# Propagation of asymmetric error bars [duplicate]

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I have two (fully independent) measurements of the same quantity X. Each of them reports a measurement $$X_{\sigma_L}^{\sigma_R}$$, where $$\sigma_L$$ and $$\sigma_R$$ are the left and right uncertainties (asymmetric error bars).

In other words, if we call the measurements $$A$$ and $$B$$, and the subscripts $$A$$ and $$B$$ stand for the measurements, we have

$$A_{\sigma_{L,A}}^{\sigma_{R,A}}$$

$$B_{\sigma_{L,B}}^{\sigma_{R,B}}$$

Now, I need to calculate the difference between those measurements, $$\Delta = A-B$$. What will be $$\sigma_{L,\Delta}$$, $$\sigma_{R,\Delta}$$?

In other words, how do I propagate independent asymmetric error bars?

## marked as duplicate by Ben Crowell, Buzz, Kyle Kanos, Aaron Stevens, John RennieDec 26 '18 at 6:45

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## 1 Answer

Well, I think as long as you find a good argument for the solution you apply, you could do a couple of things. I think the easiest, which rather overestimates the errors is to use Gaussian error propagation with:

Be $$s_\Delta$$ the error of your difference, i.e. $$s_\Delta=\sqrt{\sigma_A+\sigma_B}.$$

The asymmetric uncertainty would be

$$s_{\Delta,L}=\left\{\begin{array}{lr}\sqrt{\sigma_{L,A}+\sigma_{R,B}},&&&\sigma_{L,A}>\sigma_{L,B}\\ \sqrt{\sigma_{R,A}+\sigma_{L,B}},&&&\sigma_{L,B}>\sigma_{L,A} \end{array}\right.$$

$$s_{\Delta,R}=\left\{\begin{array}{lr}\sqrt{\sigma_{R,A}+\sigma_{L,B}},&&&\sigma_{R,A}>\sigma_{R,B}\\ \sqrt{\sigma_{L,A}+\sigma_{R,B}},&&&\sigma_{R,B}>\sigma_{R,A} \end{array}\right.$$

So you find the maximum span between them. Other solutions are likely, possible, and feasible.