# equality between 2 operations using mean [closed]

why when we do the following operation mean (10 ,20,30)= 20

is different from (mean(10,20)+30)/2=22,5

there are no equality between the 2 operations using mean ? why?

thanks a lot in advance for any help

Algebraically speaking, these are two different operations

In the first scenario you have

$$\frac{10 + 20 + 30}{3} = 20 \tag{1}$$

In the second case you have

$$\frac{(10 + 20) / 2 + 30}{2} = 22.5 \tag{2}$$

But if this is not convincing enough, consider a different set of numbers, just to make things easier take $$\{-3, -2, -1, 0, 1, 2, 3\}$$. Imagine a bar of length $$6$$, and you put a mass separated by a unit of length:

 *===*===*===*===*===*===*
-3  -2  -1   0   1   2   3


The question is, where do you need to place your finger such that bar is in equilibrium? Answer is $$0$$, which actually is the average of the points. Now let's group the first three points and replace it with the average, and the other four and replace them with the average

 ====*=============*======
-2       0    1.5


you immediately see that if you place your finger at $$0$$ the bar will topple to the left. So you changed the problem just by averaging in groups. You can return to the original situation by adding different weights to each point