Posted by Steve Moore on 1st Aug 2014
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
The Quincunx as an Educational Tool http://www.qualitydigest.com/print/9962
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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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