How to conduct a capability analysis (by calculating cpk) and check results using Minitab. Six Sigma, DMAIC etc

Question

as the title states I'm conducting a statistical analysis for the variation within a machine (a laser marker). Just as a little background info, I have started off by conducting a GR&R for the measurement system that I'll be using.

I've conducted the GR&R by measuring the same points(77 in total) a total of ten times and I've ensured that the variance within the measurements is within 10% of the maximum allowed variance. The laser marker can have a maximum variance of 0.7mm, therefore the measurement system is allowed a maximum of 0.07mm. With this in mind I have calculated the Standard Deviation of each X,Y coordinate and used this to calculate the sigma value of the measurement system which is well within the Six Sigma spec.

Now, the trouble that I am having is calculating the capability of the laser marker (cPK). I have written a small program for a cPK calculator, that I can confirm the results with in excel using the cpk formula cpk = Min(cpU,cpL). The problem seems to be that I am getting crazy high accuracy (a cpk of between 20-100) Here is a sample of my data: I'm only using 4 samples here for simplicities sake, also ignore the Y Cpk.

Now my question is, is there a way that I can calculate the same cpk in Minitab just to check that my software is performing correctly? Also, to do a complete capability analysis for the laser marker do I just need to calculate cpk for X and Y for each and every point and as long as the smallest cpk is > 2.0 the process is six sigma compliant or is there an overall cpk that I should be calculating? Sorry for the essay but any help would be appreciated. I have no previous Six Sigma/DMAIC/statistical analysis experience so have only learnt what I could online.

Thanks

Answer 1

I'm not sure exactly were your problems are, but there do seem to be some calculation errors in your analysis (although the $C_{pk}$ values do seem to be in the right ballpark).

One place to start here is that $C_{pk}$ should be calculated with a minimum of 25 points of data and about 7 is the absolute minimum (because you need reliable measures of mean and standard deviation). I will get into this more later.

In regards to checking this with Minitab, I would suggest importing your data in. You can then conduct the analysis by going to Stat->Quality Tools->Capability Analysis->Normal… and use subgroups across rows of: to define your locations to get an overall assessment of your process capability. You can also use Data->Unstack colums… to arrange your data into sets of columns by position to run the capability analysis for each position.

Going back to your data, your mean ($\bar{x}$) for the X for Position 1 should be $-0.20056\,$mm, your sample standard deviation should be $0.00376\,$mm (so $3\sigma$ should be $0.011281\,$mm). Based on your specification limits, your $C_{pk}$ should then be $44.27$. If, however, you want the capability of your measurement system, then your $USL=0.07\,$mm and your $LSL=-0.07\,$mm, and your $C_{pk}=-11.57$.

Either way, you may also want to consider the confidence interval of your results, specifically the lower confidence interval. The confidence interval is based upon your sample size. The larger your sample size, the smaller the overall confidence interval.

A lower confidence interval for $C_{pk}$ can be calcualted as: $$c=\hat{C}_{pk}\pm Z_{1-\alpha/2}\sqrt{\frac{1}{9n}+\frac{\hat{C}_{pk}^2}{2(n-1)}}$$ (or in Excel as =Cpk-normsinv(1-alpha/2)*sqrt(1/(9*n)+Cpk^2/(2*(n-1))) This can be found at NIST/SEMATECH e-handbook of statistical methods (Oct. 2013). The lower 95% confidence interval for both answers with your sample size of 4 would be $14.54$ for a $C_{pk}=44.27$ and $-19.35$ for a $C_{pk}=-11.57$.

For dealing with multiple process steams, such as this, you can refer to D. R. Bothe, A capability index for multiple process streams, Quality Engineering 11 (4) (1999) 613. doi:10.1080/ 08982119908919281.. The basic process it to calculate the upper and lower capability for each location, then develop an average (or weighted average if needed) for each location for the upper and lower capability. A final overall capability is based upon $\hat{C}_{pk}=\frac{\hat{Z}_{bench}}{3}$, with the $\hat{Z}_{bench}$ calculated based upon the average process fallouts. (Let me know if you would like me to expand that answer for you.)