# Semoi

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• #255905 Semoi
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Link break or the text get changed. So it would be helpful if you could provide a short summary describing the key points.

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#255904 Semoi
Participant

Personally, I feel that there are two parts, which are important:
1. How large are the results (=bias and non-linearity) compared to the tolerance range
2. How large are the results compared to the standard deviation of the process.
In order to combine these two considerations I would calculate the C_g and C_gk value, which judges the bias. To judge the non-linearity, I would modify the formula to account for the non-linearity. Here an example: Say your specification limits are [450, 520], the product follow a normal distribution with mean 495 and standard deviation 5. Since the tolerance range is 70 the C_g value is C_g = k* 70/(6*5) = k* 2.333 and C_gk = k*(520-495)/(3*5) = k*0.83. Usually, k is taken to be in the range [0.1, 0.3]. In addition, you have to decide which C_pk-value your products needs, and use this as the lower limit for the C_gk value — note that the two only differ by the factor k.
Since the linearity of a gauge is a bias for a specific reference value, we can use the upper formula for C_gk and plug in the largest difference between reference value and measurement result. Of course, this is a C_gk only for the specific reference value — make sure that your notation reflects this, e.g. by using C_gk(target=…). Like this you use a single quantity (the C_pk-value of your product) to determine whether or not your gauge is sufficient — in addition with a predefined k-value.

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#255784 Semoi
Participant

Hi,
suppose we have two random variables, y_1 and y_2, and that their population mean and standard deviations are mu_1, mu_2, and sigma_1, sigma_2, respectively. Then the standard deviation of the difference y_tot = y_2 – y_1 is given by sigma_tot = sqrt(sigma_1^2 + sigma_2^2) — note that we use a plus (!) in the standard deviation. Hence, the standard deviation has increased. E.g. if we simplify and assume that sigma_1=sigma_2 we have increase the standard deviation by a factor of sqrt(2) \approx 1.4.

If you are unable to follow the math argument just draw random numbers of standard normal distributions and plot a histogram for a single random variable and for the difference of two random variables. You will observe that the later is “broader”.

Hope this helps.

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#255783 Semoi
Participant

Hi,
personally, I never use specifications which are a result of what a supplier (or my company itself) is able to deliver. Instead, I define a specification due to our product requirements. Hence, my spec does not (!) include statistical quantities, but reads “the measured value must be inside the interval [x1, x2]”. Hence, I don’t have any discussion on how the supplier likes to interpret the spec.

Next, if I don’t want to perform a 100% incoming goods inspection, the supplier and I have to agree upon a sampling plan, at which AQL we are rejecting the lot, and how we proceed afterwards with the rejected lot. Doing these things upfront is important, because else you have many discussions about taking different samples from the same lot and arguing that the new measurement indicates that the hole lot is within specification. Once you understand how uncertainty and random sampling works you will see how the supplier transfers the costs to you.

Finally, if I like to skip the incoming good inspection I demand that the supplier achieves a Cpk value, which exceeds a specified value. How this value is determined is not a matter of taste, but defined by using a formula. In addition, we agree that every lot has to satisfy this Cpk value and that we will use a lot (defined by my company) to reevaluate the Cpk value after a time X.

I know that the discussions are hard and that no supplier likes this. So, if it is acceptable to us that we relax a specification, because we are able to tighten it for an other component, we do so. However, we do so because it is acceptable for our product and not because the requirements are hard for our supplier.

Hope this helps.

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