DPMO and Euler's Defect Rate Calculation | A Visual Comparison

If you read enough material on Six Sigma principles you most likely have come across the calculation of defects per unit (DPU) or defects per million of opportunities (DPMO). You have also, most likely, come across the suggested "Euler's formula" for estimating defect rates. Now, as we all know, 1 - defect rate equals yield. In other others, if you are somehow calculating a defect rate of a process you are also consequently easily computing the yield of the process. For example, a process with 10% defect rate has, mathematically, 90% yield. With that out of our way, let's look at how each is calculated and the issue with Euler's formula when applied to a defect rate level over 10%.

DPMO is simply stated, the defects (per unit) divided by one million (that is, one million opportunities of finding a defect). Note: this is any unwelcome departure from the standard, not necessarily the entire unit being defective, the distinction here is very important.

At a 4.5 sigma level (no sigma shift considered), we'll find that 3.4 defects per unit produces a DPMO of 0.0000034, or 0.00034% and consequently, a yield of 1 - 0.0000034 (or 100% - 0.00034%), which equals 99.9997%. If you're a Six Sigma Black Belt you are probably very familiar with these figures.

The following is a screen shot of how I computed and produced the plot that showcases the issue at hand (which is coming up shortly). Notice how I have computed the calculations for 100,000 records in Excel, starting at 3.4 defects, then from 20 to 1,000,000, in increments of 10. Yes, it's a big file.

You will also notice from the screen shot above that I have computed the defect rate and yield for the same rate of defects (that is, from 3.4 to 1,000,000) using the Euler's formula:

e ^ -DPU (Euler's number to the power of minus DPU)

Recall that e is the mathematical constant called Euler's number, after the Swiss mathematician Leonhard Euler. The constant has an approximated value of 2.71828.

Let us now plot these two defect rates ("Defects" and "Defects (Euler)") from the table above and see what's happening to the calculations.

We can see from the plot that at approximately 10% defect rate, the classic DPMO linear starts to veer off of Euler's curve. As many practitioners and scholars have suggested, we should not use Euler's formula to estimate defect rates (and consequently yield) at defect rates close to and definitely greater than 10%. Many books and online resources suggest the e ^ -DPU formula but not many add the caveat that we should be very careful with this approach.

You can now see why that is the case. Notice the increase in the gap (delta) on both the table and especially the plot. At about 10% defect rate, the gap becomes greater and greater.