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**The Quincunx and the Normal Distribution**

By Jim Warren

Lightning Calculator

The quincunx device was invented in the 1870’s by Sir Francis Galton (1822–1911) to demonstrate the law of error and the normal distribution. Galton was a gentleman scientist and had also a studied medicine at Cambridge, England. He went on to discover and refine many statistical tools like medians and percentiles as a way of characterizing distributions and reducing nonmetric data to a metric scale. He was also responsible for first discovering the principles of linear regression. Later in his life he would discover and validate the statistical basis of using fingerprints to uniquely identify individuals, much like DNA testing is used today. King Edward VII knighted him in 1909. While beyond the scope of this short paper, it is interesting to note that Francis Edgeworth and Carl Pearson would later present Galton’s work to many other scientists and provide even more applications of Galton’s principles to fields like astronomy. While Galton discovered the principles of the normal distribution, the latter two really advanced them and presented them to the science community in various parts of the world. (See Figure 1)

Figure 1: The Normal Distribution’s Bell Shaped Curve

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**Description of the Quincunx**

Quincunx boards have appeared in many science museums and fascinated many students. The Quincunx Model WD-7 (see figure 2) is an example of an off-the-shelf Quincunx board and is comprised of a vertical board with 10 rows of pins. Beads are dropped via a funnel into the top of the board. As they descend through the board the beads will bounce either to the left or right as they encounter each row of pins. The pins may be arranged such that a bead will come in contact with 4, 6, 8, or 10 rows of pins. As each bead leaves the final row of pins, it is captured in one of several bead-wide bins, which may be numbered for reference by the user. After a sufficient number of beads have been dropped, the height of the beads in the bins begins to resemble the classic bell-shaped curve. In reality, the distribution of beads is binomial; however, the normal distribution is approximated when n, the number of rows of pins, is large (n = 4, 6, 8, or 10 for the Quincunx referred to in this paper). (See Figure 3)

Figure 2: Quincunx Model WD-7 with adjustable pin block

Size: 28 inches high x 14 ½ wide x 1 ½ deep

Website: www.qualitytng.com (Courtesy of Lightning Calculator)

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Figure 3: Adjustable Pin Block and Pin Block Pattern

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**What can you do with a Quincunx?**

The real value of the quincunx is in being able to quickly simulate processes or tests that would be impractical to perform in real life. Some people actually refer to the quincunx ability to generate data as a “factory in box”. The quincunx performs three basic functions:

- Generating real data. Visually displaying the data along with tolerances limits or specifications in a histogram format.
- Visually displaying the data along with tolerances limits or specifications in a histogram format.
- Illustrates changes in the data’s mean by shifting of the funnel to the left or right and changes in variability by adjusting the pin block.

There are several additional demonstrations possible with a quincunx from basic concepts of variation, to statistical measures like mean, mode, median, and standard deviation. It is an excellent tool for illustrating process centering, statistical process control, and even hypothesis testing and design of experiments.

The next section will detail many of these exercises and illustrations.

**Process Tampering**

The lesson of tampering is a good first introduction to the Quincunx. When first introducing it to students, cover the front panel over the funnel and pins with a piece of paper or cardboard so that students cannot see what affects the beads. Allow the students to gain tribal knowledge as they experiment and speculate as to the functions of the funnel and pin block adjustment settings. The students are then instructed to “aim” at a particular bin and are allowed to move the funnel and pin settings after each trial of dropping 1-5 beads even though they are not quite sure of the structures under the covered panel. After 50 trials, note the shape of the distribution of the beads about the targeted bin. This distribution will tend to be a uniform distribution as the number of trials becomes large. What happens in actuality is when students see beads fall to the right or left, they try to compensate by moving the funnel the opposite direction and/or manipulating the pin block. This constant adjustment by students introduces far more variation in the process than if they had just left the adjustments alone.

To prove the over adjustment issue, place the funnel directly above the targeted bin and adjust the pin block so that the balls pass over 4 rows of pins. Leave the funnel in the same position and drop 50 beads, compare the new distribution of beads with the previous one. The lesson will be clear: Leaving the process alone will be significantly more precise than tampering with the aim.

The lesson on tampering can be extended to demonstrate the four rules of W. Edwards Deming’s famous Funnel Experiment. (DF-5 Deming Funnel Experiment is available from Lightning Calculator, www.qualitytng.com)

**Concepts of Variation**

Start this session by describing the physical situation of pouring a cup of sugar from a measuring cup at a specific height and rate. If you did this repeatedly would the mounds of sugar look the same? Yes. While you cannot predict where any one particular granule of sugar will fall, you can predict that the mounds of sugar will be similar in appearance (assuming consistency in pouring, etc.) The point is that groups of things form patterns and once you recognize a pattern, it can be helpful in predicting future events.

In statistics you are not pouring sugar, but you are dealing with numbers. Statistics is the science of dealing with numbers. A mathematical calculation like the mean is one way of observing the pattern of numbers. Calculating the dispersion or variation in a group of numbers is another way of observing the pattern of numbers. Variation can be measured in a variety of ways, e.g. observing the range, calculating the average deviation from the mean (which always is zero by definition), the variance, and standard deviation. (Students of statistics may even calculate the third and fourth moments, which correspond to the skewness and kurtosis of a distribution.) The key point to drive home is that when you notice a pattern in the numbers, it is can be helpful in predicting future events. Statistics can be described as knowing how to analyze patterns of numbers in order to make decisions.

One of the most common variation patterns in numbers is the normal distribution, often referred to as the bell shaped curve. There are numerous social and physical phenomena that fit this curve. Now is when you can to show the students the uncovered quincunx so they can see the beads falling through the funnel and over the pins. The distribution pattern of the beads in the quincunx is similar to the bell shaped curve or normal distribution. This is exactly what Francis Galton was doing in the 1870’s by applying normal curves to social data and he was used the quincunx to illustrate this fact to his audience.

Explain that the funnel directs where the center or average of the distribution pattern is on the row of channels and how the pins or the number of rows of pins determines the width of the distribution pattern. In many physical situations where we examine the average and variability of a distribution we are trying determine how to either move the funnel (move the average) or change the variability (usually reduce the variability). Both of these concepts can be illustrated with the quincunx board. This is where the old proverb applies, “a picture is worth a thousand words” and allows students to associate statistics with an actual physical model.

The advantage of the physical quincunx board versus a computerized quincunx is that students can observe the beads falling over the pins and forming normal distribution patterns. While computerized quincunx illustrations may have their place in education, there is always a question in the back of the students mind about the programming being manipulated to produce a desired result.

**Statistical Inference**

In order to perform this demonstration the channels need to be numbered or assigned some relative value. The number of beads in each channel will become a histogram of data for further calculations. For example, you can number the channels from 1 to 25 with a water based market. When you run a distribution of beads you can then count the number of each size by counting the beads in each respective channel. For example you could have three 9’s, six 10’s, ten 11’s, and so on.

Can statistics really predict an outcome based on a sample? Demonstrate this by running a sample of 35 beads and calculating the mean and standard deviation of the sample data. Once the mean and standard deviation are calculated determine the six standard deviation range. (Remember the Average +/- 3**σ **in a normal distribution encompasses 99.7% of the population.) Therefore if statistics really works one should be able to run a sample and calculate the 99.7% spread of the population. Mark the limits on the face of the quincunx and run an additional hundred or so of beads and form a large histogram. If statistics works all the beads will be within the six standard deviation (99.7%) limits.

Before running the additional 100 beads try making some sort of bet with the students that the next 100 beads will fall between the six standard deviation limits calculated. Run the next hundred beads and collect your bets. Of course this is not about winning bets, but creatively reinforcing in the students minds that statistics actually works.

**Process Centering**

Run a large distribution of beads and note that this is a histogram for a process. Draw a set of limits on the face of the quincunx with a water based marker, e.g. a lower limit and an upper limit. (These limits should be similar to the 99.7% range calculated in the previous demonstration.)

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Now move the funnel left or right and drop another distribution that has a tail of the distribution outside of a limit. Ask the students to tell you what is wrong and how to fix it. They will naturally state to move the funnel back. You can move it back to the other extreme and drop another distribution. It is easy to then illustrate that in most process situations one is trying to center the process between the limits. (see Figure 4) How do you know where the center of a process is? You calculate the average of a sample of data. You then make the point that in real life situation you cannot see a funnel, you can only see the data. In this situation we can see that the average corresponds to the location of the funnel. This is what you are trying to accomplish in statistics, i.e. determine where the funnel is located based on the statistics of the numbers.

You can then lead a discussion about how you move the funnel in real life situations. It may be an adjustment on a machine, a change in a mixture (e.g. more of an ingredient in a recipe), a different diet or exercise regime in a healthcare experiment, a change in the dosage of a medicine, harder final exam test questions in an educational setting, more study time prior to a test, different process parameters, etc.

Figure 4 Process Centering between Limits

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**The Futility of Random Checking**

This demonstration is as effective with machine operators as it is useful in showing management how they can be part of the problem. Point out that random sampling technique that may have been determined by a company. i.e. check every 20th piece, every half hour, twice per shift. etc. are not effective and can actually cause the problems they are wanting to prevent.

Set up the Quincunx as in the process centering demonstration and shift the funnel near the upper limit. Point out how many operators run to the high side of the tolerance so they can reduce scrap (you can always rerun the part if it is machined oversize but it is scrap if its undersize). Drop 19 beads then drop the 20th bead representing the inspected part and note whether it falls, inside or outside the specification. Repeat the process five or six times. Chances are every bead checked will be inside the specification with about 10% of all the beads falling outside the tolerance. If a bead does fall outside the tolerance during one of the checks remind the student that most operators would run a second piece before adjusting the process. When you finish this exercise ask the operator if any of the parts he checked were out of specification? They will answer, No. Yet when you look at the distribution of all the parts it will show approximately 10% out of specification. This demonstration drives home the reason why operators who are instructed to use random sampling techniques have trouble maintaining tolerance specifications and when the machined parts do not assemble properly because they are out of specification, all they can do is point fingers at one another because they really do not know the truth about their processes.

**Average and Range Control Charts**

Close the top gate on the Quincunx and without moving the funnel run and plot samples of five beads on an average-range control chart. Calculate and plot the control limits on the chart just as if the quincunx were a machine. After creating the control chart move the funnel approximately 1.5 channels to the left or right and plot another sample of five parts. Students will be amazed how quickly they can detect a shift in the funnel, which is analogous to a change in the process. To carry the point a little farther tape a piece of paper over the funnel portion of the Quincunx and then randomly move the funnel to different directions and let the students determine if the funnel has moved by plotting samples on the control chart. This technique is very helpful in building confidence in the value of control charting. Another point can be made by sliding the funnel completely to the right and allow the beads to drop directly onto the pins. Plot another sample of 5 beads and demonstrate how the range portion of the chart detects an out of control condition. This erratic behavior is analogous to machine conditions where bearings are worn, fixtures are loose, broken, etc.

Figure 5: Average – Range Control Chart

**Hypothesis Testing**

What you are fundamentally trying to do with most hypothesis testing situations is trying to determine if the funnel has moved or if one distribution is significantly different than another. If in real life you cannot see the funnel and must depend only on the data, what kind of things affect the data. The answer is how much variability is present. If you are trying to detect small shifts in the funnel location (like with the control chart example) and there is a lot of variation in the data it is easy to visualize that variation masks ones ability to determine true shifts in the funnel. With lots of variability one does not know if the shifts in the data are due to random variation or a true shift in the funnel location. This is why you need a measure of the variability of either the population or at least of the sample as an estimate of the population variability. This can lead into a discussion of when you use the normalized z tests versus t distribution tests.

Figure 6: Typical Hypothesis Diagram to detect Different Means

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**Design of Experiments (DOE)**

Designed experiments are designed to test different combinations of factors to efficiently determine which variables or combination of variables effects the outcome of the experiment, often called the response variable (See Figure 7). Given the general formula for any designed experiment is:

X_{abc} = X + A + B + C + AB + AC+ BC +ABC +ε (random error)

Where X is the Grand Average of all the data,

A, B, and C are the main effects of each variable

AB, AC, BC, and ABC are interaction effects

ε - Random Error is uncontrollable error throughout the experiment.

Figure 7: Three Factor Experiment Design Model Treatment Combinations

What this looks like on the quincunx is that the funnel starts out at the grand average location. It is then shifted left or right based depending on the effect of factor A in a given treatment combination. The same is true for factors B and C as they also shift the funnel some amount to the left or right. The final net funnel position (combined effect of A, B, and C for a treatment combination) is then set to produce a sample of data for the given treatment combination. But before the beads fall into the channels where the data is measured, the beads fall over rows of pins, which then adds experimental random error to the final location of the beads. A given designed experiment can have several treatment combinations with unique levels of factors A, B, C, and combinations (interactions) of same. The data analysis is then designed to filter out the effects of the factors (how much the funnel is shifted by each factor by itself) the effect of the combination of factors (interactions) in also effecting the funnel location, and finally the amount of random error (effect of pins) on the overall data set.

While there is an actual teaching kit available to perform designed experiments with a quincunx called a PROSIM (available from Lighting Calculator) just being able to associate the terminology of designed experiments with a physical object is very helpful to students to grasp the concepts of DOE.

**Process Improvement**

The quincunx has an adjustable pin block, which allows the beads to fall over 4, 6, 8, or 10 rows of pins. Obviously the more rows of pins the more variation and wider the distribution. The process centering exercise previously described illustrated the value of centering the process to the specification limits in order to achieve the best process performance. The second and more important concept is to not only center the process but to reduce the variation. (See Figure 8) This can be accomplished in real life through a variety of programs like Statistical Process Control, Designed Experiments, and more recently programs like Six Sigma are used to identify and reduce process variation to a parts per million level. This is illustrated with the quincunx by adjusting the pin block such that the rows of pins are reduced, which reduces the variability of the distribution.

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Figure 8: Reducing Variation through Continuous Improvement

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**Quincunx Construction**

The first Quincunx was designed and manufactured in 1873 by Tisley and Spiller, old time craftsmen in England under the direction of Sir Francis Galton. Since then numerous improvements have been made. Today’s boards are made of quarter sawn hard wood to prevent warping (See Figure 9) and machined on Computer Numerical Control routers. Pin blocks are precisely drilled and have adjustable pin patterns. The funnels are also now moveable and self-locking. One thing that has not changed however is the fact that practitioners still refer to the quincunx as a “factory in a box” and use it to generate data the same way that Galton did with his quincunx.

Figure 9: Quincunx Quarter Sawn and Glued Board Construction

One more thing... where did the name “quincunx” come from? Quincunx is the name of a Roman coin that had five indentions on its face, just like the five spots on a pair of dice. Four points were stamped on the coin in a square design with the fifth point in the center of the square.

Figure 10: Roman Quincunx Coin

The quincunx pattern has also been used over time in the agriculture world as a pattern for planting plant trees in order to increase pollination. Since this was the name of the pin block pattern according to Galton’s work the device has been known as a quincunx (pronounced like quinn-cux) ever since. It interesting to note that Galton originally call the device an “Instrument to illustrate the principle of the Law of error or Dispersion.” .... thank goodness that “quincunx” became the short name for this device.

Oh yes, where did the name Lightning Calculator come from? While the original Quincunx was invented by Galton in 1873. A table top Model WD-4 Quincunx was invented for industrial use in the 1950’s. Electronic calculators had not been invented yet, however there were large mechanical calculators being used at the time called "Lightning Calculators". Well you guessed it, it was named after one of these calculator because it could quickly demonstrate concepts. We occasionally get questions from antique dealers who find these devices to ask if there is any connection with our company. The only connection is that the original inventor copied the name. Since the name has been associated with quincunx boards over the decades we still call ourselves "Lightning Calculator".

**© 2015 Lightning Calculator, **

**Troy Michigan, Feb 2015**

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Jim Warren has been the owner of Lightning Calculator Since 1981. He has worked as a senior quality executive in the Automotive and Staffing Industry and as an industry and educational Consultant. He is a Lean Six Sigma Master Black Belt and has graduate degrees in Engineering and Business.