I am working through Foundation Maths seventh edition. I got stuck while trying to proof the following statement:
$∑(x - x̄)^2 = ∑(x^2) - n·x̄^2$
Is someone able to show what steps are needed to go from the absolute difference to $∑(x^2) - n·x̄^2$
Thank you!
$$\sum(x-\mu)^2=\sum x^2+\sum\mu^2-\sum2\mu x=\sum x^2+n\mu^2-2\mu\sum x$$
Recall now that:
$$\mu=\frac{1}{n}\sum x,\quad n\mu=\sum x$$
Which implies the variance is:
$$\sum x^2+n\mu^2-2\mu(n\mu)=\sum x^2-n\mu^2$$