What to Do When it's Not a Bell Curve

Description :

Statistical process control (SPC), process performance indices (Ppk), and acceptance sampling by variables (ANSI/ASQ Z1.9, or formerly MIL-STD 414) rely on the assumption that the process data follow the normal or bell curve distribution. This distribution is far more common in textbooks than in the real world. Generally accepted and off the shelf methods are however available for SPC and process performance analysis for non-normal distributions.

While we are not aware of formal sampling plans for non-normal distributions, it is however possible to get a confidence limit for the nonconforming fraction in a homogeneous production lot. As an example, we can be 95% confident that the nonconforming fraction is no greater than a certain percent depending on the sample data and sample size.

Objective :

Attendees will learn the generally accepted, and off the shelf, methods for using SPC and carrying out process performance studies for non-normal distributions, and also confidence limits for the nonconforming fraction in a homogenous production lot. The latter is a potential alternative for non-normal distributions where ANSI/ASQ Z1.9 cannot be used.

The traditional Shewhart control chart has a 0.00135 (0.135%) false alarm risk at each control limit, but this assumes that the data follow a normal distribution. When the distribution is non-normal, the risk may be considerably higher, especially when there is a one-sided specification for the quality characteristic in question. This is especially true when there is a physical limit, such as the fact that it is impossible to get less than zero impurities, particles, trace metal concentrations, and so on. If the bell curve extends below zero, it models a physically impossible situation—and the population fraction in question will instead be toward the upper control limit. If the actual distribution is known, however, control charts can be set up that have the traditional false alarm risk.

If SPC uses sample averages, the Central Limit Theorem will mitigate against the effects of non-normality, but individual measurements rather than averages are in or out of specification. This means we must identify the correct underlying distribution to perform a valid process performance study. If we assume wrongly that the distribution is normal, the estimated nonconforming fraction can be off by orders of magnitude; a purported Six Sigma process might not even be capable. Again, however, generally accepted and off the shelf methods are available to handle these situations.

ANSI/ASQ Z1.9 relies explicitly on the assumption that the measurements in the production lot follow the normal distribution. If they do not, the producer and customer cannot rely on the standard to assure the specified level of quality. It is however possible to calculate a 90%, 95%, or whatever is desired confidence limit for the nonconforming fraction based on (1) the sample data and (2) the sample size if we know the underlying statistical distribution.

Areas Covered in the Session :

Statistical Process Control
- SPC charts for non-normal distributions can be set up by calculating the 0.00135 and 0.99865 quantiles of the distribution in question, and setting the control limits accordingly. The false alarm risks will then be identical to those of a traditional Shewhart control chart.
- Traditional Shewhart charts might be usable for sample averages due to the Central Limit Theorem, which says averages for large enough sample sizes will follow a roughly normal distribution regardless of the underlying distribution.
Process Performance Studies
- The traditional calculations for capability and performance indices rely on the assumption that the process follows a normal or bell curve distribution. Application of the same mathematical formulas to non-normal data will deliver grossly inaccurate results.
- Off the shelf methods are however available with which to calculate meaningful performance indices for these processes. If for example a non-normal process has an estimated nonconforming fraction of 0.00135 beyond its upper specification limit, its PPU equals 1 (the same as for a normally distributed process with the same nonconforming fraction).
  1. This approach is described in the Automotive Industry Action Group's SPC manual.
  2. Minitab and StatGraphics perform the necessary calculations automatically.
Acceptance Sampling
ANSI/ASQ Z1.9 (formerly MIL-STD 414) relies on the assumption that the quality characteristic have a normal distribution. We cannot use this standard when this requirement is not met.
Not much literature is available for similar sampling plans for non-normal distributions. It is however possible to calculate an upper confidence limit for the nonconforming fraction in a homogenous lot that depends on (1) the underlying distribution, (2) the sample data, and (3) the sample size. Commercially available packages like StatGraphics and Minitab will do this automatically.
1. The traditional application is reliability, where we want to be (for example) 95% sure that 99% or more of the population will survive more than a certain number of hours. The same methods can however be used to state (for example) that we are 95% sure that 1% or less of the population will be outside a specification limit.

Who Will Benefit:

Manufacturing and quality engineers and technicians

About Speaker:

William A. Levinson, P.E., FASQ, CFPIM, is the owner of Levinson Productivity Systems PC. He holds professional certifications from the American Society for Quality, APICS, and Society of Manufacturing Engineers.

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