Calculating Bias from bivariate data 
My data consists of one column of real temperature, and one column of calculated temperatures. I want a single number which quantifies the 'bias' in the real temperatures.
Basically, if most of my data points lie below the ref line, then I should expect a negative bias, and if they lie above the line then I will expect a positive bias. And this bias quantified by the spread in the scatter plot. What is the best statistic to quantify this?
Any help would be greatly appreciated.
Edits:
The objective of this project is to obtain an estimate of bias in summer temperatures due  to unequal land distribution in the Northern and Southern Hemispheres. The Northern Hemisphere has about 68% of the total landmass on earth, while the Southern hemisphere has less than half of the northern hemisphere land (~32%).
The data for temperatures here is obtained from GCM simulations. Temp_Real is obtained from a simulation with real earth, and Temp_symmetric is obtained from a GCM simulation using a symmetric landmass earth model. By comparing the climates from Real Earth and a Symmetric Earth, I want to calculate a bias caused by the asymmetry in the real earth.
This scatter plot is for Northern Hemisphere.
 A: First of all, I think you have the positive and negative bias areas reversed. If Temp_Real is the true value and Temp_Symmetric is an estimate/calculation/test measurement, then you have positive bias when Temp_Symmetric is greater than Temp_Real, which corresponds to points below the diagonal.
If you need a single number, I would make the assumption that the bias does not depend on the true temperature. Then we would simply calculate
$$
\hat{b}=\frac{1}{n}\sum_{i=1}^{n}(S_i-R_i),
$$
where $n$ is the sample size, $S_i$ is the symmetric temperature, and $R_i$ is the real temperature for the $i^{th}$ observation. This gives an estimate of the expected value of the symmetric temperature minus the real temperature (bias).
However, this one-number summary may not be adequate because the bias of $S$ may depend on $R$. For example, one might interpret your scatterplot to show a clear positive bias for low real temperatures (~305) and a smaller bias as the real temperature increases (~307). There's not really strong evidence for this, but it can serve as an example. We might even see a negative bias if we got to real temperatures near 310!
If the bias is dependent on the real temperature, then I would perform a regression of Temp_Symmetric on Temp_Real, and then subtract the line $S_i=0+1\cdot R_i$ (the diagonal) from the fit. This will give you bias as a function of real temperature. The earlier one-number summary is the special case of this method where the true relationship is $S_i=b+1\cdot R_i$.
You can get a good idea of whether or not the bias depends on the true value by plotting $S_i-R_i$.
