# Thread: (Simple) proof for the arithmetic mean?

1. ## (Simple) proof for the arithmetic mean?

Hello. I am having trouble trying to figure out a way to answer this homework exercise:

"Prove that the arithmetic mean cannot be greater than the largest element in the set of numbers from which it was calculated".

Here's how I began:

Let X={x1, x2, x3,....xn} be a set of numbers out of which the arithmetic mean is x-bar.

Suppose that x-bar is greater than some xi which is the greatest element of the set X.

Then my intuition is that, maybe, I can use the fact that {(x1-xbar)+(x2-xbar)+(x3-xbar)+...+(xi-xbar)+...._(xn-xbar)} *should* add up to 0 but in this case it will not. But I'm not sure how to proceed.

I know the fact that the arithmetic mean cannot be greater (or smaller) than the greatest(or smallest) element of the set is a well-known and obvious fact, which is why I am frustrated that I cannot show this more explicitly.... particularly because it is so obvious!

Any help is appreciated!

2. ## Re: (Simple) proof for the arithmetic mean?

Denote the greatest element of the set you are considering so that . Then,

Can you continue?

3. ## Re: (Simple) proof for the arithmetic mean?

Well, if I re-express (and assume all x's are ordered from x1 the smallest to xM the greatest, following your notation):

It becomes immediatley obvious that:

Which implies that any other (individual) element of is less than

But I cannot quite see how their sum must be less than , even though I know it is true

Should be using properties of finite sums to find this out?

4. ## Re: (Simple) proof for the arithmetic mean?

Ok, we'll do it another way. Assume you have ordered your set so that :

Then you can write :

...

What happens if you sum all these inequalities?

5. ## The Following User Says Thank You to PlpPlp For This Useful Post:

will22 (02-05-2014)

6. ## Re: (Simple) proof for the arithmetic mean?

By definition you have the following n bounds / inequalities hold simultaneously:

Therefore you can apply these into the arithmetic mean which immediately yields the desired result.

7. ## The Following User Says Thank You to BGM For This Useful Post:

will22 (02-05-2014)

8. ## Re: (Simple) proof for the arithmetic mean?

OH GOD! THANK YOU BOTH!

I think I got it now. I will make it all explicity just to see if there are any problems on my reasoning.

As you two pointed out, the following constraints hold:

Which means that:

So I can do:

Thank you! I knew it had to be easy!

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