Ok. Let me start off by saying this is not something you'll read about in a textbook, nor is it standard practice.

However.

I believe the way we think about 2D coordinate data and positional capability is inadequate. Here is a good example of why that is:

So how can we assess positional capability accounting for it's 2-dimensional nature? There have been many proposed methods, but I have found the error ellipse to be the most informative approach, though it does have its drawbacks.

Drawback #1: "Bivariate Ppk" is a term I came up with to describe Ppk of a 2-dimensional nature. Customers won't know what this is. So this tool is really for internal analysis only (though I have found customers love seeing the ellipse graphed against the positional tolerance).

Drawback #2: If the centroid of the distribution lies outside the tolerance, the bivariate Ppk cannot be calculated. In 1 dimension, you can simply go left or right, so means which lie beyond the spec limit only have one direction to go. But in 2 dimensions, there are infinite directions, and since Ppk technically reports "worst case", the worst case cannot be calculated. The best case can be calculated, in the direction toward nominal, but that is not how Ppk works. However, if your centroid lies outside the tolerance, I don't think you need a bivariate Ppk to tell you that you have problems

So allow me to explain where this all came from, and how I calculated the error ellipse mathematically:

Because the data is not time-ordered, Cpk and Cp go out the window. It is what it is. I experimented with a Cpk Ellipse using sigmas calculated from Moving Range, but ultimately it didn't add much value. If your process is in control, the Cpk Error Ellipse is virtually identical to the Ppk Error Ellipse anyways. And as we all know, capability analysis is useless with an out of control process. So there ya go.

Here is what the Error Ellipse looks like. Click the graph to auto-resize. You must enter the nominal (x,y) and tolerance for the TP, then paste the data underneath. If it's using material condition, you must also enter the USL and LSL for diameter, and paste that data underneath. Because the expanded tolerance would be different for every part, and that would make the graph too messy, i use the minimum of either "mean minus 3 st dev" and the "min data point". Whichever is lowest, that is used for the expanded tolerance calculation, as a safety net.

The 3 sigma ellipse is based off the square of the mahalanobis distance which follows the Chi-Square distribution, and is set to cover exactly 99% of the process.

I've used this for many years now, and was always reluctant to share it because it's a novel approach, and I feared criticism from statistical gurus who would pick it apart. But, it's been so useful I just can't keep it to myself any longer.

However.

I believe the way we think about 2D coordinate data and positional capability is inadequate. Here is a good example of why that is:

So how can we assess positional capability accounting for it's 2-dimensional nature? There have been many proposed methods, but I have found the error ellipse to be the most informative approach, though it does have its drawbacks.

Drawback #1: "Bivariate Ppk" is a term I came up with to describe Ppk of a 2-dimensional nature. Customers won't know what this is. So this tool is really for internal analysis only (though I have found customers love seeing the ellipse graphed against the positional tolerance).

Drawback #2: If the centroid of the distribution lies outside the tolerance, the bivariate Ppk cannot be calculated. In 1 dimension, you can simply go left or right, so means which lie beyond the spec limit only have one direction to go. But in 2 dimensions, there are infinite directions, and since Ppk technically reports "worst case", the worst case cannot be calculated. The best case can be calculated, in the direction toward nominal, but that is not how Ppk works. However, if your centroid lies outside the tolerance, I don't think you need a bivariate Ppk to tell you that you have problems

So allow me to explain where this all came from, and how I calculated the error ellipse mathematically:

Because the data is not time-ordered, Cpk and Cp go out the window. It is what it is. I experimented with a Cpk Ellipse using sigmas calculated from Moving Range, but ultimately it didn't add much value. If your process is in control, the Cpk Error Ellipse is virtually identical to the Ppk Error Ellipse anyways. And as we all know, capability analysis is useless with an out of control process. So there ya go.

Here is what the Error Ellipse looks like. Click the graph to auto-resize. You must enter the nominal (x,y) and tolerance for the TP, then paste the data underneath. If it's using material condition, you must also enter the USL and LSL for diameter, and paste that data underneath. Because the expanded tolerance would be different for every part, and that would make the graph too messy, i use the minimum of either "mean minus 3 st dev" and the "min data point". Whichever is lowest, that is used for the expanded tolerance calculation, as a safety net.

The 3 sigma ellipse is based off the square of the mahalanobis distance which follows the Chi-Square distribution, and is set to cover exactly 99% of the process.

I've used this for many years now, and was always reluctant to share it because it's a novel approach, and I feared criticism from statistical gurus who would pick it apart. But, it's been so useful I just can't keep it to myself any longer.