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Re: Cpk and Sample Size/frequency determination

From: Stan Hilliard
Date: 15 Feb 2001
Time: 14:07:02

Comments

Hi Nick,

I found the article by LeRoy A. Franklin on the web. I had previously seen the JQT material that he references. These references estimate the process capability with the Cpk metric.

You said that your goal is to use "the Cpk indices to determine sample size...". I assume that the purpose of the samples is to make decisions for acceptance/rejection and/or to know when to adjust. For this you need to develop a sampling plan that can discriminate between defined acceptable (AQL) and rejectable (RQL) levels of the variable. If you like to think in terms of Cpk, you could define these sampling criteria in terms of that metric. This only applies, however, to lots or processes that consist of individual items.

1) PROCEDURE FOR USING THE Cpk METRIC:

1A) Define the acceptable Cpk [AQL(Cpk)] and the rejectable Cpk [RQL(Cpk)]. Interpret AQL and RQL as per www.samplingplans.com/modern3.htm#EVALUATE

1B) Convert the AQL and RQL from Cpk units to fraction defective.

2) PROCEDURE TO ADJUST FOR NON-NORMAILTY:

2A) Cpk automatically assumes normality, so if the variable is not normally distributed between items, you can adjust the sampling criteria by the method that I describe below. I do not recommend adjusting for non-normality regularly -- only when there is evidence of serious non-normality. You will see that taking non-normality into account will add to the complexity of designing the sampling plan.

2B) Measure the variable on historical lots and plot it on normal probability paper. Use multiple lots. Plot the value of (X - Xbar) instead of X, where Xbar is the average of the lot that measurement X is from. This will remove the effect of possible between-lot differences in the mean. With a french-curve, draw a line through the points. (Not a straight line, as that would be to assume normality) If the line is not very curved, assume normality by drawing a straight-line.

2C) Mark the AQL and RQL fraction defective on the probability chart. From the chart, read the number of standard deviations that the mean must be from the specification limit for AQL and RQL. Calculate the value of the AQL(mean) and RQL(mean).

2D) Design a fixed-n sampling plan for the mean. Sampling plans for the mean do not assume normality - because of the central limit theorem. Use Alpha, Beta, AQL(mean), RQL(mean), and within-lot standard deviation. (All sampling plans for the mean are based on known within-lot standard deviation.) Example: www.samplingplans.com/outputvariablesmean.htm

3) PROCEDURE TO MINIMIZE THE SAMPLE SIZE:

3A) Reduce the sample size by converting the fixed-n plan to a sequential sampling plan. I prefer the TSS sequential. Examples: www.samplingplans.com/modern3.htm#MATCHING

3B) A sequential plan will additionally enable you to use the ASN curve to re-target the process to attain an average sample size of any value. (I wouldn't go below n=2 so that you have a chance of detecting a measurement error.) www.samplingplans.com/outputvariablesisl.htm#ASN-Curve

4) WITHIN-LOT VARIABILITY:

If the "known" within-lot standard deviation changes on a future lot, that would throw off the ability of the sampling plan for the mean to discriminate as planned. To protect against this, you should calculate limits for the range of the sample.

5) SOFTWARE:

The software program TP414 can do the calculations and facilitate the type of design that I described here. www.samplingplans.com/modern3.htm#Software

6) DESIGN JOURNEY:

If you want examples email me your mailing address and fax number, and I will send you a copy of Design Journey. (Other visitors to this forum may do the same.)

Sincerely, Stan Hilliard


Last changed: November 20, 2007