Solving for variable in Summation problem

Question

$$\sum_{i=1}^n \frac{(x_i- \mu)}{\sigma^2}=0$$

I need to solve for $$\mu$$, and I am having trouble trying to justify the step where you go from $\sum_{i=1}^n (x_i- \mu)$ to $\sum_{i=1}^n (x_i- \mu n)$ to $\frac{1}{n}\sum_{i=1}^n (x_i- \mu)=0$

Answer 1

You've gotten things a little wrong, which is probably why you're confused. We can split the sum $\sum (x_i - \mu) = \sum x_i - \sum \mu$, and $\sum \mu$ is adding $n$ copies of the constant $\mu$ together, i.e. it's $\mu + \mu + \ldots + \mu = n\mu$.

So then we have $\sum (x_i - \mu) = \sum x_i - n \mu$ where the $n \mu$ is not inside the summation, and if the whole thing is equal to zero then you get $\sum x_i - n \mu = 0 \implies n \mu = \sum x_i \implies \mu = \frac{1}{n} \sum x_i$.