Captain Dallas—propagation of error isn’t related to standard deviation v. standard error though, it is how you translate uncertainty through a function of what you’re measuring.
“for the ocean surface temperature range you could have a maximum variation with zero change in energy of about 1.5 C”
It’s not clear what we’re talking about now, since you say “change in energy”. Your 1.5˚C refers to variance, I think I understand that, perhaps of the same measurement that Kennedy et al. include; but “change” makes me think you’re also talking about trends. Do you merely mean the difference between the ship and buoy measurements when you say “change”? If so, I agree: we can have large variances compared to the mean value. But the confidence in the mean is not equal to that variance, you have to take into account sample size, which decreases the uncertainty range.
“What is likely is a fraction of that, but definitely greater than 0.02 C degrees.”
So, what equation should we use? Again, we’re not converting any units, still just talking about temperatures and mean temperatures (and temperature differences), so propagation of error does not come into play here. Do you think the equation I gave, √(var(x) / (n-1)), is incorrect here? I guess it seems that’s what you’re saying, but these other matters don’t play in here; so, why so?
“Now is the objective to create a thermodynamics relevant metric…”
So it seems you object to using either heat content (so, Joules) or temperature (I’m not sure which). Either can be useful, but it depends on what you want to know. For the surface, which is for all intents and purposes supposed to be 2D, I don’t think it’s possible to calculate heat storage because we don’t have an actual mass to use heat capacity calculations on.
We might just ask how much heat the first meter of ocean water is storing, and just assume that the temperature variation at the surface is the same through the first meter, in which case we could do that. However, since the equation is linear to calculate heat capacity change, ∆Q = mc∆T (where m is the mass of the 1-meter layer, c is the specific heat capacity of sea water, ∆T the change in temperature with time, and ∆Q the change in heat stored; and m and c remain constant), then the error propagation will be linear too. So you wouldn’t see it blow up like you might if you were using non-linear (i.e. exponential, higher power polynomial) equations.