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 [email protected].

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 [email protected].


Dr. Ron

Calculating Confidence Intervals on Cpks

Let’s look in on Patty, it’s been awhile.

Patty was looking forward to sleeping in.  Normally she was up very early, sometimes before 5:30 am, after usually getting to bed too late, so she was looking forward to an alarm set at 7:45 am. The kids were off from school and Rob was taking them skiing, so all agreed a 7:45 am wake up time was reasonable.  Since she had no early meetings, her scheduled 9 am arrival at her Ivy University office was also in the cards.

Patty was sleeping soundly when she heard her seven-year old twin sons shouting, “Mom! Dad! Come quickly.”   At the same time, their two-year old beagle, Duchess, started barking.

Her heart pounding, Patty raced to the racket now being produced by this energetic trio.  As she arrived she saw her sons and Duchess looking out of their back window to see a beautiful female deer eating from their bird feeder, just 30 feet away. The entire family was involved in a bird counting exercise and had noticed, several times, that the bird feeder was “wiped out” overnight. This mystery was now solved.

The entire family agreed that it was hard to be angry at the doe, as deer are such beautiful creatures.

Figure 1.  A Female Deer at the Bird Feeder at Patty’s House


It was 6:15 am and it didn’t seem to make sense to go back to bed.  So, Patty stayed up and was off to Ivy U in less than 30 minutes.

Patty had a rather light week as she had guest speakers for her two lectures.  However, she was sitting in for one of the engineering school’s senior professors later in the day.  This fellow prof had asked her to sub for him as he was called to an emergency meeting overseas.  Her topic was manufacturing processes; one with which she felt very comfortable.  But, she had to admit to being a bit nervous sitting in for one of Ivy U’s most famous professors.

As was her usual practice, Patty checked her email first.  After going through the first 5 or 6, she saw an email with the subject header, “Ivy U Professor Wins Prestigious Queen Elizabeth Prize for Engineering.”  As she opened the article, she was stunned as she saw a photo of the professor for whom she was substituting later in the day.  The article went on to explain that this prize was like the “Nobel Prize” for engineering.

As she finished her emails she was relishing the thought of having a less hectic day and week ahead.  Maybe she would even have time to read the Wall Street Journal during a relaxing lunch.  Suddenly, her phone rang, startling her a little.  She picked up the receiver to hear a familiar voice.

“Professor Coleman, this is your most faithful student Mike Madigan,” Madigan cheerfully said.

Madigan was CEO of ACME at large electronics assembly contractor. Patty worked at ACME before becoming a professor at Ivy U. Her husband, Rob, and sidekick, Pete, were also ACME employees, but were now all at Ivy U.  Pete was a research assistant and Rob was just becoming a research professor.  Although they all enjoyed their time at ACME, they were much happier at Ivy U.  All three had a part-time consulting contract with ACME and Madigan was typically their main contact at their former employer.

“Mike! What’s up?” Patty said cheerfully.

“We are evaluating a new solder paste and I’m concerned we might make a mistake if we switch,” Mike responded.

“How so?” Patty asked.

“Well, we agreed that consistency in the transfer efficiency (TE) of the stencil printed deposits was the most important criteria,” Madigan began.

“That sounds reasonable as most of our past work has shown that a consistent TE is a strong determinant of high first-pass yields,” Patty responded.

“Right! But the difference between the pastes is only two percent. The old paste has a Cpk of 0.98 and the new paste 1.00,” Mike went on.

“I sense there is more to the story,” Patty suggested.

“Yeah. The new paste has a poorer response to pause,” Madigan said.

“Yikes!” Patty almost shouted.

Patty had shown, time and time again, that poor response-to-pause in the stencil printing process can hurt productivity and lower profitability considerably.

“My sense is the two percent difference in Cpk, might not be significant,” Mike suggested.

“Mike, I think you are on to something. What printing specs were you using and how many samples did you test?” Patty asked.

“The lower TE spec was 50% and the upper 150%. We tested 1,000 prints,” Madigan answered.

“Let me do some homework and I’ll get back to you,” Patty said.

“One problem. Can you get back by 3 pm today? The new solder paste supplier is coming for a meeting at 4PM and is pressing us,” Mike pleaded.

“OK. Will do,” Patty said, sighing a bit.

“There goes my somewhat relaxing day,” she thought.

It was a good thing she had already prepared her lecture and that it was scheduled for 4:30PM.

For several hours Patty thought and searched through some textbooks on statistical process control.  Finally, she came upon the solution to the problem in Montgomery’s Introduction to Statistical Quality Control.

“Perfect!” she thought.

She did finish early enough that she could read the WSJ over lunch, marveling, as always, that she was the only person her age that enjoyed reading a real newspaper.

She called Madigan at 3 pm.

“Mike, I think I have your answer.  I found a formula to calculate the confidence intervals of Cpks,” Patty started.

“And the answer is?” Madigan asked expectantly.

“The Cpk 95% confidence interval on the new paste is 0.95 to 1.05, however the old paste is 0.93 to 1.03,” Patty began.

“So, even I can sense that they aren’t different,” Mike commented.

“Yes, since the confidence intervals overlap, they are not statistically different,” Patty agreed.

Figure 2. The Confidence Interval of the Cpk on the New Paste is 0.95 to 1.05.


They chatted for a while and Madigan asked if Patty could join the first 20 minutes of the meeting by teleconference.  It was a bit close to her lecture start time, but she agreed.

Patty had met Madigan’s son at West Point when she visited there to be an evaluator for a workshop two years ago.  She decided to ask how he was doing.

“Mike, how is your son doing at West Point?” she asked.

“Thanks for asking. He is now a Firstie and was in the running for First Captain, but he just missed it.  It’s a good thing he takes after his mom,” Madigan proudly responded.

“Wow! That’s great,” Patty replied.

“I have to admit though, my wife and I are a bit nervous. He has chosen armor as his branch and there is a good chance he will see combat sometime in his career,” Madigan responded with a bit of concern in his voice .

They chatted for a while more and Patty was touched to see so much humanity in Mike Madigan.  He seemed much changed from his gruffness of earlier years.


Dr. Ron

As always, some of this story is based on true events


Cpk is Still King in Evaluating an SMT Solder Paste Printing Process


If you think about it, to evaluate any process you typically want to know its precision and accuracy. Look at the dart players in the Figure 1 below. The yellow player has good precision, but his accuracy is off. The green player has such poor precision, it is hard to tell if his accuracy is good. The yellow player will typically be easier to correct, as she just needs to change her aiming point.

Figure 1. The yellow player has greater precision. She only needs to change her aiming point.











Recently I was asked to evaluate several solder pastes to determine which printed better. We used transfer efficiency (the volume of the stencil printed solder paste “brick” divided by the stencil aperture volume) as the evaluation metric, expressed in percent. So 100% would be the target. The lower specification limit we choose was 50% and the upper specification at 150%.

Figure 2. Data from Pastes A and B.


A good result would be an average of 100% with a “tight” distribution. The “tightness” of the distribution being determined by the standard deviation. Figure 2 shows data from two pastes. Note that Paste A has an average of 100% and a standard deviation of 16.67%, whereas Paste B has an average of 80% and a standard deviation of 30%. Clearly, Paste A is superior to Paste B in both accuracy and precision. But what is the best way to express this difference? Is there one metric that will do it? Cpk is the answer.

Cpk is one metric that is sensitive to both the accuracy and precision. Cpk is defined as:




Where x is the average and S is the standard deviation.

Using these equations, we see that the Cpk of Paste A is 1.0, whereas the Cpk of Paste B is 0.333. Note that Paste B has a significant number of data points (about 17%) outside of the specification limits, however, Paste A has almost no data points out of specification.
So when evaluating most processes, Cpk tells it all!

Dr. Ron


In Electronics Manufacturing, Does Cpk =1 Yield 66,800 DPM?

As Patty was walking past the Professor’s office on her way to see Pete and Rob, she decided to drop in.

“Professor, I got the strangest phone call. A man claimed he had invented a machine that could create energy,” Patty began.

“Tell me about it,” the Professor chuckled.

“Well, he correctly noted that, when he took his kids to the beach, a submerged beach ball pushed up with a lot of force. So, he developed a technique to extract the energy produced when the ball is released,” Patty explained.

“Let me guess,” the Professor offered. “He then developed a technique to continuously extract energy; an energy producer of sorts.”

“Exactly! How did you know?” Patty responded.

“Well, I have been here about 40 years, and I have had forty such calls,” the Professor said.

“Tell me the details of your call,” he continued.

“There would be a box of small mass with a generator and pump inside; the generator and pump occupying little of the volume of the box. The box would be filled with water at the top of a lake and would then would sink to the bottom. Once the box was at the bottom, the water would be pumped out and the buoyancy would cause the box to rise. A rope would guide the box on its up and down journey and the generator would spin as it travels up the rope, hence generating electricity. The cycle would be repeated over and over and, in a sense, become a power plant,” Patty explained.

“And the problems are?” the Professor asked.

“I told him that it violates the laws of thermodynamics, and that I could make some calculations that would show that it would not work. Basically, the amount of energy required to pump the water out is greater than what the buoyancy would generate, considering friction, etc.,” Patty replied.

“His response?” the Professor led.

“My sense is that he thought he could make it work, in spite of the physics,” Patty answered.

“In my experience, that is always the response. Probably my most troubling experience was a chap who convinced a small venture capital firm to advance him about $3 million. He had a machine that, he claimed, continuously extracted energy out of the earth’s magnetic field. The biggest shock to me was that the leader of the venture capital firm was a graduate engineer who had retired as COO of a Fortune 50 company. I still haven’t figured out how such an accomplished person could not see that an energy-producing machine is not possible,” the Professor expounded.

“What was the upshot of all of this?” Patty asked.

“Well, they didn’t pay my consulting fee when I explained how it couldn’t work,” he chuckled. I checked a few months ago and the company’s website is down,” the Professor replied.

“The people that are into this folly don’t even realize that, if an energy-creating machine could be made, it would be the greatest discovery in history,” the Professor went on.

After a few more minutes of this discussion, Patty resumed her short walk to Pete’s office. Rob was already there.

“Looks like Mike Madigan needs us again. Did you see the email he sent us?” Pete asked.

“No, what’s up?” Patty and Rob said in unison.

“Something about Cpk,” Pete answered.

Patty reached for the phone to set up a conference call to Mike.

As she dialed, Patty admonished, “Now remember you two, good manners. No laughing at any of Mike’s questions.”

“Yes, ma’am,” Pete and Rob said in unison.

Mike’s secretary answered and said she would put them right through.

After a few pleasantries, Mike got to the point.

“Remember the tolerance analysis and specification that you did for passive resistor and capacitor length?”  Mike began.

“Yes. We were all involved in that project,” Patty answered.

“So, it is a Cpk = 1, or a Three Sigma spec, right?” Mike asked.

“Sure,” Patty, Rob, and Pete answered in unison.

“So, what percent of parts should be out of spec?” Mike asked.

“Let’s see … Three Sigma is 99.73% of parts in spec … so that would be 0.27% out of spec,” Pete calculated.

“Well, they are shipping us 5% out of spec parts and claiming they are better than Three Sigma, or a Cpk of 1, because they used a recently published graph, that said a Three Sigma, or Cpk = 1, process was 6.68% of parts out ot spec. I just sent it to all of you,” Mike said.

Pete opened the email and showed it to Patty and Rob.

“I’ll be darned! It does say that a Cpk = 1, or Three Sigma, has a defect rate of 66,800 defects per million or 6.68%,” Rob groaned.

“I’ll bet it has to do with the definition of ‘Six Sigma,’” Patty opined.

A look of recognition came over Pete and Robs eyes.

“What do you mean by the definition of ‘Six Sigma?’” Mike asked.

“We have all heard people claim that ‘Six Sigma’ is 3.4 ppm out of spec. Actually that’s a 4.5 sigma process. This definition allows a drift in the average of 1.5 Sigma that knocks the Cpk down to 1.5.  True Six Sigma is a Cpk = 2 and is 0.002 ppm parts out of spec,” Patty replied.

“I’m a bit confused. But, let me show you some of the length data for 0402 passives,” Mike said.

“We measured them metrically so the length should be 1mm +/-0.1, Three Sigma.  Instead, it is more like 1mm +/-0.1, Two Sigma. That’s a little more than 5% outside of the spec,” Mike continued.

A Minitab Analysis of the 0402 Length Data.

“Give us some time to sort it out,” Patty suggested.

Is a Cpk of 1, or a Three Sigma, process really 66,800 ppm (6.68%) out of spec?  Will Patty and the crew figure out what’s going on?

Stay tuned…


Dr. Ron

An Example of Cpk and Non Normal Data In Electronics Assembly Soldering


Let’s see how Patty is doing after teaching her first class at Ivy U…

Patty arrived at home after teaching her first class at Ivy U and she couldn’t contain her excitement. For the next couple of hours her husband, Rob, had to politely listen to her talk about how amazing it was to teach the young, bright, enthusiastic, future engineers.

Time went quickly and, before she knew it, she was standing in front of the class for her second lecture.

Patty reviewed quickly the fact that it was incorrect to average Cpks and that, to calculate a Cpk, the data should be normally distributed.

“The question was asked last time, how one can tell if the data are normal? Minitab can be used to plot the data on a normal probability plot. By eye, we can get a good sense if the data are normal or not. In addition, Minitab will perform various tests, one of them being the Anderson-Darling Normality Test,” Patty began.

“Let me show you some real data to demonstrate this,” Patty continued.

“When assembling a smartphone like the new Druids, the mechanical tolerances for the many tiny capacitors and resistors are very precise. One common capacitor size is only 0.6mm long.” Patty paused as she saw a student’s hand raised.

“Yes, Martin?” Patty asked.

“Professor, you mean 0.6cm, right?” Martin asked.

“No, 0.6mm,” Patty answered.

Patty’s answer caused quite a bit of murmuring, finally Patty had to ask for order.

“Why is everyone surprised?” Patty asked.

“Professor, that is much smaller than a grain of rice, it is more like the size of a grain of sand,” Alison March responded.

“This is fun,” Patty thought, “and good timing.”

Patty showed a slide of passives on a match head and passed around a teardown of Druid smartphone with a magnifying glass so that the students could see how small the passives were.

It took a while for the class to calm down.

Patty then said, “And about 200 to 500 capacitors and resistors of this size are individually placed and soldered in the electronics assembly process in each smartphone.”

The student’s mouths were agape.

“OK, let’s discuss how these little rascals relate to SPC. My company orders billions of these electrical components each year. We are sent a sample lot to approve a larger order. For the components of interest, we already know that the mean length is 0.6mm, the 3 sigma (standard deviation) tolerance is +/-0.03mm. So, the lower spec limit is 0.57mm and the upper spec limit is 0.63mm. This equates to a Cpk = 1.00,” Patty went on.

Patty put a PowerPoint slide up that showed the data.

“The data for the sample lot is on the left. What is the problem?” Patty asked.

Charles Parsons raised his hand.

“Yes, Charles?” Patty asked.

“Well, Dr. Coleman, the standard deviation for the data is 0.15 and the resulting Cpk is only 0.67, so the targets are not met” Charles answered.

“Precisely,” Patty replied.

“We then went back to the supplier and asked them to fix the problem. The graph on the right is from the capacitors they sent us 3 weeks later, after they claimed to have solved their manufacturing process problems,” Patty explained.

“What do you think?” Patty asked.

There was a lot of murmuring. Finally DeShaun Martin raised his hand.

“Yes, DeShaun?” Patty acknowledged him.

“Well, Professor, it looks like they simply sorted out the parts to make the sigma lower and Cpk higher,” DeShaun responded.

“Precisely,” Patty said.

“I don’t see what is wrong with sorting,” Sandy Lisle commented.

“You see from the Druid smartphone that I passed around that it is so densely packaged with components that there is hardly any room in it. To achieve this density we have to perform tolerance analyses to assure everything will fit. In all of these analyses we model with normal distributions. With a sorted distribution we will likely have more tolerance interferences,” Patty answered.

“Look at the red arrow. There will be an excess of components with this size and a lack of components of the size where the green arrow is pointing. These differences will cause some tolerance interferences with the pads on the printed wiring board where the passive will be assembled,” Patty continued.

“Can you review? How we can tell that the distribution on the right is not normal?” Conor Stark asked.

“Sorry, I almost forgot. Look at the normal probability plot for the sorted data. Note how it diverges from the straight line on the ends. Also the Anderson-Darling value for p is <0.05. These two criteria are cause to reject the hypothesis that the data are normal,” Patty finished.

Patty was just wrapping the class up when someone raised their hand.

“Yes, Natalie?” Patty asked.

“What was the final outcome?” Natalie asked.

Patty chuckled, obviously she should share he result of this adventure.

“Oh, yes. We could not use the parts for the Druid smartphones, but they were OK for some toys we were assembling. In addition we insisted on a 30% discount, since the passives did not meet the specification,” Patty answered.

As she was cleaning up, two of the female students came up to Patty. One them, Justine Randall spoke for the two.

“Professor Coleman, you are an inspiration for us. We hope in 20 years we can be just like you,” Justine said with emotion.

Patty was indeed touched, but as she left the classroom, she decided that she had to start dyeing her hair.


Dr. Ron


Cpk Can Only be Calculated from Data that are Normally Distributed


Let’s look in on Patty….

Patty was really nervous. As a matter of fact, there was no time she remembered being this nervous. The cause of her nervousness? She was going to teach a series of classes at Ivy University.

This opportunity came about because one of the professors at IU was in a serious accident. A full recovery was expected, but there was no way the prof could finish the last three weeks of the statistics course she was teaching. Things are hopping in the Engineering Department at IU, and the Dean could not find another prof to take over.

The Professor himself was too busy, however, when the Dean asked his advice he immediately recommended Patty. Patty was honored by the request and would be more so if she knew the behind-the-scenes story. IU has an unwritten rule that all teaching profs had to have a Ph.D., which Patty did not. However, if Bill Gates wanted to teach, an exception would be made, as he is a world class technical executive. Patty was hired under that exception. She was stunned to see she was on the front page of “The Ivy U Review,” under the headline, “Famed Executive to Teach at IU.”

Well, her first class was tomorrow and she took comfort in the fact that her husband, Rob, told her she shouldn’t be nervous. Pete wasn’t much help. He told her he would be so nervous that he wouldn’t be able to eat. It was just unnerving teaching the best and the brightest. She was proud of her academic accomplishments at Tech, but this was IU, arguably one of the top 10 universities in the country.

“Rob, I have to tell you, even though I’m still nervous, it comforts me to know that you wouldn’t be nervous,” she said to her husband.

“I never said I wouldn’t be nervous, I said you shouldn’t be nervous. After all, you’re Patty Coleman,” Rob replied.

At this Patty burst into tears and Rob came over and gave her a big hug.

“But, you’re smarter than me,” Patty insisted.

“No way,” Rob replied.

They then spent the next 10 minutes arguing that the other was smarter. Patty always felt she had a good business sense, but for understanding deep technical things, she believed Rob was her superior. After a while they looked at each other and laughed.

“Not too many couples would get in to an argument, saying that the other person was smarter,” Rob teased.

Time passed quickly and Patty was soon in front of the 35 students in the class. The topic was Cpk as the most important metric to determine the quality of a lot of material or product. She asked the students if it was OK to average Cpks from different lots. A student raised her hand.

“Yes Emily.” said Patty. (Patty had ask the students to use nametags.)

“No, Professor Coleman. One can’t average Cpks. The reason being that Cpk goes as one over the standard deviation and standard deviation is a squared term. So one can’t average two lots and get the same result as taking the Cpk of all of the two lot data.

Patty responded, “Emily is 100% correct. Remember, when you get out in the working world, to always check to see that your suppliers are not averaging Cpks. This might happen when half of the lot is below the spec, say the Cpk is 1.4 and the spec is 1.5 and the other half is over the spec, say 1.7. The supplier will say that the lot is in spec because the average Cpk is 1.55. This isn’t necessarily so, as Emily points out. You should only accept a calculation of Cpk for the entire lot.”

Patty chuckled that Emily thought she was a Professor, at Ivy U. Right!

The class continued to go well and Patty began to relax. As the class began to end, she mentioned another important point.

“What important criteria must the data have to be considered acceptable to calculate Cpk?” Patty asked.

There was a bit of murmuring and finally a boy (man?) raised his hand. He looked 12 years old to Patty.

“Yes, William,” Patty acknowledged him.

“Dr. Coleman, I think the data must be normal,” William answered.

“Dr. Coleman?” Patty thought.

“Absolutely correct, William,” Patty responded.

“Class, remember this point when you get out into industry. Almost no one checks to see that the data are normal before calculating Cpk. The data must be normal to calculate Cpk. I can’t tell you how many times I have rejected a lot of incoming material because the Cpk was calculated from non-normal data. In some cases non-normal data can be transformed so that the data are normal, “ Patty continued.

“Professor, how do we know if the data are normal?” a student named Kathy asked.

“Stay tuned for the next lecture,” Patty chuckled and dismissed the class.

As she was gathering her laser pointer, lap top etc., a number of students came to talk to her. Emily was with a group of about six of them.

“Professor, we just wanted to tell you that we are thrilled to have you as our instructor. We appreciate your practical, real world perspective on statistics,” Emily said.

Patty responded warmly and was close to being choked up by this show of respect and appreciation. She decided she would walk to The Professor’s office to tell him how it went.

On the way out, she heard one of the male students say to his friend, “You know, she is quite attractive for an older woman.”

Patty didn’t know whether to laugh or cry.


Dr. Ron

The Law of Averages


Patty had been working with engineering on a new product that needed a very precise and controlled volume of the stencil printed “brick” of solder paste on the PWB pads. The product had many 01005 passives and CSPs with 0.030″ spacings and the application was “mission critical.” So solder joint integrity was critical. The critical factor in obtaining this solder joint integrity was a consistent volume in the stencil printed brick. Her favorite solder paste gave a Cp and Cpk of 1.5 in 500 prints. The upper and lower spec limits were 60% and 140% of the aperture volume.

Purchasing called to tell her that XLK Company just announced a solder paste with a Cp and Cpk greater than 3, under the same printing conditions that this product required. Needless to say Patty was skeptical. When she looked at the report, she groaned. The data were collected by Mort Bittler. She had seen him give several presentations and he always seemed to misrepresent the data to make his company’s solder paste look better than it was. She was on her way to a team meeting and expected that this new “break through” would be discussed.

As the meeting came to order, the VP of Engineering, Todd Hamilton, spoke.

“I saw this new data from XLK with a printing Cpk = of 3.72, we will use this paste,” Todd commanded.

“Wait a minute,” Patty responded, “the decision on which solder paste to use is with my group.”

“But your group has dropped the ball. How could you not know about this superior paste?” Hamilton challenged.

“We have evaluated their pastes continuously, they have always been second rate,” Patty shot back.

“Well things have changed. Get with it Coleman; this project is too important,” Todd shouted.

Patty was really angry. Technically Todd was her superior, but she found his attitude and words insulting. Using her last name was a bit unfriendly too. “I’ll travel to XLK tomorrow and review their data,” Patty responded, her voice shaking more from anger than anything else. She called Mort Bittler and he was available, so he agreed to meet with her the next day.

As she hung up, Pete showed up at the door.

“Hey kiddo, how’s it going?” Pete asked.

“You were at the meeting, so what do you think? Hamilton impugned all of us,” Patty said flatly.

“Any way I can help?” Pete asked. “Why don’t you go with me to XLK tomorrow, it might be good to have two people check the data.

Fortunately XLK was only 120 miles south of their southern New Hampshire office. Pete had become one of her best friends in the past year. They spoke in Spanish the whole way to XLK to get their skill level up. Patty had also taught Pete some Mandarin, but it was slow going.

After 120 minutes of discussing the PGA Tour vis a vis Tiger Woods, in Spanish, they arrived at XLK. Mort was waiting. Mort was 45 years old, with a thick Boston accent. He came across as being knowledgeable … to someone who wasn’t knowledgeable. After brief pleasantries, Patty asked to see the raw data.

“Patty, I already made the calculations, why do you need to see the raw data?” Mort asked.

“The Professor always told me to ‘look at the raw data,’ ” as often one can glean things that the final calculations don’t show,” Patty answered evenly.

“Well, maybe later. Let me show you how we took the data first,” replied Mort evasively.

Patty and Mort went to the printing lab and Patty noticed that Pete was not with them. After verifying that the printing process was reasonable, Patty asked if she could have a little time with Pete … if she could find him.

Patty and Mort found Pete in the break room. “Pete, let’s pow-wow for a while,” Patty said. Mort said he would go answer some emails and they would meet in 30 minutes.

“Pete, where have you been? You’re not going to embarrass me again are you?” Patty pleaded.

“Me embarrass anyone?” Pete sheepishly replied. “I found the person who took the data, Beth Thompson,” he went on, “and she told me they average Cpks.”

“Not again,” Patty groaned. “We just went through that with a vendor last week. When will they learn that it’s wrong to average Cpks?”

In 30 minutes they went to Mort’s office. All agreed to go lunch. After ordering, Patty asked, “Mort what are your thoughts on averaging Cpks?” Mort seemed defensive, and squirmed a little before he finally he said, “seems OK to me, it’s just like averaging golf scores.”

“What about the nonlinearity of the standard deviation in the Cpk equation?” Patty asked.

Mort was clearly not grasping the issue, so Patty continued, “If you have two sets of data and calculate the Cpk of each and average them, you will not get the same result as if you calculated the Cpk of the data added together. One of the reasons is that the standard deviation is nonlinear. For the same reason it is wrong to add Cpks together.”

Then Patty came right out and asked, “Did you average the Cpks?”

“Yes,” Mort said glumly. “Let’s look at the data when we get back from lunch,” Patty insisted.

When they looked at the data, it showed Patty’s point, four runs, of 100 samples each, had Cpk’s of around 1.2 to 1.3 and one run had a Cpk of 15.56. The average Cpk was 3.73, but if one takes the data together, the Cpk is 1.58.

Patty had calculated the total Cpk on the spot with Minitab (below).

Cp Cpk
Run 1 1.26 1.23
Run 2 1.3 1.3
Run 3 1.39 1.39
Run 4 1.21 1.2
Run 5 15.56 15.54
Average 3.73 3.72
Altogether 1.59 1.58

The correct results, calculated by Patty, are in the last row.

“But the 1.58 still is quite good,” Mort pleaded.

“But the data suggest that run 5 is a fluke; it is clearly not from the same population as runs 1-4. Let’s go out to the lab and run another 100 data points to see if we can reproduce run 5,” Patty insisted.

They ran another 100 and the Cpk was 1.28. On the way over Pete whispered in Patty’s ear that he had more vital intel to share with her on the way home. With the end of the data collection, Patty and Pete were done, so they headed home.

“OK, what is the vital intel you need to share with me? she asked Pete in Spanish.

“While you were collecting data with Mort, I visited Beth again. She told me that Mort had her collect 150 data points on run 5 and he threw out the 50 points furthest from the mean. You were right, run 5 was from another population, a cheating one,” Pete chuckled.

“Well, I guess we will still use our favorite solder paste,” Patty summed up.

Best Wishes,

Dr. Ron