Meaning of calculating the mean

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Discussion Overview

The discussion revolves around the concept of calculating the mean of a set of numbers, specifically comparing the traditional method of finding the mean with an alternative method of calculating the mean of pairs. Participants explore the implications and interpretations of these different approaches.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes the standard method of calculating the mean of the numbers 1, 2, 3, 4, 5, resulting in a mean of 3, and contrasts this with a method of calculating the mean of each pair sequentially.
  • Another participant questions the validity of the pairwise mean calculation, pointing out that the intermediate values used in the calculation are not part of the original set of numbers.
  • A different participant challenges the rationale behind the chosen order of summation in the pairwise method, suggesting that any sequence could yield different results, raising questions about the meaningfulness of the approach.
  • One participant suggests that under certain circumstances, a mean can be calculated by assigning weights to values, implying that the pairwise method may reflect a weighted mean approach.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity and meaningfulness of the pairwise mean calculation. There is no consensus on whether this method provides useful insights or is simply incorrect.

Contextual Notes

The discussion highlights the ambiguity in defining what constitutes a valid mean calculation, particularly when considering different methods and their implications. Participants do not resolve the mathematical validity of the pairwise approach.

King
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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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