Questions about Capability Statistics - Part 2
Another thing you may notice when performing a Capability Analysis in Minitab is the option to change whether you want to see capability statistics (Cp, Pp…) or Z.Bench. From Minitab Help:
“A disadvantage of the Cpk index is that it only represents one side of the process curve, and tells you nothing about the other extreme. For example, the two graphs below display processes with identical Cpk values. However, one violates both specification limits, and the other only violates the lower specification limit...”
The Z in Z.Bench refers to the standard normal distribution with mean 0 and standard deviation 1. This statistic is considered a "benchmark" because it is a standard by which the process can be measured (i.e., the statistic is a report of the sigma capability of the process). It’s an attractive feature because it also takes into account information on both sides of the curve, which, as mentioned above, Cpk/Ppk does not.
To delve into these calculations further, Z.Bench is the z-value you get from a standard normal distribution if you place the probability of defects entirely on the right tail. So even though P(-3 < Z < 3) = 0.9973, it is also true that P(Z < 2.78) = 0.9973, and thus the last probability defines the sigma level (Z.Bench), not the first one. You can calculate these probabilities under Graph > Probability Distribution Plot.
There is another connection to be made here. Let’s say your Z.Bench is 4.5. You can then determine that P(Z > 4.5) = 0.0000034. It also happens to be one step closer to representing a value in units PPM, if we multiply it by a million. Thus, a Z-bench equaling 4.5 is equivalent to 3.4 PPM defective. It is up to you whether you prefer to use Z.Bench or PPM since they will always agree with each other. Whether you select to see Cpk or Z.Bench, you’ll always see the PPM tables below your Capability Analysis histogram. Here are more examples:
Z.Bench = 3 corresponds to 1300+ PPM defective, which corresponds to a 4.5 sigma process
Z.Bench = 4 corresponds to 30+ PPM defective, which corresponds to a 5.5 sigma process
Z.Bench = 4.5 corresponds to about 3.4 PPM defective, which corresponds to a 6-sigma process
How does a Z.Bench of 4.5 equal a 6-sigma process? Well, this is what we call a Z-shift. This 1.5 sigma shift is an empirical observation made at Motorola and based on some of the initial studies they made about their processes. The 1.5 sigma shift can be used to calculate the long-term capability of a process when only few observations have been collected by using the relation: Zbench (Long) = Zbench (Short) - 1.5. Minitab's Normal Capability Analysis (under Quality Tools) does not consider this equation, and in fact uses sigma-overall and sigma-within to calculate each of these two statistics independently.