Hypothesis and Confidence Interval Calculations for Cp and Cpks

Folks,

I am reposting an updated blog post on Cp and Cpk calculations with Excel, as I have improved the Excel spreadsheet. If you would like the new spreadsheet, send me an email at rlasky@indium.com.

One of the best metrics to determine the quality of data is Cpk. So, I developed an Excel spreadsheet that calculates and compares Cps and Cpks.

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Folks,

It is accepted as fact by everyone that I know that 2/3 of all SMT defects can be traced back to the stencil printing process. A number of us have tried to find a reference for this posit, with no success. If any reader knows of one, please let me know. Assuming this adage is true, the right amount of solder paste, squarely printed on the pad, is a profoundly important metric.

In light of this perspective, some time ago, I wrote a post on calculating the confidence interval of the Cpk of the transfer efficiency in stencil printing. As a reminder, transfer efficiency is the ratio of the volume of the solder paste deposit divided by the volume of the stencil aperture. See Figure 1. Typically the goal would be 100% with upper and lower specs being 150% and 50% respectively.

Figure 1. The transfer efficiency in stencil printing is the volume of the solder paste deposit divided by the volume of the stencil aperture. Typically 100% is the goal.

I chose Cpk as the best metric to evaluate stencil printing transfer efficiency as it incorporates both the average and the standard deviation (i.e. the “spread”). Figure 2 shows the distribution for paste A, which has a good Cpk as its data are centered between the specifications and has a sharp distribution, whereas paste B’s distribution is not centered between the specs and the distribution is broad.

Figure 2. Paste A has the better transfer efficiency as its data are centered between the upper and lower specs, and it has a sharper distribution.

Recently, I decided to develop the math to produce an Excel® spreadsheet that would perform hypothesis tests of Cpks. As far as I know, this has never been done before.

A hypothesis test might look something like the following. The null hypothesis (Ho) would be that the Cpk of the transfer efficiency is 1.00. The alternative hypothesis, H1, could be that the Cpk is not equal to 1.00. H1 could also be that H1 was less than or greater than 1.00.

As an example, let’s say that you want the Cpk of the transfer efficiency to be 1.00. You analyze 1000 prints and get a Cpk of 0.98. Is all lost? Not necessarily. Since this was a statistical sampling, you should perform a hypothesis test. See Figure 3. In cell B16, the Cpk = 0.98 was entered; in cell B17, the sample size n = 1000 is entered; and in cell B18, the null hypothesis: Cpk = 1.00 is entered. Cell B21 shows that the null hypothesis cannot be rejected as false as the alternative hypothesis is false. So, we cannot say statistically that the Cpk is not equal to 1.00.

Figure 3. A Cpk = 0.98 is statistically the same as a Cpk of 1.00 as the null hypothesis, Ho, cannot be rejected.

How much different from 1.00 would the Cpk have to be in this 1000 sample example to say that it is statistically not equal to 1.00? Figure 4 shows us that the Cpk would have to be 0.95 (or 1.05) to be statistically different from 1.00.

Figure 4. If the Cpk is only 0.95, the Cpk is statistically different from a Cpk = 1.00.

The spreadsheet will also calculate Cps and Cpks from process data. See Figure 5. The user enters the upper and lower specification limits (USL, LSL) in the blue cells as shown. Typically the USL will be 150% and the LSL 50% for TEs. The average and standard deviation are also added in the blue cells as shown. The spreadsheet calculates the Cp, Cpk, number of defects, defects per million and the process sigma level as seen in the gray cells. By entering the defect level (see the blue cell), the Cpk and process sigma can also be calculated. 

Figure 5. Cps and Cpks calculated from process data.

The spreadsheet can also calculate 95% confidence intervals on Cpks and compare two Cpks to determine if they are statistically different at greater than 95% confidence. See Figure 6. The Cpks and sample sizes are entered into the blue cells and the confidence intervals are shown in the gray cells. Note that the statistical comparison of the two cells is shown to the right of Figure 6.

Figure 6. Cpk Confidence Intervals and Cpk comparisons can be calculated with the spreadsheet.

This spreadsheet should be useful to those who are interested in monitoring transfer efficiency Cpks to reduce end-of-line soldering defects. It is not limited to calculating Cps and Cpks of TE, but can be used for any Cps and Cpks. I will send a copy of this spreadsheet to readers who are interested. If you would like one, send me an email request at rlasky@indium.com.

Cheers,

Dr. Ron