. This gives 3. This is not the same as calculating the mean of each pair, which would be performed as follows:
1 + 2 = 3 / 2 = 1.5
1.5 + 3 = 4.5 / 2 = 2.25

Why are you adding 1.5 to 3, it's not one of the numbers. The sum of 1+2 is being divided by 2 again.



2.25 + 4 = 6.25 / 2 = 3.125
3.125 + 5 = 8.125 / 2 = 4.0625

My question is, what does the above (calculating the mean of each pair) show us?

It whows us that there is an infinite number of ways to do something wrong.
 
Haha, nice. So what's wrong with it?
 
Well it's not the right way to calculate the mean. What do you expect it to show you? Why did you sum them in that order, and not, for example

(5+4)/2=4.5

(4.5+3)/2=3.75
(3.75+2)/2=2.875
(2.875+1)/2=1.9whatever

Given a bunch of numbers you can do whatever sequence of operations you want on them, it's just not clear why you would
 
King said:
Hi,

Given the numbers: 1, 2, 3, 4, 5; to calculate the mean [as we all know] we would sum them up and divide, in this case, by 5. This gives 3. This is not the same as calculating the mean of each pair, which would be performed as follows:
1 + 2 = 3 / 2 = 1.5
1.5 + 3 = 4.5 / 2 = 2.25
2.25 + 4 = 6.25 / 2 = 3.125
3.125 + 5 = 8.125 / 2 = 4.0625

My question is, what does the above (calculating the mean of each pair) show us?

The usual way of calculating the mean is (as you noted) adding up the numbers and dividing by the number of entries. However under some circumstances, depending on the underlying problem, a mean can be obtained by assigning weights to the different values (as long as the weights add to 1) and summing. This is essentially what you are doing in the second part.
 

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