Sum of Deviations: Proving $\sum_{i=1}^Nv_i(v_i - \langle v \rangle) = 0$

[kubaanglin]

Homework Statement

The average value of N measurements of a quantity $v_i$ is defined as
$$ \langle v \rangle \equiv \frac {1}{N} \sum_{i=1}^Nv_i = \frac {1}{N}(v_1 + v_2 + \cdots v_N)$$
The deviation of any given measurement $v_i$ from the average is of course $(v_i - \langle v \rangle)$. Show mathematically that the sum of all the deviations is zero; i.e. show that
$$\sum_{i=1}^Nv_i(v_i - \langle v \rangle) = 0$$

Homework Equations

$?$

The Attempt at a Solution

I understand that this is simply describing an average, but I am not sure how to express this mathematically. It makes sense to me that the sum of the deviations would be zero.

 

[Mark44]

Homework Statement

The average value of N measurements of a quantity $v_i$ is defined as
$$ \langle v \rangle \equiv \frac {1}{N} \sum_{i=1}^Nv_i = \frac {1}{N}(v_1 + v_2 + \cdots v_N)$$
The deviation of any given measurement $v_i$ from the average is of course $(v_i - \langle v \rangle)$. Show mathematically that the sum of all the deviations is zero; i.e. show that
$$\sum_{i=1}^Nv_i(v_i - \langle v \rangle) = 0$$

Your formula above is incorrect, as it has an extra $v_i$.
The sum of the deviations is
$$\sum_{i = 1}^N (v_i - \bar{v})$$
Here $\bar{v}$ is the mean of the measurements $v_i$.

Homework Equations

$?$

The Attempt at a Solution

I understand that this is simply describing an average, but I am not sure how to express this mathematically. It makes sense to me that the sum of the deviations would be zero.

Simply expand the summation.

 

[kubaanglin]

$$((v_1 - \langle v \rangle) + (v_2 - \langle v \rangle) + \cdots (v_N - \langle v \rangle))$$
$$((v_1 - \frac {v_1 + v_2 + \cdots v_N}{N}) + (v_2 - \frac {v_1 + v_2 + \cdots v_N}{N}) + \cdots (v_N - \frac {v_1 + v_2 + \cdots v_N}{N}))$$
Is this sufficient to "show mathematically" that the sum of all the deviations is zero? I am just not sure what I am being asked to do.

 

[Ray Vickson]

$$((v_1 - \langle v \rangle) + (v_2 - \langle v \rangle) + \cdots (v_N - \langle v \rangle))$$
$$((v_1 - \frac {v_1 + v_2 + \cdots v_N}{N}) + (v_2 - \frac {v_1 + v_2 + \cdots v_N}{N}) + \cdots (v_N - \frac {v_1 + v_2 + \cdots v_N}{N}))$$
Is this sufficient to "show mathematically" that the sum of all the deviations is zero? I am just not sure what I am being asked to do.

You are being asked to show that the summation you wrote above evaluates to $0$ for any possible inputs $v_1, v_2, \ldots, v_N$.