It is very good that you explicitly state your goal, i.e. "I want ... to understand the influence sea surface temperatures (x-axis) have on land temperature over a particular region (y-axis)". Too often this aspect is ignored in these sorts of questions!
First, as always it is important to understand that correlation does not imply causation.
Now, the two approaches to line fitting differ statistically in that OLS treats $x$ as "error free", while TLS (a.k.a. "errors in variables" linear regression) treats uncertainty in both $x$ and $y$. (These are treated symmetrically in the case of orthogonal least squares.)
The two approaches also differ in their goals: Orthogonal least squares is similar to PCA, and is essentially fitting a multivariate Gaussian joint distribution $p[x,y]$ to the data (in the 2D case, at least). Ordinary least squares is more oriented to fitting a set of conditional Gaussian distributions $p[y \vert x]$ to the data.
Now, as your $x$ and $y$ variables have the same units (both are temperatures), and similar ranges, then orthogonal least squares is certainly reasonable. It is difficult to tell (given the large size, low transparency, and high density of over-printed points), but the TLS line appears to better capture the data as well.
A summary of the usefulness of the two approaches might be as follows:
- If your goal is to constrain the distribution of $y$ given a precise value of $x$, then the OLS curve is what you want. (For example $R^2$ gives the reduction in variance for $y|x$ vs. $y$).
- If your goal is to constrain the "independent components" of the 2D $(x,y)$ data, then TLS is better. For example this first principle component may have a common cause, in terms of the system dynamics.
Given your stated goal, it appears that the OLS line ($p[y|x])$ is what you are probably after.
However, note that OLS assumes that the residual variance is independent of $x$ (i.e. $\sigma^2_{y|x}\neq f[x]$), a condition known by the colorful term "homoskedastic". This is something you should check (e.g. by plotting residuals). As noted above, your plot is difficult to judge by eye, but it appears the ($y$) spread around the OLS line may have some variations in the $x$ direction. (So, as noted above, the TLS line may be a more reliable fit.)
Best Answer
Linear regression ends up being a lot more than this, but when you plot a “trend line” in Excel or do either of the methods you’ve mentioned, they’re all the same. The formula you give is a simple way of finding the regression equation that works in the particular case that you’re considering where there’s only one predictor variable. We use matrix algebra when the regression is more complicated with multiple predictors, and what you have is a special case of that full model.
I do not follow what you’ve written about a football, but the least squares regression line passes through $(\bar{x},\bar{y})$, yes.