1. 3. Use an in-line sensor to sense when the effects of tool wear...
Tool wear - an X - will result in systematic variation - “drift” - in The Y:
1. Monitor The Y and change the tool when the value of The Y nears the spec appropriate limit.
2. Determine the mean number of uses before The Y is out of spec and change the tool after the appropriate number of uses.
3. Use an in-line sensor to sense when the effects of tool wear...
In this example the variation in the X cannot be controlled. However, it’s resulting variation in The Y is systematic and distinct and relatively easy to measure. By monitoring the Y and changing the X when appropriate, we minimize our costs and reduce the variation in the Y to the point where The Problem doesn’t occur.
Tool wear does not exhibit a Normal distribution. It has a Uniform distribution - similar to a deck of cards. Any attempt to use “standard Cp/Ppk calculations to estimate yields or provide a Cp/Ppk value will result in serious overstatement of the ppm level and understatement of the Cp/Ppk values. This may yield to a “forced” uneconomical “over-control” of the process
Beverly Daniels

2. Initial Tool Wear Data Collection
Initial data must be collected across multiple tool changes to assess the stability and pattern of ‘normal’ tool wear
si is the standard deviation of the initial results of the tool.
It may be estimated by se, the standard deviation of the results in the Y dimension about the average (best fit or regression) line.
Beverly Daniels

3. If the tool wear results in smaller trending values:
Tool Wear Calculation
The initial cut is targeted at the lower target limit, and the tool is changed or adjusted once the first unit measures at or above the upper target limit.
If the tool wear results in smaller trending values:
The initial cut is targeted at the upper target limit, and the tool is changed or adjusted once the first unit measures at or below the lower target limit.
Beverly Daniels

4. Ppk for the Uniform Distribution

5. An Alternate Approach to Calculating Capability
Since one of the primary purpose of Capability Indices is to quantify the process capability to produce conforming parts (Yield), one can calculate the Yield of a process once stability has been established.
Stability should be established via both variables and attributes control charts to account for variation and error based defects.
The Ppk value can then be calculated by working backwards thru the Z table.
Yield Z score
Or in Excel use Ppk = NORMSINV(Yield)/3
The other purpose of a capability index is to quantify the reduction in total variation compared to the tolerance limits. Given actual shape differences a more informative way of conveying this information is to provide the actual frequency distribution plotted against the tolerances.

6. Estimating Yield & Capability When np=0
When The Y is a real attributes characteristic (and not a variables characteristic that is only measured as pass-fail or categorical classification for manufacturing efficiencies), estimating the process yield is traditionally dependent on the detection of at least one defect.
When the process is of relatively high quality, this can take considerable to accumulate enough units. There are two alternative calculations that may be utilized until such time as actual defect rates can be calculated.
Both calculations require the number of sequential units with no detected defects.
The Bayesian estimate* :
Since d = 0,
The Poisson Estimate:
Using a 50% confidence limit assuming a Poisson distribution, the Upper Limit of np when x=0: -ln(1-g), where g is the confidence coefficient for the maximum expected number of defective units, np. If g is .50, then 50% of the time the observed number will be no greater than: -ln(1-0.5) = .7  the average number of expected units. So the probable average rate of occurrences for a process can be estimated by:
A process that generates 10,000 sequential units with out a defect can be estimated to have a long term defect occurrence rate:
Bayesian: p =
Poisson: p =
The Bayesian method provides a more conservative estimate.
*An explanation or derivation of the Bayesian method is beyond the scope – or purpose – of this course

7. A Final Caution on Capability Indices
There are 4 dimensions that fully describe the variation of any distribution:
The Location (Average, Median or Mode)
The Dispersion (Standard Deviation, Range )
The Shape
The Trend (Time sequence with families of variation)
A Cp/Pp value only provides information on the dispersion in relationship to the tolerance width.
A Cpk/Ppk value only gives limited information on the location and dispersion of a tail of the distribution: how much of the distribution is outside the tolerances…
Neither of these single parameters provides sufficient information to describe any distribution.
Each of these is a point estimate from sample data.

8. ENDNOTES, REFERENCES AND RECOMMENDED READING
Bhothe, Davis R.., “Measuring Process Capability”, McGraw-Hill, 1997
Gunter, Bert, “The Use and Abuse of Cpk, Parts 1, 2,3 and 4”, Quality
Progress, January 1989, March, 1989, May 1989, July 1989
Pignatiello, Joseph J., Ramberg, John S., “Process Capability Indices: Just Say “No!””, ASQC Quality congress Transactions – Boston, 1993