Hypothesis and Confidence Interval Calculations for Cp and Cpks


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.



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.


Dr. Ron

The Area Ratio for Odd-Shaped Stencil Apertures

Joey writes:

Dear Dr. Ron,

I have a stencil aperture with an unusual shape. See Figure 1. How do I calculate the area ratio? The stencil thickness is 5 mils. The dimensions of the aperture are also in mils.

Figure 1. Joey’s Stencil Aperture


The area ratio is simply the area of the stencil aperture opening divided by the area of the sidewalls. For common aperture geometries such as circles, squares, etc. it is easy to derive formulas. See Figure 2.

Figure 2. Formulas can be developed for common aperture shapes.

For an unusual shape like yours, it is easiest to simply calculate and divide the areas. From Figure 1, we get that area of the aperture opening as: 40*24+ the area of the two triangles. A little geometry (can you do it?) shows each triangle to have an area of 89 sq mils. So, the total area is 960 + 2*89 = 1138 sq mils. The perimeter is 40+24+16+16+28+12+16+16 = 168 mils, hence the area of the sidewalls is 168*5 = 840 sq mils. Therefore, the area ratio is 1138/840 = 1.355. Experience has shown that an area ratio of > 0.66 is needed for good solder paste transfer efficiency, so this stencil aperture will do well for transfer efficiency.

Careful thought would suggest that the triangular protrusions alone do not have a good area ratio. Calculations show their area ratios to be 0.37. So, the transfer efficiency in this part of the aperture might not be good. However, the area of the rectangle is so great, more than five times that of the triangles, as to alleviate this concern.

Dr. Ron

GKG: Westward Ho?

Southeast Asian assembly process equipment companies have approached Western markets in fits and starts.

A few have made inroads: From time to time, we have seen JT and Fulongwin soldering equipment at US plants, usually smaller ones (Flextronics is an exception) and often on the US West Coast. But while we’ve been reporting for more than a decade on the availability of literally scores of Chinese-made brands, some of which are very popular in Taiwan and China, it’s still highly unusual to see any make it across the ocean.

Many have been stymied by patent issues that effectively have blocked them specifically from the US and European markets. Another problem is finding good channel partners. From time to time, firms ranging from independent reps like FHP Reps and Bill West to solder paste vendors like Qualitek have tried, with limited success. Service and access to spare parts have been limiting factors.

That’s what makes Friday’s announcement from GKG so interesting. GKG has named Juki as exclusive distributor of its screen printers in the Americas. Known primarily for its placement equipment, Juki has been inching toward a full-line offering for the past couple years, having begun distributing Intertec’s selective soldering equipment in 2009.

For years, DEK and Speedline have dominated the Western printer markets, with Asys/Ekra in third with an estimated 10% share. Juki’s track record and never-say-die approach to selling makes it a formidable competitor. However, Juki has many of the same distributors as Speedline, and it is unclear that they will give up the latter for a new player.

But the real prize may be the emerging South America market. As Juki CEO Bob Black told us, “In Latin America, out major competitors are offering complete lines. To be competitive, we need to do the same.” And Juki has the breadth and depth in its service department that many standalone reps have not.

Keep an eye on this.