The wierd concept behind taking the AVERAGE of a set of number

Question

good morning,

when we take the average of an odd set of numbers that have the same rate of change, for ex ( {1+2+3+4+5}/5 ), how and why does it ALWAYS work that the number in the middle is the Average? without even adding then dividing by 5, i can be sure that "3" the middle number WILL be the average ?

and why when the set is even we add the 2 middle numbers then divide by 2: why specifically the 2 middle numbers ?

thanks in advance for your time and effort

Answer 1

Hi Reem,

To see why this works, we can pair up the numbers in the set in a clever way. Imagine folding your list in half and pairing up the numbers that line up. If there's an odd number of them, the middle one is just on its own.

So in your list (1, 2, 3, 4, 5) we get the pairs (1,5) and (2,4) and (3) on its own. Now we can see that the average of each of these groups is the same! (1+5)/2 = (2+4)/2 = 3.

Similarly, if we have an even number of elements like (3, 6, 9, 12). We get the pairs (3, 12) and (6, 9). Both of these will have the same average 15/2.

Here's another way to think about it.

Imagine we just start with the list (3, 3, 3, 3, 3) so obviously the average is 3. Now pick two of the numbers, add 1 to one of them and subtract 1 off the other. The total stays the same, so the average must be the same. We get (2, 4, 3, 3, 3). Now pick two more numbers and add and subtract off 2, so we get (2, 4, 1, 5, 3) and the average must still be the same.

I hope this answers your question.