I have a question in mind and would be thankful for any help:
Consider some values:
$a_1, a_2, \cdots, a_n$. Assume that these values are the true, correct values.
Now consider two other sets of values:
$b_1, b_2, \cdots, b_n$ and $c_1, c_2, \cdots, c_n$. The $b$ values are determined experimentally with one method, and the $c$ values are determined experimentally with a different method. Both attempt to be as close as possible to the $a$ values.
If I want to find out which method produces the closest values to $a$ overall, is it mathematically sound to graph $b$ vs $a$, and separately $c$ vs $a$, and then compare the slopes? Is it fair to say that whichever graph ($b$ or $c$) has a slope closer to 1 is the more accurate method?
If so, how should I compare the slopes? (E.g., by finding the ratio between them?)
If not, how should I compare the $b$ and $c$ values to the $a$ values in order to determine which method is most accurate?
I think the most common way to deal with your problem is to see $a,b,c$ as elements on $\mathbb R ^n$ and check if $b$ or $c$ is the closest element to $a$. In a general setting (i.e. if you replace $\mathbb R^n$ by an arbitrary set) this problem is very difficult because you have to tell what it means to be far or close. A way to do so is to introduce a distance (or a metric) on your set. That is, $d(x,y)$ tells you how far is $x$ from $y$.
Here in $\mathbb R^n$ we have norms, which produce nice distances via $$ d(x,y) = ||x-y|| $$ and we know they are all equivalent. That is, if you pick a norm $||.||$ on $\mathbb R^n$, compute $||a-b||$ and $||a-c||$ and select the smallest result, your conclusion wont depend on the choice of the norm. So the answer to your first question would be : choose your favourite norm.