# The relation between least-square estimation in two seemingly related problems

I am presenting here a very simplified problem, because I have a feeling that if I'll understand this I will be on my way to understand my original problem. I tried googeling this and also searched this site but couldn't find the answer I needed. Hopefully you'll be able to point me to the right direction.

I am solving two seemingly related problems using least square minimization given a set of random measurments $\{x_i\}$.

The first problem is finding $b$ to minimize $$\sum_i^n (x_i - b)^2$$

The second problem is finding $c$ to minimize $$\sum_i^n (\frac{x_i}{c} - 1)^2$$

The solution is obviously the average of $x_i$ for $b$ $$b = \frac{\sum_i^n x_i}{n}$$ while for $c$ it is $$c = \frac{\sum_i^n x_i^2}{\sum_i^n x_i}$$

Naively I would expect $c$ to be the average as well. My logic: I expect $c$ to answer the question "which number, when dividing each number in the set, make the average of the resulting set closest to $1$". Obviously I'm wrong and that's not the way least-square works. Why is $c$ not the average? What does $c$ mean?

• Well, $\sum_i^n (x_i - b)^2 = b^2\sum_i(x_i/b - 1)^2 \neq \sum_i(x_i/b - 1)^2$, so in each of these problems, you're minimizing totally different losses. I don't see a relation to these two problems. Can I ask why you're trying to find $c$? Jan 19, 2018 at 2:43
• Basically you are beeing mislead by your intuition which seemingly ignores the ^2 Jan 19, 2018 at 9:45