I am doing a convergence analysis in on a finite element model. The converged result is the result when the element size is zero which is also when there are an infinite number of elements in the model. Therefore, there are two ways to identify the converged result:

  1. Linear extrapolation of the result vs. element size data.
  2. Exponential extrapolation of the result vs. number of elements data.

Both methods produce an estimate of the converged value with a corresponding uncertainty (i.e. the standard error of regression). My initial thought was that I should use the result with the smallest uncertainty as the actual result. However, I just realized that if I were to find a means of combining the uncertainty, then the combined uncertainty would be smaller. For example, the image below shows a red probability distribution overlapping a blue probability distribution. The combined probability distribution is shown in purple.

Overlapping Probability Distributions

Now, my question is, knowing the statistical means and the standard deviations defining the red and blue curves, how do I go about calculating the statistical mean and standard deviation for the purple curve?

I know the equation for the purple line is obtained by multiplying the equations for the red and blue lines together, but that doesn't tell me what the statistical mean and standard deviation is that defines the purple curve. At the very least, it's pretty clear that multiplying the standard deviation of the two results together would give vastly different results depending on the units being used (i.e. standard deviations of +/-3 mm and +/-2 mm would give vastly different results than standard deviations of +/-0.3 cm and +/-0.2 cm).

Ultimately, I'm not extremely experienced with statistics, so I may have made mistakes in my description above, but basically, I'm trying to leverage to two separate means of extrapolation at my disposal to reduce my uncertainty.

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    $\begingroup$ Not sure if I understood the problem. So you're trying to multiply two arbitrary, single-peaked continuous distributions and now you want to find the mean and standard deviation represented by the resulting product? $\endgroup$ – Andreas Steimer Apr 27 at 21:14
  • $\begingroup$ I have two means of predicting a single result. I assume that actual result is in the region where the two predictions overlap. Therefore, I want to calculate the what the probability is associated with the overlapping region. $\endgroup$ – tlewis3348 Apr 27 at 21:38
  • $\begingroup$ to put it another way, what does the purple curve represent? $\endgroup$ – Andreas Steimer Apr 27 at 22:07
  • $\begingroup$ The probability distribution where both the red and blue probability distributions overlap. I think I solved my problem, so I'll post an answer. Hopefully that will clear things up a little. $\endgroup$ – tlewis3348 Apr 28 at 14:08

I think I was overthinking this. For purposes of an example, below are some numbers similar to what I might see as results from the two different methods:

Linear Extrapolation: 26.4+/-0.4 ksi
Exponential Extrapolation: 26.8+/-0.6 ksi

The maximum value from the linear extrapolation is 26.4 + 0.4 = 26.8 ksi. The minimum value from the exponential extrapolation is 26.8 - 0.6 = 26.2 ksi.

Therefore, the estimated result is (26.8 + 26.2) / 2 = 26.5 ksi. The associated uncertainty is (26.8 - 26.2) / 2 = 0.3 ksi.

Therefore, the reported result is 26.5+/-0.3 ksi.

This can be generalized by saying the reported result is the average of the maximum end of the range for the minimum result and the minimum end of the range for the maximum result. The associated uncertainty is half the difference of these two values.

Obviously, the assumption is that both predictions are sufficiently close together that their ranges overlap. Someone more experienced with statistics may be able to explain cases where this assumption is invalid. Additionally, they may be able to explain what I said in more technical terms. In any case, any input/feedback would be welcome.


I just did some more research on this. The plot below shows the curves from the original post with vertical lines defining the +/-1 standard deviation. The location of the vertical lines for the purple curve was calculated by first determining the percentage below the peak the +/-1 standard deviation results were for the red and blue curves. The same relative location on the purple curve is where the vertical purple lines are located. As you can see, the results are not exactly the same as what I described above. The actual result from this calculation is 26.52+/-0.33 ksi. However, when the significant digits are reduced such that there is only one digit of significance in the uncertainty, the result is the same as what I described previously.

Ultimately, if you need a quick approximation, the method provided initially should be pretty close, but if you're concerned about being precisely correct, use the method described in my edit above.

intersecting bell curves

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