Conditions for Gaussian long term manufacturing processes

The manufacturing community defines capability indices for manufacturing processes applicable time-wise for both long term and short term processes. The long term process distribution can be constructed from the consolidation of the data sets that was used to estimate the multiple short term distributions. There is a tendency for Gaussian distributed short-term processes to exhibit time sensitive random variation in the mean, while the short-term standard deviation remains unchanged. A mathematical model for construction of the long-term distribution from consolidation of short-term histograms will be discussed. Also, it can be shown the long term process will be Gaussian distributed if the mean for the short term processes is a random variable which is distributed Gaussian. A simple rule for predicting the long term (composite) variance is derived.