Does Solder Paste’s “Five Ball Rule” Remain Valid in SMT Today?


My good friends, Phil Zarrow and Jim Hall, in their audio series “Board Talk,” were recently asked about the “Five Ball Rule”. In the comments section for this session, one listener asked if this rule, created in the 1990s, was still valid. After all, the 1990s was the era of 0603 and 0402 passives; 01005 and even 008004 passives have arrived.

First, let’s consider what a “rule” is verses a “law.” As an example of a law, consider Newton’s Laws of Motion. At everyday speeds, these laws are shown to be accurate to within our capability to measure. As we will recall from Physics 101, these laws were superseded by Einstein’s Theory of Relativity, at speeds close to those of the speed of light. However, in our everyday world, Newton’s Laws are well … laws. They are, for practical purposes, exact.

What is a “rule” then? A rule is an expression that approximately fits some empirical data or the experience of experts. Moore’s Law is actually a rule, as it is not precise. The doubling of transistor density has varied from every 18 months to every two years. That’s why I call it a rule, a very useful rule indeed!

The “Five Ball Rule” is clearly a rule. It was likely developed a generation ago by some of the first SMT pioneers. It may be backed up by experiment, but I think it was likely more a consensus of SMT industry authorities from the 1980s and 1990s.

What is the “Five Ball Rule?” It states that the solder paste’s largest solder particle diameter should be such that at least five of these particle diameters would span the width of a rectangular stencil aperture (Figure 1).

Figure 1. The Five Ball Rule








When this rule was developed, stencil apertures were much coarser than today, and the finest solder powder was a Type 3, with Type 4 on the horizon. While it is true that stencil aperture widths are much finer today, solder pastes of Type 4.5, 5, and even 6 are now in use.

The particle sizes of different “Type” solder pastes are shown in Figure 2. Note that, for Type 4 powder, 80% by weight of the particle diameters are between 20 and 38 microns. 38 microns is considered the “largest particle.” So, from Figure 2, for Type 5 powder, the “largest particle” is 25 microns. For the sake of the Five Ball Rule, the “largest particle,” for each powder type are those shown in Figure 2.

Figure 2. Solder Powder Sizes.










So, is the Five Ball Rule still valid? It would be hard to argue that it is not. Hundreds of experiments have been performed using the Five Ball Rule, combined with the aperture ratio being >1.5 for rectangular apertures or the area ratio being > 0.66 for square or circular apertures, with successful results.

StencilCoach software now includes the newer (finer) solder powder sizes to 1) tell the user the fineness of solder paste powder for the Five Ball Rule, as well as 2) help with calculating aperture or area ratio. By the way, some have suggested that, for a square or circular aperture, an “Eight Ball Rule” is more appropriate. So, StencilCoach uses the Eight Ball Rule for such apertures.


Dr. Ron


5 Traps to Avoid When Changing from Hand to Machine Assembly

In the past, it was usually pretty easy to find chips in both surface mount and through-hole packages. Somewhere in the past decade or so, component manufacturers stopped introducing through-hole versions of their newest chips as standard practice. In many cases, new components can only be found in tiny QFN (quad flatpack, no leads), or wafer-scale BGA (ball grid array) packages.

The maker community, never shying away from a good hack, found ways to work with many of these parts while still hand building. There are very few components used in the pro-design world that are still unusable by a creative DIY maker.

But what happens when a maker has a great design and wants to mass-produce it?

Sometimes the techniques that make things work when hand soldering, will completely break a machine assembly process. To cure that ailment, I’ve compiled five common traps to avoid when moving from hand to robotic assembly.

5. Consider moisture sensitivity. 
It may not seem logical, but plastic does absorb moisture. And, it doesn’t have to be dropped in the sink for it to happen. Just sitting around exposed to the air, plastic chips will absorb humidity. In a reflow oven, these parts can end up acting a bit like popcorn. The moisture turns to steam, and if it can’t outgas fast enough, may split the chips open. Often, the damage isn’t visible to the naked eye, but with show up as an unreliable product in the field.

When we DIY folks hand-build boards, we tend to open the component packages and then just let the parts lie around without giving thought to proper storage. If you are going to send your project off to be machine assembled, you can do two things with moisture sensitive parts.

First, order the parts when needed, not before, and keep the packages sealed. Alternately, you can send in parts that have been exposed to the air; if you inform your assembly house that the parts are moisture sensitive, and ask that they be baked prior to assembly. Prebaking will remove the moisture safely.

4. Don’t skimp on solder mask. Some board fabricators offer reduced prices if you order your boards without soldermask or silkscreen. That’s not a problem when you’re hand building — you can regulate the amount of solder by eyeball.

However, when a stencil is used to apply solder paste and the board is run through a reflow oven, the solder will spread back on the exposed copper traces. This may leave your parts without enough solder on the pins to create a reliable connection.

Solder mask may add a bit of cost up front, but will increase reliability and reduce cost in the long run. Creative choice of solder mask color can also add some personality to your boards.

3. Silkscreen is important too. Lack of sIlkscreen isn’t a reliability issue, but it can make accuracy of assembly more difficult to achieve. In a perfect word, the CAD files would tell the assembly machines exactly where each part is supposed to go and what angle and orientation it needs.

Unfortunately, we don’t live in a perfect world (who knew?). It’s far too common to have footprints with errors in them, or components with ambiguous marking, to depend on the CAD files alone. Clear silkscreen will help to ensure that any errors in the data are caught visually.

If you don’t want to clutter your PCB with reference designators and polarity markings, put the designators and any other important markings in the document layer in your layout software. Then, tell your assembly house to look on that layer for the information.

2. No need to fear surface mount. One of the easiest ways to ensure that a board can be hand-built is to stick with through-hole parts. But doing so puts many limits on a design, and rules out a lot of new technologies.

Little breakout boards — a small surface mount chip pre-mounted on a PCB, with hand-solderable headers — are available for a lot of new parts, but not all. That’s helpful, but they take up a lot of extra board real estate and cost more that the part alone.

If you’re hand building a prototype, or a small number of boards for your own use, go ahead use a breakout board. But, when it’s time to get a thousand built up to sell, re-layout your PC board to use the chip without the breakout board. Just don’t forget the bypass capacitors or any other required support components.

As a bonus, many breakout boards are open source, so you may be able study and use a proven schematic and layout for that part of your design.

1. No open vias in pads. QFNs and BGAs have pins/pads under the part, often completely inaccessible. That’s fine for a reflow oven, but what if you’re soldering it by hand?

A common hand-soldering practice is to put large vias in the pad. Fix the part onto the board with tape. Then, turn the board over and stick solder and a small tipped soldering iron through the via. By doing this, you can hand solder almost any leadless surface mount part.

You can probably guess that I’m going to tell you open vias in pads will not work with automated assembly. The solder will flow down the via and end up on the back side of the board. You may end up with shorts on the back side, and parts that fall off of the front side, or just don’t connect with all their pads.

If you use the open via hand solder technique, you’ll need to re-layout your board without any open vias in the pads before sending it for manufacture.

0. Go for it! It wasn’t that many years ago when the tools and services necessary to get an electronic product manufactured were so complex and expensive as to pretty much make it impossible for DIYers to turn a hobby project into a small business. Times have changed, and with those changes, the hardware startup is back — and within just about anyone’s reach.

Duane Benson
Breaker, breaker, one nine, clear the line, we’ve got boards to build

To Minimize BTC Voiding, Start with the Right Solder Paste

Let’s see what’s up with Patty ….

Patty was just dropped off at O’Hare airport after finishing a 3 day workshop on Lean Six Sigma statistics, design of experiments, and statistical process control. Interestingly, the students were lawyers. In recent years more and more service-based organizations were adopting lean Six Sigma and it was a long time since Patty had taught such a workshop to engineers. She noted that although the lawyer’s math skills were a bit rusty, they were very good listeners and picked up the math behind lean Six Sigma topics very quickly.

After paying the cab driver, she entered the terminal and went to see an agent. She was early enough to get an early flight home, so she had called the people at the online ticket agency during the cab ride. They said the change fee would be over $300, she felt that was just too much to pay. She was delighted to see that it was only $75 at the terminal.

She looked at her paper boarding pass and saw that she had more than two hours, just enough time for a relaxed lunch at Wolfgang Puck while she read USA Today. Patty was the only person her age that she knew who enjoyed reading a paper newspaper, she guessed that she picked the habit up from her dad.

The two hours went by quickly and she was standing in line waiting to board the flight to Boston’s Logan Airport. She had now been at Ivy U for a few years and traveled much less than when she worked at ACME. She had forgotten how stressful and unpleasant traveling was. As she stood in line, the man in front of her put his smartphone on the scanner and the scanner could not read the QC code. He and the agent fumbled for a while before they got it to work. This was another place where, in her opinion, paper was still king.

Patty got on board and settled into her middle row seat. She groaned a little bit at how uncomfortable and cramped it was. Patty was reminded of what her dad used to say in situations like this; “I know it is a bit uncomfortable, but just think what the 49ers went through to get to California,” he would tease.

After takeoff, she turned on her laptop. She absolutely had to send some emails, so she signed on to the onboard WiFi. She got sticker shock when she saw that it cost $18.95!  Even though Ivy U would pay for it, the high price galled her.

After she finished the emails, a wave of fatigue swept over her and she needed a break.  She chuckled to herself when she thought of a recent event. She had taken two of her best teaching assistants (TAs) to lunch and the conversation somehow came to discussing people who hid Jews from the Nazi’s in World War II. Patty mentioned to her two young protégés about an excellent book and movie she read and saw as a teenager, The Hiding Place. The story is about Corrie Ten Boom and her family and how they hid, and hence saved, many Jews from the Nazis in Holland during WWII. Although the movie was made before she was born, it was shown at Patty’s church every few years, for the new sets of youngsters who came along. Patty mentioned to her two superstar TAs that the film was produced by Billy Graham’s organization.

“Who is Billy Graham?” they both asked in unison.

Patty struggled to keep her composure as she explained who he was. How could they not know this?  She decided to examine the situation a bit further.

“OK, you two. Who was Mickey Mantle?” Patty asked.

The youngster’s both looked at each other.

“We have no clue,” they chuckled.

Patty though she would try a few more, “Nikita Khrushchev?”


Roy Orbison?”


Patty started humming a few bars of Orbison’s most popular song.

“Oh, Pretty Woman,” the boys said in unison.

Patty thought to herself, “Each of these young lads are the best student in every class that they take and yet they don’t know these ‘celebrities’?”

The next day Patty arrived at her office early to meet with Rob and Pete to discuss how the presentations that they were making for Mike Madigan on voiding were coming. Patty had arrived so late the night before, that Rob was already asleep. She did not see him in the morning as it was her turn to get the boys ready for school and he was off early to get in his 90 minutes of exercising. So, they had no chance to discuss the progress of the presentation.

“Pete, your presentation of BGA voiding is terrific. How is my hubby doing on BTC voiding?” she chuckled as she looked at Rob.

“I feel like I’m going to get yelled at ’cause I didn’t do my homework,” Rob said sheepishly.

“Yikes! We only have a few days,” Patty responded. “And I have yet to do my part on using solder preforms to minimize voiding,” she went on.

“I’m only teasing. I have quite a bit of info,” Rob said.

“We have been out of the mainstream for a while and one thing is for sure, voiding is the number one issue among assemblers today.  So many people are assembling QFNs and are struggling with voiding. Voiding with some solder pastes can be over 50% of the area,” Rob went on.

“Wow! With 50% voids, think of how poorly the heat is being transfer away for the BTCs,” she looked at Rob and chuckled. “Remember, ‘BTC’ not ‘QFN,’ Patty went on.

“Yes ma’am,” Rob jokingly replied.

“Can you imagine the effect on reliability and field issues with so little heat being removed? The ICs inside the BTCs must be frying” Pete added.

“Voiding at this level has got to be really costly,” Patty mused.

“One of the things that really helped me was that I found quite a few experiments on voiding,” Rob added.

“What were some of the key points?” Pete asked.

“Well, as you might expect, the solder paste is typically the most critical part of the process. Some pastes have voiding lower than 10% with others above 50%,” Rob replied.

“What about the process?” Patty asked.

“Well, the reflow profile can be very important, as is controlling the PWBs and components. But, with the best pastes, it has been found that you can control the voiding content even if you can’t change the reflow profile and the PWBs and components have some issues,” Rob responded.

“Look at the x-rays of poor and good voiding between two pastes,” Rob said.

“What a difference,” Patty and Pete said in unison.

“What about the stencil design and venting?” Pete asked.

“Chris said that stencil design for venting is not as critical as once thought, although a window pane design is usually used,” Rob replied.

Figure 1.  The window pane design for the stencil is used to permit venting.

Figure 1.  The window pane design for the stencil is used to permit venting.

“So it sounds like starting with the best solder paste solves 90% of the problem and adjusting the process, say with the right reflow profile, helps refine the result,” Patty summed up.

With this Rob went off to put the finishing touches on his PowerPoint® slides for his part of the presentation, while Patty started working on her part of the presentation on using solder preforms to reduce voiding.

Two weeks later.

Patty’s mom and dad came for a visit on a Sunday. Her mom had graciously offered to bring a complete Sunday dinner. Patty, Rob and the boys were grateful for the delicious meal. As they began to eat, Patty shared the story of her best students not knowing Billy Graham, et al.

“But, what was even more surprising was that I ended up asking 10 or 20 more students and only one had ever heard of any of these four ‘famous’ people,” Patty sighed.

“It’s your age,” Patty’s mom replied.

Thirty years old was not that far in the rear view mirror for Patty and she really didn’t consider herself old.

“These youngsters were born in the late 1990s, a generation after these people were prominent,” her mom went on.

“Mom’s right.  Do you know Billy Sunday, Ty Cobb, Glenn Miller, and Trotsky?”  her dad asked.

“Who?” Patty asked.  And then she chuckled, getting the point.

After a brief pause, she said, “I do know who Trotsky was; tell me about the others.”


Dr. Ron

As always, this story is based on true events.


Wicked Wicking

This PCB assembly challenge involved attaching a solar panel to one side of a pad using solder paste with a pass through an SMT reflow soldering oven.

Figure 1.

Solder wicking through the unmasked vias to the back side forms unacceptable “bumps” on top of the vias.

The attachment or bond itself wasn’t the issue; but after the first trial runs, it was clear that solder wicking through the unmasked vias was going to be. Solder would wick through the unmasked vias to the back side and form “bumps” on top of the vias.

These bumps made the surface nonplanar and of course were unacceptable. It wasn’t an issue of using excess solder paste. But the “wicked wicking” had to be stopped, or at least prevented.

Figure 2.

Kapton tape is applied to cover the unmasked vias; it will block the molten solder from leaking through.

But how? Clearly, to keep the solder where we wanted it to remain during reflow, we had to find a way to prevent it from wicking up, collecting at the opposite ends of the vias and forming bumps. We had to find a solution that was simple, temporary, and tolerant of reflow soldering temperatures. The answer was Kapton polyimide tape, a familiar product to PCB assemblers for many years, and a material that does not degrade at reflow temperatures.

Kapton tape was applied to cover the unmasked vias in order to block the molten solder from leaking through the vias to the back side during reflow. After reflow and cooling, it was a simple matter to peel off the tape. This temporary masking solution worked; there were no more solder bumps on the back side of the assembly, and the cost of the fix in terms of time and material was very low.

Figure 3.

Figure 3. This temporary masking solution worked; there are no more solder bumps on the back side of the assembly.

Roy Akber


Too Close for Comfort

This is a little bit like the old college prank of trying to see how many kids can squeeze into a telephone booth. Pretty soon everyone’s too close for comfort!

In this PCB assembly challenge, someone made a mistake and created a layout for rows of dual-flat no-leads (DFN) SMT packages without taking into account the size of the component bodies. The footprints are too close together, and the bodies of the components are touching.

Because they don’t all fit, as the packages are lined up there isn’t enough room, and alignment issues develop for some of the IC locations. They’re forced off their footprints, while others appear to be acceptable.

Figure 1

Figure 1. With DFN footprints too close to one another, component bodies are actually touching and causing alignment issues, literally forcing others off their footprints.

Figure 2

Figure 2.

As can be seen from the photos (Figures 1 and 2), the crowding causes alignment issues for locations IC1, IC5, IC7, IC9, IC13, and IC15. Locations IC3 and IC11 seem fine.

What can be done? It’s too late to redesign and order new PCBs, and there is no possibility of shrinking the dimensions of the components.

Figure 3

Figure 3. Removal of components in locations IC5 and IC1 have allowed the rest to fit properly.

Luckily, the customer had a solution that worked: removal of the components in locations IC5 and IC1 (Figure 3). This permitted the remaining parts to fit correctly; it made “breathing room” for the rest, and best of all, was accomplished without compromising the functionality of the circuit.

Roy Akber


Component Footprint Rotation

Before we (or any old assembly house) go about putting surface mount parts on a board, we need to program our assembly robots. I’m oversimplifying, but essentially, the machine program needs to know the X / Y coordinates, relative to the board origin (which is the lower left-hand corner), the part rotation, and the side of the board.

In years past, we needed a centroid file (AKA pick-and-place file) containing all of that information. In some cases, we still need the centroid, but not always. Today, we can get the same information from ASCII CAD files, ODB++ CAD files or Eagle .brd files. You only need a centroid if you send us your board files in Gerber format.

If you do send us a centroid file, you no longer need to worry about rotation. The IPC has defined the zero degree orientation, as well as proper rotation direction, but too many part footprints set the zero degree at different angles. We can’t rely on the data.

While we have to ignore rotation and figure it out with other means, we still do strongly recommend that you follow IPC standards when you make your own footprints. The illustrations below show how footprints are supposed to be oriented.

Duane Benson
There’s no earthly way of knowing
which direction we are going
There’s no knowing where we’re rowing

Package origins

Passives orientation r2

Chip rotation

Quad and BGA

Three-pin parts

Cost Reduction in Design — More Advice

If you’re looking for the absolute, cheapest possible assembly service, you’ll need to look outside of North America. If you really need a decent price with good quality and good service, you can keep your gaze West of the Atlantic and East of the Pacific.

Like everything else in the modern world, design decisions can have a pretty big impact on your cost. So, lets take a look at some design decisions that can make your manufacturing more affordable.

  • Accept longer lead times

Lead times are one of the biggest factors in electronics manufacturing. Screaming Circuits can turn a kitted assembly job overnight, but it costs a lot of money to do that. Screaming Circuits also has a 20 day turn-around that is much, much more affordable. Accepting longer lead times on PCB fab will drop your cost as well.

  • Avoid leadless packages like QFNs and BGAs

We build tons of QFN and BGA boards – even down to 0.3mm pitch micro-BGAs. That’s great if you need those packages. However, since all of the leads are underneath, we have to x-ray every part. That adds a bit of cost to the process. If you can, stick with TSSOPs and other parts with visible leads.

  • Use reels, or 12″ or longer continuous strips

We will gladly assemble parts on strips of almost any size. But, to save costs, use full or partial reels or continuous strips of at least 12″ long. It costs us less time to work with reels and continuous strips, and we pass those saving on.

  • Stick with surface mount

These days, through-hole components tend to be hand-soldered. That costs more than machine assembly, so use surface mount wherever possible. Surface mount components tend to be less expensive than through-hole too. If you do need a few through-hole parts, this is an opportunity to put in a little sweat equity by soldering the through-hole yourself and save a bit of money.

  • Panelize small boards

We can work with really tiny boards individually, but sticking with a larger size makes the job easier, and, again, we’ll pass those saving on. If your PCB is smaller than 16 square inches, panelize it. We put in less labor and you get a price break.

By following these guidelines, you get a decent price and really good quality and service.

Duane Benson
That would be telling

Hitachi’s SMT Exit

And then there were … 27?

Hitachi’s board today announced plans to exit the SMT component placement business, selling off certain parts of the division and closing the rest. In a press release, the firm said it would transfer the sales organization to Yamaha and cease its development and manufacturing activities.

Japan has always been the major provider of the world’s component mounters, headed such major conglomerates as Fuji, Yamaha, Juki, Sony and Panasonic. And while Hitachi’s competitors will welcome one fewer player in the market, this in all likelihood won’t shake up the industry.

Over the years it’s been widely assumed consolidation was inevitable, yet it’s taken more than a decade since the Great Tech Recession of 2001-03 for any major moves to be made.

There have been several transactions and reshufflings, of course: ASM bought Siplace (Siemens), Universal was acquired by a private equity group, as was Assembleon. Mydata was acquired by a fellow Swedish OEM. And earlier this week Dima, a small European player, was snatched up by Nordson. None of these deals has truly changed the shape of the market.

In fact, the June 2013 merger of Juki and Sony was the first major deal in which a serious player ceased to exist. Hitachi’s will be the second.

The 27 (at least) remaining players will welcome the chance to grab Hitachi’s roughly $68 million in equipment sales now in play as result of this decision. Someone’s bottom line will look at least marginally better in the coming year. But more moves will be needed before the SMT market can truly regain the types of margins needed to inspire significant commitments to innovation that were standard fare in the 1990s.




Calculating Optimal Solder Paste Printing Aperture Parameters


A while ago, I developed an Excel-based spreadsheet, StencilCoach, that calculates optimal stencil aperture parameters for several common SMT solder paste printing applications. These applications include standard apertures, passive apertures, pin-in-paste (PiP) apertures, and preforms with pin-in-paste (PiP+) apertures. These algorithms are now online at Over the next several posts, I will review the use of this software tool.

Let’s first look at standard apertures. After logging into to the website, click on “Stencil Coach” and then “Standard Apertures.” The page gives the definitions for “Aspect Ratio” for a rectangular aperture and “Area Ratio” for circular and square apertures. The aspect ratio, which is defined as the width of the rectangular aperture divided by the stencil thickness, should be greater than 1.5. Whereas the area ratio, for circular or square apertures, is given as the area of the opening divided by the area of the sidewalls. This formula simplifies to D/4t, where D is the diameter of a circular aperture or the width of the square aperture. The area ratio should be greater than 0.66. These recommendations are not standards, but are good rules of thumb.

Let’s consider a situation where a PWB has rectangular apertures with a pitch of 35 mils and circular and square apertures with pitches of 40 mils each. The stencil thickness is 6 mils. See if you can develop the pad and aperture sizes and reproduce the figure below. Hopefully this tool will help you design your stencils.


Dr. Ron

PS: If you need a hand, feel free to contact me at rlasky@ indium com.

Demonstrating Zero Defects In SMT Production?

Folks, let’s see how Patty is doing at Ivy U …

Patty had to admit that she really liked being a professor at Ivy University. No, that wasn’t strong enough; she was ecstatic. The combination of the stimulating and collegial environment and the flexible schedule was terrific. She was able to play a little more golf and spend more time with Rob and the boys. 

In addition to developing a course on manufacturing processes, she was asked to teach an additional offering on statistics. Engineering enrollments had increased so much that another stats class was needed. Teaching stats gave her an opportunity to delve into topics she was interesting in learning more about, such as non-parametric analysis, cluster analysis, and numerous other statistical concepts.

She was also happy for Pete. As much as he enjoyed working with her at ACME, he, too, was thrilled to be at Ivy U. As a research associate, he spent a lot of time working with students on projects for their classes. He was surprised at how grateful the students were for his practical experience.

As Patty was thinking these pleasant thoughts in her office, suddenly Pete was at the door.

“Hey, Professor Coleman! The folks we left behind at ACME are being asked, forced really, to guarantee zero defects by examining a small sample size, say 20 samples,” Pete announced. Pete had stopped calling her “kiddo” and now teased her by calling her “Professor Coleman.”

“We both know that’s impossible. Tell me more,” Patty answered.

“Well, ACME just hired our favorite SMT engineer … after Hal Lindsay,” Pete responded.

“Oh, no! Not Reggie Peirpont,” groaned Patty.

Reggie was a well-meaning sort of chap who had some good ideas. But, his follow through was often sloppy and only touched the surface of what was needed from an engineering perspective. He was a very good salesman of his ideas and had a following in some SMT circles.

“What is he foisting on ACME?, Patty asked.

“A zero defect program,” Pete replied.

“Sounds like a worthy goal. But, let me guess, he has convinced everyone that they can demonstrate with 95% confidence they have zero defects by only sampling 20 units,” Patty said.

“Precisely,” Pete chuckled.

“I’ll contact Mike Madigan,” Patty said.

Patty had agreed to Mike’s request that she be available to consult for a year or so. And, he also made her promise to contact him if she knew they were doing something foolish.

Patty sent Mike an email with her concerns, and with some analysis. She suggested a teleconference.

Time passed quickly and, before they knew it, Patty and Pete were on a telecon with Madigan, Peirpont, and a few staff people.

Their discussion started with the good points of a zero defects program. On this topic, everyone was in agreement. Eventually, Madigan grew impatient.

“Peirpont! According to Coleman, your assessment that we only need to sample 20 units to demonstrate zero defects with 95% confidence is bull s__t.” Mike began, always getting quickly to the point.

Patty then said, “Let’s let Reggie explain his analysis.”

“Well it’s simple,” Reggie began. “All you have to do is recognize that 1 is 5% of 20, so if you sample 20 and don’t get a defect you can be 95% confident you have no defects,” he finished.

“Yikes,” Patty thought.

“Well, Coleman?” Madigan asked.

“That approach is not correct. A correct method is what I sent to Mike in an email” Patty answered.

“Before we begin the analysis, look at the photo I sent. The red bead is one bead in 2,000 white ones. Ask yourself how you could detect this one “defect” by sampling only 20 beads?” Patty said.

There was some murmuring and groaning, Patty could tell this visual really help to define the issue.

“OK, Patty. Please explain your analysis,” Mike asked.

“Let’s say that the defect level is 1 in a thousand. If I sample the first unit, the chance it is good is 0.999. What is that chance that the first two units would be good?” Patty began.

“0.999*0.999,” Pete answered.

“Correct!” Patty said.

“Let’s say I keep sampling until the likelihood that I have still found no defects is 0.05,” Patty went on.

“Let me take this one,” Madigan said.

“You now have 0.999^n = 0.05. So there is only a 0.05 chance you would not have found a defect if the defect rate is one in a thousand,” Madigan continued.

“So what could you say about the defect rate if you found no defects in n units? Patty asked.

“I got it! I got it!” Madigan shot back enthusiastically.

Patty was incredulous. Mike Madigan, CEO of multibillion dollar ACME Corp, was like a second grader excited to show the teacher he understood.

“You can say that the defect rate is 1 in a 1000 with a confidence of 1–0.05, or 95%,” Madigan said with excitement.

“Actually, you can say that the defect rate is 1 in a thousand or less,” Patty said.

“But we need to know n,” Madigan implored.

“Well, let’s solve for n with logarithms,” Patty suggested.

Groaning was heard over the telecon. No one likes logarithms!

Since their telecom was on GoToMeeting, Patty showed the solution:

n = log 0.05/log .999 = 2994.23

“Man! So we have to sample almost 3,000 units with no defects to demonstrate 1 defect per 1,000 or less?” Madigan asked with disappointment in his voice.

“Yes,” Patty responded.

She continued, “It ends up with a good rule of thumb. Since n is close to 3,000, let’s say that is the number we need to analyze. To demonstrate 1 in 10,000 defects or less, n is 30,000, one in a million or less, and n is 3 million.”

“So, n is 3 times 1 divided by the defect level you are trying to establish?” Madigan asked.

“Exactly,” Patty answered.

Patty wrote it on the PowerPoint slide:

To establish a certain defect level or less with 95% confidence, one must sample n units with no defects

n = 3 x 1/defect level

“That means to establish zero defects, we need an infinite sample,” Madigan sighed.

“Yep!” Patty replied.

“Peirpont! What do you have to say for yourself?” Madigan barked.

“Well, in the first case, Patty said 1 defect per thousand or less. It still could be zero defects,” Peirpont responded glumly.

Patty was going to respond, but Madigan beat her to it.

“But, you can’t prove it is zero. Only 1 in a thousand or less. So, to be conservative, we would say that the defect level would be 1 in a 1,000. That’s what is proved,” Madigan opined testily.

The meeting ended with Madigan expressing his thanks, an unusual thing for him. Peirpont said little else. It was clear he was probably going to get a talking to by Mike Madigan.

Patty was a little wistful after the meeting. She missed ACME and the folks there, even the occasionally cranky Mike Madigan. But every day she felt more like her home was at Ivy U.


Dr. Ron

Epilogue. As with all Patty and the Professor posts, this one is based on a true story. After sharing this concept with a colleague who had to get FDA approval for drug trials, she decided to ask statistician job applicants: “Do you think you could develop a sampling plan that could assure with 95% confidence that there were no defects in a population?” The last I talked to her, most job candidates had said yes.