The Quincunx as an Educational Tool

By: Steve Moore, Wausau Papers

Sir Francis Galtonʼs invention can do more than just demonstrate

normal distribution.

The purpose of this article is to give you an appreciation of the

Quincunx as an educational tool for teaching some of the theory behind

the tools and concepts of so-called modern quality management. The

Quincunx is often seen in the possession of organizations practicing

in-house education of statistical process control (SPC); however, it is

seldom utilized for anything beyond a demonstration of the normal

distribution. Indeed, the literature itself is virtually devoid of references

to the Quincunx beyond this use.

Sir Francis Galton (1822–1911) invented the Quincunx in the 1870s to

demonstrate the law of error and the normal distribution. He was a

prolific inventor, scientist, and mathematician and was knighted in

1909. Beyond the Quincunx, Galton conceived the standard deviation,

invented the use of the regression line, and was the first to describe the

phenomenon of the regression toward the mean.

Description of the Quincunx

The Quincunx Model WD-7 (see figure 1) is an example of a 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.

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

article).

Figure 1: Quincunx Model WD-7 (Courtesy of Lightening Calculator,

www.qualitytng.com [1])

When a bead is dropped, it will bounce to the right from zero to k times

(and to the left for the remaining pins). It then lands in the kth bin. The

bins may be numbered zero to k either from left to right or right to left.

The number of paths a bead can take to land in the kth bin is given by

the standard binomial coefficient for n choices taken k at a time.

Quincunx and process behavior charts

There is much literature that contains the misleading notion that data

must be normally distributed for a process behavior chart (control

chart) to work. Quincunx data is actually binomially distributed,

especially as n decreases toward four (the minimum for the Quincunx

model used here). As seen in figures 2 and 3, XmR control charts for

100 bead drops when n=4 and n=10 are both quite successful. Note

that, as expected, the control limits are farther apart as n increases

because the distribution will be wider as the beads contact more rows

of pins.

Figure 2: Quincunx process behavior chart with n=4

Figure 3: Quincunx process behavior chart with n=10

As a further demonstration of the fallacy of the normal distribution

requirement, an experiment was performed with the Quincunx to obtain

a significantly skewed distribution. This was achieved by placing a

vertical barrier of heavy card stock strips in the pins so that no bead

could travel to the left more than twice. Then 100 beads were dropped

and the results plotted on an XmR control chart. The histogram and

resulting control charts are seen in figures 4 and 5. They clearly

demonstrate that the chart is fully functional in spite of being far from

normally distributed.

Figure 4: Histogram of skewed Quincunx distribution with n=10

Figure 5: Process behavior chart of Quincunx skewed distribution with

n=10

Another area of dispute regarding control charts is the number of data

required for control limits to be useful. Figure 6 shows the results of an

experiment in which control limits of an XmR chart were calculated

after every bead was dropped, beginning with the fourth bead and

continuing until the fortieth. The upper control limit quickly stabilizes

around a median of 17.52 after the ninth bead is dropped. This

demonstration dispels the rumor that at least 20 to 30 data points are

needed to calculate useful control limits and confirms the Donald

Wheeler arguments.

Figure 6: Upper control limit as the number of beads dropped

increases.

I have successfully utilized the Quincunx to teach the principles of SPC

to “students” from the shop floor to upper management for many years.

I have used two major exercises: A demonstration of “tampering,” and

a demonstration of a stable process when a change has been made to

the system.

Tampering

The lesson of tampering is a good first introduction to the Quincunx.

When introducing it to students, have the front panel over the funnel

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and pins covered with a piece of paper or cardboard so that they

cannot be seen. Allow the students to gain tribal knowledge as they

experiment and speculate as to the functions of the funnel and pin

settings. The students are then instructed to “aim” at a particular bin

and are allowed to move the funnel and pin settings 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 target

bin. This distribution will tend to be a uniform distribution as the number

of trials becomes large.

Next, place the funnel directly above the targeted bin and set the pins

so that n=4. After 50 beads have been dropped, compare the new

distribution of beads with the previous one. The lesson will be clear:

Leaving the process alone will be significantly more accurate and

precise than tampering with the aim.

The lesson on tampering can be extended to demonstrate the four

rules of W. Edwards Demingʼs famous Funnel Experiment. Figures 7,

8, and 9 are process behavior charts demonstrating Rules 2, 3, and 4,

respectively with n=10. Figure 3 demonstrates Rule 1. It is instructive to

note the change in the control limits from those generated by Rule 1 as

Rules 2 and 3 are invoked. The process behavior chart for Rule 4

shows a complete lack of statistical control.

Figure 7: Process behavior chart of Quincunx funnel rule 2 with n=10

Figure 8: Process behavior chart of Quincunx funnel rule 3 with n=10

Figure 9: Process behavior chart of Quincunx funnel rule 4 with n=10

Learning about process behavior charts with a Quincunx

With the funnel held stationary and n=10, have students drop 50 beads

and construct an XmR chart of the numbered bins that the beads fall

into. Note the upper and lower control limits and discuss the sources of

variation. Next, move the funnel 1.5 pins to the left or right and take 10

more data points. The control chart will quickly show one or more

signals of lack of control against the original control limits as seen in

Figure 10. This demonstration may also be performed with an X-R bar

chart, but this, of course, requires more class time.

Figure 10: Quincunx before and after funnel moved 1.5 pins.

Summary

The Quincunx is a valuable learning tool that has been mostly

overlooked in classroom settings. With a little imagination, this tool can

be effectively utilized to teach many lessons in the understanding of

variability and process behavior charts.

Source:

The Quincunx as an Educational Tool http://www.qualitydigest.com/print/9962

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