# The Use of PPK to Determine If Confidence and Reliability Statement is met.

##### Involved In Discussions
Just because you can doesn’t mean you should.

The manipulation of mathematical formulas is no substitute for thinking.

By the way, most real statisticians made fun of Cpk when it was first introduced in the eighties. It simply wasn’t serious theoretical statistics. There are many articles from that time rebutting the use of Cpk for anything.
i know that, but qualitative arguments fall on deaf ears.

#### Bev D

##### Heretical Statistician
Super Moderator
You can start with my article Statistical Alchemy in the resources section. The article includes references for further quantitative arguments.

#### Desmond23

##### Registered
I just realised that I did not describe the math behind my calculations. Here it goes ...

To calculate the Cpk value, we use the concept of tolerance intervals. For a single-sided specification we use the non-central t-distribution,
View attachment 29763
As you can see, we need three inputs to calculate the k_1:
(a) the coverage p = 1 - RQL,
(b) the confidence level 1-alpha, and
(c) the sample size n.
Hence, we just pick a sequence of sample sizes and redo the calculation until we are satisfied. E.g. for n=20, p=0.99, alpha=0.05 we get k_1=3.295157.

Next, we user the relationship Cpk = k_1/3. This yields the required (minimal) Cpk-value. In the above example we get Cpk=1.098386. This number is always rounded upwards. I round to the second digit, which yields Ppk >= 1.10.
Note, that translating the RQL=1% value directly into a Cpk-value would yield Cpk=0.7754493. By using the tolerance interval method we "add the required 95% confidence" into the result. Hence, the Cpk-value 1.10 is an upper limit of the corresponding tolerance interval.

Finally, I use the "common approximation" of the uncertainty of the Cpk value (see NIST link above), and calculate the single-sided confidence interval. This yields 1.416078 as an upper limit. This corresponds to the value you stated in your pre-PQ column.

Please note, that k_1 is for a single sided specification. The calculation is similar for two-sided specifications, but we are not allowed to use the non-central t-distribution.
Hi,

Would it be possible to show the workings for one of the line items
I just realised that I did not describe the math behind my calculations. Here it goes ...

To calculate the Cpk value, we use the concept of tolerance intervals. For a single-sided specification we use the non-central t-distribution,
View attachment 29763
As you can see, we need three inputs to calculate the k_1:
(a) the coverage p = 1 - RQL,
(b) the confidence level 1-alpha, and
(c) the sample size n.
Hence, we just pick a sequence of sample sizes and redo the calculation until we are satisfied. E.g. for n=20, p=0.99, alpha=0.05 we get k_1=3.295157.

Next, we user the relationship Cpk = k_1/3. This yields the required (minimal) Cpk-value. In the above example we get Cpk=1.098386. This number is always rounded upwards. I round to the second digit, which yields Ppk >= 1.10.
Note, that translating the RQL=1% value directly into a Cpk-value would yield Cpk=0.7754493. By using the tolerance interval method we "add the required 95% confidence" into the result. Hence, the Cpk-value 1.10 is an upper limit of the corresponding tolerance interval.

Finally, I use the "common approximation" of the uncertainty of the Cpk value (see NIST link above), and calculate the single-sided confidence interval. This yields 1.416078 as an upper limit. This corresponds to the value you stated in your pre-PQ column.

Please note, that k_1 is for a single sided specification. The calculation is similar for two-sided specifications, but we are not allowed to use the non-central t-distribution.
Hi,

Would it be possible to show the workings from one of the line items from above ,
i.e. [1] "nSample = 15 => Ppk >= 1.18 and 95% (single-sided) CI = [1.18, 1.57]"

as im not coming up with the same as yourself, so am going wrong somewhere.

Thanks you

#### Mustapha

##### Starting to get Involved
Could you expand on where this comes from? What's the statistical basis for this relationship?
The only reference I've seen that comes close to that statement is in Wayne Taylor's book. The screenshot below shows how the two are equated.

Also, Desmond, some background of how the original table was calculated is located here: STAT-12: Verification/Validation Sampling Plans for Proportion Nonconforming - Taylor Enterprises. In the comments section, there are a few posts that talk about how the values were calculated, you may be able to get some ideas of how to reach the correct formula there.

#### Johnnymo62

##### Haste Makes Waste
FWI In July Ford updated their GRR requirements for one sided tolerances in their IATF 16949 CSR and their PPAP CSR.

#### Bev D

##### Heretical Statistician
Super Moderator
The only reference I've seen that comes close to that statement is in Wayne Taylor's book. The screenshot below shows how the two are equated.

View attachment 29967

Also, Desmond, some background of how the original table was calculated is located here: STAT-12: Verification/Validation Sampling Plans for Proportion Nonconforming - Taylor Enterprises. In the comments section, there are a few posts that talk about how the values were calculated, you may be able to get some ideas of how to reach the correct formula there.
So much obscure math with requirements (aka ASSumptions) which get ignored and so little insight. My advice: Forget the voodoo doodoo and plot your (raw) data, then think about what the data are telling you.