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Non-Normal
Distributions in the Real World
by
Thomas Pyzdek
Introduction
One day, early in my career
in quality, I was approached by Wayne, a friend and the manager of the
galvanizing plant. "Tom" he began, "I've really been pushing
quality in my area lately and everyone is involved. One of the areas we're
working on is the problem of plating thickness. Your reports always show
3% - 7% rejects and we want to drive that number down to zero."
I was, of course, pleased.
The galvanizing area had been a perennial problem child of the company.
"How can I help?" I asked.
"We've been having
meetings, trying to discover the cause of the low thicknesses. But we're
stumped. I want to take copies of the quality reports to the team so they
can see what was happening with the process when the low thicknesses were
produced."
"No problem,"
I said, "I'll have them for you this afternoon."
Wayne left and I went
to my file of galvanizing reports. The inspection procedure called for
seven light poles to be sampled each hour and plotted. I took the reports
and computed the daily average and standard deviation by hand (this was
in the BPC era, "Before Personal Computers"). Then, using a
table of normal distribution areas, I found the estimated percent below
the low specification limit. This number was reported to Wayne and to
a number of other important people. As Wayne said, the number tended to
be between 3% and 7%.
But, after searching through
hundreds of galvanizing reports, I found not a single thickness below
the minimum! My faith in the normal distribution was not shaken, however.
I concluded that the operators were "adjusting" their results
by failing to record out-of-tolerance thicknesses. I set out for the storage
yard with my thickness gauge in hand to prove my theory.
Hundreds of parts later,
I was forced to admit defeat. There simply weren't any thickness readings
below the minimum requirement. News of this was met with shock and dismay
by the hard-working galvanizing teams. Wayne's comment was "How could
you people do this to us?"
This embarrassing experience
led me to begin a personal exploration of just how common normal distributions
really are. After nearly two decades of research involving thousands of
real- world manufacturing and non-manufacturing operations, I have an
announcement to make: Normal Distributions are Not the Norm!
I say this knowing full
well that you've heard otherwise from one or more "experts."
Well, you heard wrong. You can easily prove this to yourself by collecting
data from real live processes and evaluating it with an open mind. In
fact, the early pioneers of quality (such as Shewhart) were fully aware
of the scarcity of normally distributed data; since then, somehow, we've
forgotten. Today the prevailing wisdom seems to be "if it ain't normal,
something's wrong." Well, that's just not so.
The Normal
Distribution
Figure
1: Sum of 10 Dice Rolled 1,000 Times
Consider figure one, which
is a histogram of the sum of the spots showing when ten six-sided dice
were thrown one thousand times (and you thought a quality consultant's
life was exciting). The shape of the histogram is approximately normal.
This means:
- A curve that is computed
from the normal distribution function passes near the midpoints of the
histogram cells.
- Both halves of the
histogram look about the same.
- There is a single mode,
or peak.
- Either the histogram
is unbounded on both ends, or the bounds are so far away as to not matter.
Think for a moment about
the "process" used to create this near-normal histogram, its
features are:
- There are a large number
of variables (ten dice).
- The variables are completely
independent of one another (die number 5 doesn't change if some other
die changes).
- The value of any given
variable (die) is determined by random chance.
- The value of a given
variable (die) on a given roll is completely independent of any previous
rolls.
Processes which possess
similar features will produce distributions that are approximately normal.
Nature may well produce many such processes if left alone, I am not conversant
enough about such processes to know. However, I do know that business
processes don't produce normal distributions. There are many reasons
why this is so. One big reason is that the objective of most management
and engineering activity is to control natural processes tightly, eliminating
sources of variation whenever possible. The result of this control is
often value added to the customer; for example, who would want to stay
in an outdoor hotel? Other distortions occur when we try to measure our
results. Here are some examples of "de-normalizing" activities:
Human Behavior
Patterns
Figure
2: Days receive payment from time invoice is sent
Figure 2 shows a histogram
of real data from a billing process. A control chart of days-to-pay for
non-prepaid invoices showed statistical control. The histogram indicates
that some customers like to prepay, thus eliminating the work associated
with tracking accounts payable. Customers that don't prepay tend to send
payments to arrive just after the due date. There is a second, smaller
spike after statements are sent, then a gradual drop off. The high end
is literally unbounded because a few customers will never pay their bill.
This pattern suggests a number of possible process improvements, but the
process will probably never produce a normally distributed result. Human
behavior is rarely random and processes involving human behavior are rarely
normal.
Management
Systems
The measurement system
may create a normal distribution where none exists! I've seen measurement
systems with such crude repeatability that virtually all of the variability
in the numbers was from measurement error. As Gauss observed nearly two
hundred years ago, measurement error seems to tend toward normality (Kotz
& Johnson, 1985). At least the measurement errors studied by Gauss.
More often measurement
systems produce distortions which make the data appear less normal.
Figure 3a shows a process distribution that is normal. Figure 3b shows
a histogram obtained when using a gauge with a resolution of two measurement
units. It might be tempting to simply use a "better" gauge.
But what if the cost goes up by a factor of 100 (I've seen this much and
more). Or what if the tolerance is ± 50 measurement units? Sometimes
it's simply not worth the effort just to get the results some expert says
you're "supposed" to get.
Figure
3. (a) Actual process distribution; (b) measurement distribution (gage
resolution = 2 units)
At times the measurement
is non-normal because of the scale of measurement. For example, pH or
decibel scales. These data can sometimes be made normal by using the logarithm
of the result. But why confuse everyone just to get a normally distributed
result?
Physical
Laws
Nature doesn't always
follow the "normal law" either. Often natural phenomena produce
distinctly non-normal patterns. The hot-dip galvanizing process discussed
earlier is an example. A metallurgist described the process to me (alas,
too late to prevent the aforementioned debacle) as the creation of a zinc-iron
alloy at the boundary. The alloy forms when the base material reaches
the temperature of the molten zinc. Pure zinc will accumulate after the
alloy layer has formed. However, if the part is removed before the threshold
temperature is reached, no zinc will adhere to the base metal.
Such parts are so obviously defective that they are never made. Thus,
the distribution is bounded on the low side by the alloy layer
thickness, but (for all practical purposes) unbounded on the high
side since pure zinc will accumulate on top of the alloy layer as long
as the part remains submerged. Figure 4 shows the curve for the process
--- a non-normal curve.
Figure
4: Distribution of Zinc Plating Thickness
Many other natural processes
are non-normal. The breaking strength of a metal is a smallest extreme
value distribution (the break occurs at the weakest point). A gap is bounded
at zero. A hole can't be smaller than the punch that made it. The list
goes on and on.
Engineering
Activity
When nature doesn't impose
constraints, people often do so. Such things as locating pins, positive
stops or automatic sensors often bound or truncate the distribution. Blanking
dies or pin/punches can doubly bound a distribution.
Numerically controlled
(NC) machines often have backlash in their positioning mechanisms that
create one distribution when moving away from the home position,
and a completely different distribution when moving toward home,
resulting in a bimodal distribution with a "dead zone" in the
middle.
Chemical and electrical
engineers have their own bag of tricks for bounding, truncating, or otherwise
manipulating nature. It's what engineers do. It's where products and processes
come from. The fact that the result isn't normal doesn't matter.
Inspection
Sometimes inspection itself
can create non-normal data. ANSI Y14.5, a standard for dimensioning and
tolerancing used by all aerospace and defense contractors, describes a
concept called "true position." The true position of a feature
is found by converting an X and Y deviation from target to a radial deviation
and multiplying by two. Even if X and Y are normally distributed (of course,
they usually are not), the true position won't be. True position is bounded
at zero and the shape often depends solely on the standard deviation.
Many other inspection
procedures create non-normal distributions from erstwhile normal data.
Perpendicularity might be normally distributed if the actual angle was
measured and recorded. But quite often perpendicularity is measured as
the deviation from 90 degrees, with 88 degrees and 90 degrees both being
shown as 2 degrees from 90 degrees. The shape of the resulting distribution
varies depending on the mean and standard deviation. It's shape can be
anything from a normal curve to a decidedly non-normal curve. An example
is shown in figure 5 (note the "comb tooth" pattern indicating
rounding off of the numbers, yet another cause of non-normality). This
discussion also applies to flatness, camber, and most other form callouts
in ANSI Y14.5. The shape of the curve tells you nothing about your control
of the process.
Figure
5: Perpendicularity in Thousandths of Inches
Implications
A purist might say at
this point "So What?" After all, any model is merely an abstraction
of reality and in error to some extent. True enough, but when the error
is so large that it has drastic consequences, the model should be re-evaluated
and perhaps discarded. Such is often the case with the normal model.
Process capability analysis
(PCA) is a procedure used to predict the long-term performance of statistically
controlled processes. Virtually all PCA techniques assume the process
distribution is normal. If it isn't, PCA methods, such as Cpk, may say
an incapable process is capable, or vice versa. It may predict high reject
rates, when in reality no rejects ever appear, as with the galvanizing
process discussed earlier, or vice versa.
If you are among the enlightened
few who have abandoned the use of "goal post tolerances" and
PCA, you will find that assuming normality hampers your efforts at continuous
improvement. If the process distribution is skewed, the optimal setting
(target) will be somewhere other than the center of the engineering tolerance;
but you'll never find it if you assume normality. Your quality improvement
plan must begin with a clear understanding of the process and its distribution.
Failure to understand
non-normality leads to tampering, increased reject rates, sub-optimal
process settings, failure to detect special causes, missed opportunities
for improvement, and many other problems. The result is loss of face,
loss of faith in SPC generally, and strained customer - supplier relations.
Solutions
The most important solution:
THINK! Don't use statistics-by-rote, blindly plugging numbers into formulas
and accepting the results. Plot a histogram of the data after you've attained
statistical control and think about how the histogram pattern relates
to the real world.
If you decide that the
non-normal process distribution really should be non-normal, use your
new knowledge to help you manage and improve the process. Adjust your
target values and control limits to minimize rejects and to improve product
performance.
Software can help you
perform PCA with non-normal data. Such software will base yield predictions
and capability index calculations on either models or the best fit curve,
instead of assuming normality. One software package can even adjust the
control limits and the center line of the control chart so that control
charts for non-normal data are statistically equivalent to Shewhart control
charts for normal data (Pyzdek, 1991). With modern software, proper analysis
of on-normal data is as easy as assuming normality.
Another
View
Not all authors agree
that it is necessary or wise to modify process capability analysis (PCA)
or control limits to compensate non-normality. The major objections to
the practice are (Wheeler, 1995, 46-54):
- Non-normal adjustment
are based on the third and fourth moments of the distribution, which
contain little information compared with the mean and variance.
- Estimates of the higher
moments are unstable and therefore unreliable unless sample sizes are
unreasonably huge.
- Subgroups with similar
skewness and kurtosis statistics may have dramatically different distributions.
- Higher moments confuse
users.
Let’s consider these
objections one by one.
Non-normal adjustment
are based on the third and fourth moments of the distribution, which contain
little information compared with the mean and variance
True, but a little information
is still information. Most of the added information describes the tails
of the distributions, which is the primary area of interest for quality
analysis. This is especially true when conducting PCA studies, but it
also applies to control charts. The added information is virtually cost
free when using computers. If you get something for nothing, why not take
it?
Estimates of the higher
moments are unstable and therefore unreliable unless sample sizes are
unreasonably huge
To support this contention,
one author provides a graph showing that the skewness and kurtosis statistics
converge only slowly towards their theoretical value (this is much more
so for Kurtosis than skewness). However, the same approach can be used
to show that the mean and variance are "useless." Figure 6 plots
these two statistics for the author’s data and sample sizes of 100,
200, 300, 400, and 500. Note that the mean estimate moves away
from the true value for quite some time, and it is always on the high
side. The author provides no parameter value for the variance, and one
would be hard pressed to find it using figure 6. Does this imply that
the sample average and variance have no value and should not be used?
Of course not. The problem is with the method used to display the estimates
rather than with the estimates themselves.
Figure
6: Convergence of Location and Spread Statistics to Theoretical Parameters
The author presents abstruse
mathematics showing how the various moments are derived, but the math
merely confuses the issue of the value of these moments for quality analysis.
The method of derivation is irrelevant. The truth is that the higher moment
estimates are reasonably stable for samples of sizes usually recommended
for PCA (n >= 125), especially skewness. Because of modern data processing
and storage technologies, many processes have thousands of historical
data points on file. For decidedly non-normal data, PCA estimates using
the skewness and kurtosis estimates from even as few as n=100 samples
usually provides better results than PCA that assumes normality. Bear
in mind that the Western Electric Statistical Quality Control Handbook
recognized the need for dealing with skewed data at least as early as
1956, and they also provided a crude formula for the purpose of curve
fitting.
Subgroups
with similar skewness and kurtosis statistics often have dramatically
different distributions
The author presents a
graph showing a scatterplot of skewness and kurtosis for 100 subgroups
and notes that few of the subgroups are close to the theoretical value.
He also presents "histograms" of several subgroups with similar
skewness and kurtosis statistics, showing that the "distributions"
are different. Once again, if this approach were applied to the mean and
variance it would also lead to the conclusion that these statistics are
without value (figure 7). But this ignores the fact that no one is suggesting
that we create control charts of the higher moments. They are only to
be computed from aggregate data from a large number of subgroups. Plotting
scatter plots and histograms of skewness or kurtosis for subgroups of
n=5 is just plain silly, for that matter plotting histograms and scatter
plots of any statistics for samples of 5 is silly. I know of no qualified
person who would recommend doing such a thing.
Figure
7: Two Subgroups with the Same Mean and Variance
Higher Moments Confuse
Users
This hasn’t been
my experience. In fact, I explain the control limits for classic Shewhart
charts and non-normal control charts in precisely the same way: they are
limits on common cause variation. True, the calculations are (sometimes)
more involved for non-normal data, but the calculations are handled by
the computer and are therefore transparent to the user. I’ve found
that users are far more confused if I insist on using classical control
charts even if they produce, for example, negative numbers for length
measurements and waiting times, defective rates greater than 100% or less
than 0%, special cause indications where no special cause can be found,
etc. Also, when classical control charts produce control limits beyond
the bounds of a measurement or process, there is no way the control chart
can indicate process improvement. As the story I opened this paper with
indicates, users are more often angry when they learn that they’ve
been wasting their time because someone used the wrong statistical technique
to analyze their quality data, causing them to waste their time.
About the
Author
Thomas Pyzdek is an ASQC
Fellow, CQE and CRE. He is the founder of Quality America, Inc. and the
founder of Quality Publishing, Inc.. He is an internationally known consultant,
lecturer and author on quality subjects and recipient of the ASQ Edwards
Medal. Mr. Pyzdek has published numerous papers in such respected journals
as Quality Engineering, The Quality Management Journal and
Quality Progress and he has a regular column in Quality Digest
magazine. He has presented papers at several ASQ Annual Quality Congresses.
Pyzdek served on the board of examiners for the first Malcolm Baldrige
National Quality Award and on the panel of judges for the first Arizona
Governor’s Award for Quality. He is listed in the International
Who’s Who in Quality and Who’s Who in the West.
References
Kotz, S. and Johnson,
N.L., editors-in-chief, (1985), Encyclopedia of Statistical Sciences,
Vol. 6, Wiley Interscience, New York, pp. 347-359.
Pyzdek, T., "Johnson
Control Charts," Quality, February 1991.
Pyzdek, T., "Process
capability analysis using personal computers," Quality Engineering,
4(3), 1992, pp. 419-440.
Wheeler, D.J. (1995),
Advanced Topics in Statistical Process Control: The Power of Shewhart’s
Charts. Knoxville, TN: SPC Press.
Charts courtesy Quality
America's SPC-PC IV
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