# Sampling Plans and Cpk - Ways to Calculate the Sample Quantity

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#### Angela at SKF

We currently check 100% of our parts and produce about 1500 per shift. If we want to hold a Cpk to a certain level is there some way to calculate the sample quantity needed each shift? Tolerance is (lower bound) 0 to +10 microns.

Say our Cpk is 2.0 and at PPAP the Cpk was 1.7 can we justify reducing our sampling? Ultimate goal.

#### Tim Folkerts

Super Moderator
Sample Size shouldn't directly affect Cpk

Cpk is based on the mean and standard deviation of the product, combined with the specs (and the assumption that the process stays in control). Cpk doesn't directly relate to sample size.

The one place where sample size comes in would be your confidence in the Cpk you calculate. For example, suppose you checked just N=25 parts and had
x-bar = 7
sigma = 0.5
USL = 10
LSL = 0

You are (10-7)/0.5 = 6 sigma from the closer spec so Cpk = 6/3 = 2. But the standard error in x-bar is 0.5/(25^0.5) = 0.1, so the true mean could easily be 7.1 or 7.2. Furthermore, there is considerable uncertainty in sigma: 0.71*sigma/N^0.5 = 0.071, so sigma itself could easily be 0.57 or 0.64. You might easily be just (10-7.1)/0.57 = 5.1 sigma from the spec limit, or Cpk = 1.7.

Or simplified, if you take a small sample, you can't be as sure about the accuracy of the Cpk calculation.

Tim F

#### Miner

##### Forum Moderator
Staff member
You want to control process drift

As Tim said, sample size is not what you are after. You are really trying to control the process drift to assure a minimum Cpk.

While the most statistically pure method is some form of SPC, you could also calculate modified control limits that allow some drift, but limits it to a level that assures a minimum Cpk.

Be advised that if a customer is driving this, you will probably be challenged if you take this route. If you are doing it for yourself, you have a lot of latitude.

R

#### Ricardo SKF INTERN

Re: Sampling plans and Cpk - Some way to calculate the sample quantity

Hello Angela,

I'm currently working in a project in SKF to caculate a sampling plan frequency and size procedure based on Cpk of the machine assuming a Cp target constant in time.

I've been surfing for similar cases and found your post. Given that is from 2003 I figure you may give me some advice.

Would appreciate very much your help,

Ricardo.

#### bobdoering

Trusted Information Resource
Re: Sampling plans and Cpk - Some way to calculate the sample quantity

Ricardo,

What kind of process are you looking at? What is its current capability? What is the actual distribution? What does its ongoing variation look like? Is any SPC being done on the process? Do you have any sample data?

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Re: Sampling plans and Cpk - Some way to calculate the sample quantity

There is a formula for the confidence level of Cpk (See attached file). This is the only way I know to have a sample size for a given Cpk level.

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#### Ricardo SKF INTERN

Re: Sampling plans and Cpk - Some way to calculate the sample quantity

Ricardo,

What kind of process are you looking at? What is its current capability? What is the actual distribution? What does its ongoing variation look like? Is any SPC being done on the process? Do you have any sample data?
Hello, Bob,

I'm looking at a grinding process of an external profile. Actual distribution is normal. Ongoing variation is very small, quality specifications are measured in micro millimeters and tolerances are considerately away from control limits. Currently SPC are made but not based on Cpk, which is my task. Sample data is available.

R

#### Ricardo SKF INTERN

Re: Sampling plans and Cpk - Some way to calculate the sample quantity

There is a formula for the confidence level of Cpk (See attached file). This is the only way I know to have a sample size for a given Cpk level.

#### bobdoering

Trusted Information Resource
Re: Sampling plans and Cpk - Some way to calculate the sample quantity

I'm looking at a grinding process of an external profile. Actual distribution is normal. Ongoing variation is very small, quality specifications are measured in micro millimeters and tolerances are considerately away from control limits. Currently SPC are made but not based on Cpk, which is my task. Sample data is available.
If you are grinding a profile, I recommend reading this. Your true distribution should not be normal. If it is, you are likely picking up measurement error, or the process is being overcontrolled. Tradiitonal Shewhart SPC has no relation to capability. If you are using X-bar-R chart or I-MR, your control limits are going to be too tight, because those chart calculations are wrong for grinding - and will cause overcontrol.

For grinding, the correct controls using the X hi/lo-R chart would be based on capability, as the control limits are a function of the tolerance.

If you are grinding, and you are using very little of your tolerance, do you do any adjustments at all during your run? How long is your run?

#### optomist1

##### A Sea of Statistics
Super Moderator
Re: Sampling plans and Cpk - Some way to calculate the sample quantity

Bob,

I didn't think you would let that one get by you.....thanks for your continued isight.

Regards,
Marty