Meaning of calculating the mean

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SUMMARY

The discussion centers on the calculation of the mean using two different methods: the traditional method of summing all numbers and dividing by the count, and an alternative method of calculating the mean of pairs. The traditional method for the numbers 1, 2, 3, 4, 5 yields a mean of 3. In contrast, the pairwise mean calculations lead to a series of intermediate values, ultimately resulting in a final mean of 4.0625. The conversation highlights that while both methods yield numerical results, only the traditional method is recognized as the correct approach for calculating the mean.

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  • Understanding of basic arithmetic operations (addition, division)
  • Familiarity with the concept of mean in statistics
  • Knowledge of weighted averages and their applications
  • Ability to follow sequential calculations and their implications
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  • Research the concept of weighted averages and their significance in statistical analysis
  • Explore different methods of calculating means in various statistical contexts
  • Learn about the implications of using pairwise calculations in data analysis
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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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