Confidence Intervals

Confidence Intervals for Capability Indices Using MINITAB By Keith M. Bower, M.S. Introduction  The concept of using a single summary statistic, or statistics, to assess the capability of a process is currently widely practiced. In addition, the use of confidence intervals based upon the distributions of appropriate capability statistics may be integrated. This paper addresses the use of a MINITAB™ macro to compute approximate confidence intervals for Cp, and Cpk by use of an example. The requirements of a stable process and approximate Normality are necessary for such calculations to be justifiable. Capability Indices  Frequently, manufacturers and others are required to estimate the capability of a process, i.e. they require a reasonable estimate of what proportion of their products will conform to certain specifications (or “specs”). Certain manufacturers look to the results provided by some wellknown capability indices, namely Cp and Cpk. With given spec limits, a process may be investigated to assess its capability. If the process appears to be stable over time and we may use the Normal distribution to usefully model the process (perhaps after transforming the original data) we may attempt to get a handle on the capability of the process using the aforementioned indices. For more information on performing capability analysis, refer to Montgomery1 (2000) and examples using MINITAB by Bower2 (2001). [Post-publication note from author – examples available via http://www.minitab.com/company/virtualpressroom/Articles/CapabilityPart1.pdf and http://www.minitab.com/company/virtualpressroom/Articles/CapabilityPart2.pdf ] Importantly, as was shown by Somerville and Montgomery3 (1996) if the assumption of Normality is violated, the actual capability estimates may be misleading. The simplest index to use is Cp, the formula of which is Cp = (USL – LSL)/6σ where USL = Upper Specification Level, LSL = Lower Specification Level and σ is the process withinsubgroup standard deviation. In practice, of course, we will never know the true value of σ, which is why we have to make use of the estimated within-subgroup standard deviation (e.g.

R ˆ . Similarly, we may also estimate the process mean µ by using the grand ) in C p d2 ˆ where Cpk = min  USL - , - LSL  . average, X for use in C pk 3  3  by

         

 Example Suppose we have a particular situation in which the process may be judged as in statistical control, and the measurements from this process are well modeled by a Normal distribution. This information is reflected in the capability sixpack in Figure 1. Figure 1.

Note that the specs for the process are LSL = 37, USL = 43, and subgroups of size 5 are being taken. With these specifications, and in order to compute an approximate 95% confidence interval, one may use the macro available from the MINITAB website at http://www.minitab.com/support/macros/index.asp?cat=QC_DOE#33 by saving it into the MINITAB macro folder. One would enter the commands through the editor window by entering the column number (in this case, column 2), followed by the LSL, the USL, the subgroup size, and the desired confidence level. Therefore, to obtain 95% confidence intervals for Cp and Cpk using the data in this example, one would enter via the Command Line Editor (CTRL+L): %capaconf c2 37 43 5 .95 This results in the output as shown in Figure 2.

Figure 2.

The approximate 95% confidence interval is 0.975 to 1.163 for Cp and 0.959 to 1.163 for Cpk. Importantly, values less than 1 are included in these intervals, which would be of great concern to many practitioners. In conclusion, therefore, this author argues that the inclusion of confidence intervals with capability analysis is an appropriate and necessary step. It is now possible to utilize this procedure along with capability analysis using the MINITAB macro indicated. References 1. Montgomery, D.C. (2000). Introduction to Statistical Quality Control, 4th edition; John Wiley & Sons, Inc. 2. Bower, K.M. (January, March 2001). “Capability Analysis Using MINITAB (I & II),” ExtraOrdinary Sense; http://www.isssp.org/ 3. Somerville, S.E., Montgomery, D.C. (1996). “Process Capability Indices and Nonnormal Distributions,” Quality Engineering, Vol. 9, No. 2.

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