Denote the greatest element of the set you are considering so that . Then,
Can you continue?
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!
Denote the greatest element of the set you are considering so that . Then,
Can you continue?
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?
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?
will22 (02-05-2014)
will22 (02-05-2014)
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