3. Production Process Characterization
3.1. Introduction to Production Process Characterization
3.1.3. Terminology/Concepts

## Propagating Error

The variation we see can come from many sources When we estimate the variance at a particular process step, this variance is typically not just a result of the current step, but rather is an accumulation of variation from previous steps and from measurement error. Therefore, an important question that we need to answer in PPC is how the variation from the different sources accumulates. This will allow us to partition the total variation and assign the parts to the various sources. Then we can attack the sources that contribute the most.
How do I partition the error? Usually we can model the contribution of the various sources of error to the total error through a simple linear relationship. If we have a simple linear relationship between two variables, say,

$$y = \mu + \alpha y_{1} + \beta y_{2} + \epsilon$$

then the variance associated with, y, is given by

$$\mbox{Var}(y) = \alpha^{2}\mbox{Var}(y_{1}) + \beta^{2}\mbox{Var}(y_{2}) + 2 \alpha \beta \mbox{Cov}(y_{1},y_{2})$$

If the variables are not correlated, then there is no covariance and the last term in the above equation drops off.  A good example of this is the case in which we have both process error and measurement error. Since these are usually independent of each other, the total observed variance is just the sum of the variances for process and measurement. Remember to never add standard deviations, we must add variances.

How do I calculate the individual components? Of course, we rarely have the individual components of variation and wish to know the total variation. Usually, we have an estimate of the overall variance and wish to break that variance down into its individual components. This is known as components of variance estimation and is dealt with in detail in the  analysis of variance page later in this chapter.