We must understand the sigma of a process is merely an equivalent unilateral figure-of-merit. As such, it is always regarded (and reported) as a short-term measure of capability. The metric can be directly computed when continuous data is at hand, or it can be statistically synthesized from discrete data. In the latter context, a reported sigma is only analogous to the Z statistic. In other instances, the sigma metric can be made to represent the normalized capability of several critical-to-quality characteristics (CTQs). To better facilitate our understanding, consider the following example.
For the first scenario, we will postulate a single random normal CTQ with a symmetrical bilateral performance specification. Respectively, we compute Z.usl.st = (USL T) / S.st = +3.5 and Z.lsl.st = (LSL T ) / S.st = -3.5. Of course, each Z must be transformed into a tail-area probability (by way of a table or Excel macros). Doing so reveals that p(Z.usl.st) = 1 – normsdist(Z.usl.st) = 1 – normsdist(+3.5) = 1 – .99976733 = .00023267. On the other end, we compute p(Z.lsl.st) = normsdist(Z.lsl.st) = normsdist(-3.5) =.00023267.
Combining both probabilities exposes the total probability of defect. Doing so reveals 00023267 + .00023267 = .00046535. Given this, the short-term first-time yield was presented as Y.ft.st = 1 – .00046535 = .99953465. With this as a backdrop, the equivalent unilateral capability was given as Z.bench = Z.st = normsinv(Y.ft.st) = normsinv(.99953465) = 3.31071537. In rounded form, we accept the quantity 3.31 Sigma. Consequently, the equivalent long-term unilateral capability can be approximated as Z.lt = Z.st 1.50 = 3.31 1.50 = 1.81. Translated, this is equivalent to DPMO = 35,148.
Let us now postulate a second application scenario. Suppose we examined a certain discrete CTQ and discovered the long-term defect rate to be DPMO = 4,200. Converting this quality metric to an equivalent long-term first-time yield discloses the approximation Y.ft.lt = 1- (DPMO / 10^6) = 1- (4,200 / 10^6) = 1 – .00420 = .99580. Of course, such a long-term first-time yield can be statistically converted to an equivalent long-term unilateral standard normal deviate. So doing provides the quantity Z.lt = normsinv(1 Y.ft.lt) = normsinv(.99580) = 2.63626982, or 2.64 in rounded form. Since no short-term information or data were available, the instantaneous (short-term) capability was approximated as Z.bench = Z.st = Z.lt + 1.5 = 2.64 + 1.50 = 4.14. Thus, we assert the process has an inherent capability of 4.14 Sigma.
Pooling the two scenarios, we must convert the respective sigmas into first-time yield values. For the first scenario, we have Y.ft.st.1 = .99953465. For the second scenario, we observe Y.ft.st.2 = .99998234. The short-term rolled-throughput yield would then be computed as Y.rt.st = Y.ft.st.1 * Y.ft.st.2 = .99953465 * .99998234 = .99951700. The normalized yield would be given as Y.norm = Y.rt.st^(1/K) = .99951700^(1/2) = .99975847.
Converting Y.norm to an equivalent unilateral short-term Z value reveals Z.norm.st = 3.49, meaning that the overall sigma of the process should be reported as 3.49. The normalized long-term unilateral capability was approximated as Z.norm.lt = Z.norm.st 1.50 = 3.49 1.50 = 1.99. Interestingly, such a level of capability translates to a quality level of DPMO = 23,293. Thus, we are able to merge heterogeneous metrics for the purpose of quality reporting.