# Thread: why is the average of a set of numbers alway smaller than the largest number in the s

1. ## why is the average of a set of numbers alway smaller than the largest number in the s

why is the average of a set of numbers alway smaller than the largest number in the set?

2. Not true, take the set 5,5,5. The average is 5, which is not greater than the largest element 5.

3. Originally Posted by m3clubracer
why is the average of a set of numbers always smaller than the largest number in the set?
I assume smaller is in the weak sense rather than in a strict sense.

Let us do it with three elements so you can see the idea. Say $\{a_1\leq a_2\leq a_3\}$.

The average $a_1=\tfrac{a_1+a_1+a_1}{3}\leq \tfrac{a_1+a_2+a_3}{3} \leq \tfrac{a_3+a_3+a_3}{3} = a_3$