tolerance dimensions arithmetic or how to add or subtract values with probability distributions?

Question

This is a follow up to this question. Imagin two physical objects A and B:

with nominal heights of $l_A$ and $l_B$. These parts are being mass-produced so their heights have a probability distribution (e.g., Natural/Gaussian distributions), with standard deviations of $\sigma_A$ and $\sigma_B$. Now I want to know what are the expected probability distributions if the blow measurements:

In other words, what are the standard deviations of the values $l_1$ and $l_2$ from their nominal values $l_A + l_B$ and $l_A - l_B$? Is it correct to assume $\sigma_1 = \sigma_2 = \sigma_A + \sigma_B$?

P.S. Dear moderators, I have no clue if I have used the right terms or if I have chosen the right tags here. Your help in that regard would be highly appreciated.

Answer 1

I was slightly wrong. The correct value for the standard deviations is:

$$\sigma_1 = \sigma_2 = \sqrt{\sigma_A^2 + \sigma_B^2}$$

• Reference

Answer 2

Just be sure to distinguish between tolerances (absolute max and min values) - which would simply add and subtract - and the measured and calculated standard deviation for a set of data - which would mean 99.7% of data falls with 3x the std deviation (give a normal distribution).

I.e. 5 ± 1 means min 4 and max.6. 5 with σ = 1 implies the mean value is 5 and most of the data would sit between 2 and 8 given a normal distribution.