The following is an approximate propagation of uncertainty through Dessler et als. equation for estimating the strength of water vapor feedback λ. We have been looking at the error-bars in his recent paper Water-vapor climate feedback inferred from climate fluctuations, 2003-2008, not calculated in the published paper. Assumptions made are noted. Refer to wiki for propagation of error equations.

Here R is the top of atmosphere IR, q is the specific humidity and T is the temperature.

1. $$\lambda = \Sigma \frac{\partial R}{\partial q}\frac{\Delta q}{\Delta T} =K\frac{\Delta q}{\Delta T} $$

Rolling up the summation over the earths surface into K.

2. $$\lambda = K\frac{q_1-q_0 \pm \sqrt{2}\sigma_q}{T_1-T_0 \pm \sqrt{2}\sigma_T} $$

Substituting the values two endpoint years used in calculating the differences, and their uncertainties, using propagation of errors for differences, assuming independence.

3. $$(\frac{\sigma_\lambda}{\lambda})^2 = (\frac{\sqrt{2}\sigma_q}{\Delta q})^2 + (\frac{\sqrt{2}\sigma_T}{\Delta T})^2 $$

Substituting uncertainty of q and T into equation for propagation of errors through ratios, assuming independence.

4. $${\sigma_\lambda}^2 = 2{\sigma_q}^2 + 2{\sigma_T}^2 $$

Assuming λ, q and T are the same magnitude. This is an underestimate if λ=2.

5. $${\sigma_\lambda} = 2\sigma_{qT} $$

Assuming uncertainty of q and T are equal, and squaring.

So according to these rough calculations, the actual uncertainty in λ could be roughly twice the uncertainty observed in the Dressler figures. This increase is due to the use of a single year, 2008 as the reference point, for calculating the change in humidity and temperature relative to other years. The uncertainty in the arbitrary choice of this point increases the uncertainty when propagated through the calculations for water vapor feedback.

Our calculated standard deviation of the mean was 0.37 W/m²/K. The confidence limits of the mean are then 1.96*2*0.37 or 1.45, giving a lower limits to the estimated 2.04 W/m²/K value of vapor feedback of 0.59 W/m²/K.

If we substitute values into step 3 of λ=2, q=2, T=1 we get an even higher uncertainty reflecting the effect on the ratio of dividing by a smaller number.

6. $${\sigma_\lambda}^2 = 2{\sigma_q}^2 + 8{\sigma_T}^2 $$

7. $${\sigma_\lambda} = \sqrt{10}\sigma_{qT} $$

The confidence limits of the mean are then 1.96*3.16*0.37 or 2.29, giving a lower limits to the estimated 2.04 W/m²/K value of vapor feedback of -0.25 W/m²/K. Being less than zero, this indicates that zero feedback is within the limits of uncertainty. This is very similar to the CI obtained be a t-test of difference of means in the previous post.

Dessler rambles on about the large influence temperature has on the uncertainty of the feedback here.

[20] Figure 4 also helps explain the large year-to-year
variability in our calculated values of lq in Table 1. It is
tropical q that primarily determines the size of the water
vapor feedback, and tropical q is primarily regulated by the
tropical surface temperature [e.g., Minschwaner and
Dessler, 2004]. The definition of lq, however, uses
changes in global-average surface temperature. While
changes in global and tropical temperatures are related,
there are often variations in the global average that are
not reflected in the tropical average and vice versa. Such
variations lead to large variations in lq.

[21] Consider, for example, the small feedback lq
inferred between January 2007 and January 2008. The
difference in the global average surface temperature DTs
between these two months was 0.60 K. Much of this,
however, was due to extreme changes in the northern
hemisphere mid- and high latitudes. The tropical average
surface temperature difference DTtropics was a milder 0.37 K.
The relatively small change in tropical surface temperature
leads to a relatively small change in q, and therefore a
relatively small value of (@R/@q)Dq of 0.57 W/m2.
Dividing that by the large DTs leads to the small value
of 0.94 W/m2/K inferred for lq over that period.

[22] The months with the largest inferred values of lq, on
the other hand, are the months where DTs is smaller than
DTtropics. For example, DTs between January 2008 and
January of 2006 was 0.28 K, while DTtropics between these
months was 0.33 K. This arrangement contributes to a large
value for the inferred lq between these months. Given
enough data, such variations should average out. In a short
data set such as the one analyzed here, however, such
variations can be significant.

You got that right. It would seem that three almost equal contributions to overall uncertainty are as follows:

Total uncertainty = measurements + reference point + ratio amplification