Figure 3
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Chapter 5 Part 5
Quality Control and Pareto Charts
"You
cannot inspect quality into a product." Harold F. Dodge (1893-1976).
Statistical
Quality Control has been growing in importance within American business for the
past few years. However, it is not a new idea. The concepts of quality control
have been around for several decades. Mistakenly, some people believe that
quality control is a final
inspection of a product at the end of a production line. Many companies are now
recognizing that quality control is much more -- that it can contribute to all
aspects of a product's design and production.
Instead
of being a final check to make sure a product has an acceptable number of flaws,
quality control is more correctly used as a tool to discover and correct current
problems along the entire life of a product, from conception to customer
support. Strategically devised statistical quality control techniques can
trigger warnings about impending problems and suggest changes in design,
manufacture and support so a product or service can be continually improved.
Most firms that correctly implement a quality control system not only improve
the product, but also improve consumer loyalty, employee pride, and reduce
overall costs. When quality is designed into a product, the need for one overall
final inspection (called acceptance sampling) is cut down to a minimum or
eliminated.
The
Purpose of Quality Control
Statistical
quality control methods should be used to identify unexplained variation in a
process. When such variation is identified, that variation can then often be
controlled, corrected, or eliminated. When statistical methods are used to
produce these kinds of improvements, it should, in turn, mean greater
productivity and superior products.
One
of the pioneering statisticians who is often credited with initiating the modern
quality movement is W. Edward Deming. In 1949 Deming was invited to Japan, where
he introduced the country's businesses to the concept of quality and statistical
methods. In the February 1981 issue of Nation's Business, Deming is quoted as
saying, "I told them Japanese quality could be the best in the world
instead of the worst. I was the only man in Japan then who believed that
Japanese industry, with its magnificent work force, engineers and statisticians,
and its unexcelled management, could do that. Many of them totally disagreed
with me." Deming's original 8-day lecture series began a quality revolution
in Japan that has had economic repercussions in virtually every country in the
world. The Deming Award for quality in Japan is one of the country's most
treasured prizes. In recent years, America has responded with the Malcolm
Baldrige Award. As a result of the interest among American businesses, Deming's
principles (as well as those of other quality gurus) have been taking hold in a
number of businesses. Until his death in
1993, Deming took his message directly to American companies.
Other
quality gurus include Dr. Joseph M. Juran and Dr. Kaoru Ishikawa . Dr. Juran has
noted that 85% of quality problems are the result of faulty management
(including production) systems and 15% can be accredited to production workers.
He is also an advocate of Pareto chart analysis. Dr. Kaoru Ishikawa is
credited for the creation of Quality Circles in 1960 (in Japan). These circles
consist of small groups of people who use knowledge of the process and
statistical data to analyze and solve problems. In the US, some quality circles
have not been as successful when the participants where not given the
statistical tools or data to locate and solve production problems. Although
there are many facets to quality improvement, the WINKS program only
concentrates on control chart calculation and presentation and Pareto Charts.
Pareto Charts are covered in a following section in this manual.
WINKS'
QC Features
WINKS'
QC tools can be a valuable part of making an overall QC plan work.
Generally, the portion of the quality control system that is addressed by
WINKS is that of estimating process variability. A process is examined by taking
samples of the process over a period of time. Using certain criteria that
produce limits of acceptability, a process is said to be in statistical control
as long as the measured item stays within acceptable limits. If the item
measured goes beyond acceptable limits, the process is said to be out of
statistical control.
For
example, in the production of a microprocessor chip, the company is interested
in how many of the chips are acceptable, and how many fail an initial test.
If the proportion of failed components grows beyond an acceptable limit,
it may signal a failure of the process along the production line.
Other tests along the line may be instituted to locate the source of the
problem.
Another
example might be a critical measurement on a bearing. The company buying the
bearing demands that the bearing meet a certain tolerance--that is the size of
the bearing may not vary by more than a very small amount. If the bearing is
shipped, and the customer finds that some bearings are too small or too large,
the entire shipment may be refused. Therefore, it is imperative that the
manufacturer keep a close eye on the process. It would be too costly to measure
every bearing, so a sampling scheme is devised, and a chart is produced that
shows the average dimension of the sample bearings over time. If the chart
reveals that some of bearings are beyond the statistical limits, manufacturing
may be halted until the problem is found and corrected.
The
WINKS Quality Control module allows you to create five basic kinds of control
charts. They are the X-Bar chart,
the R-Chart, the S-Chart, the P-Chart and the Individual Measurements Chart.
The
X-Bar Chart derives its name from the symbol for the mean (average), an X with a
bar on top. The X-Bar chart is a
chart of the mean value of a sample taken over time.
For example, in the case of the bearing manufacturer mentioned above, a
sample of 5 bearings is taken from the manufacturing process once every 15
minutes during the day. The
diameter of the bearings are immediately measured, and the average of the five
bearing diameters are plotted on a chart. Statistical Control limits describe
the limits of natural variation (i.e., based on within subgroup variation). If
the average diameter of the bearings goes above or below the statistical limits,
the manufacturing process is immediately halted, and engineers begin examining
the machines to find the source of the problem.
Sometimes
the average is not a sufficient measure for a process.
For example, if the bearings are "on the average" okay, but the
size varies widely, then this may give the manufacturer important information
about the variability of the production process. Two charts provided to evaluate
"within" sample variance over time are the R-Chart (Range Chart) and
the S-Chart (Standard Deviation Chart). In the example above, along with X-Bar
chart, an R-Chart or S-Chart may be produced.
The Range chart shows the range (max-min) for each sample subgroup of 5
bearings. An S-Chart would show the variability of the sample based on the
standard deviation of the 5 diameters. If the variability is large (even if the
process is "in control" according to the X-Bar Chart), the management
may want to examine the process and discover the cause of the excess
variability. You might want to include some sort of sampling further down the
production line to catch errors at their point of origin rather than at a later
checkpoint. See "X-Bar and R-Chart Example" later in this topic.
Note:
R-charts
and S-charts are a graphical form of the test for "constant variance",
which is a requirement for an Analysis of Variance test.
The
P-Chart measures a proportion rather than a mean.
It is intended for use on a process in which there are counts of
"failure". For example,
in the microprocessor example, a failure is the inability of a chip to pass a
performance test. Since the
manufacturer may be producing thousands of chips an hour, it would be very
costly to have a large proportion of bad chips installed in a component, only to
find that the component does not work because of the bad chip.
A process of testing the chips can be devised so that the number of bad
chips is kept to a minimum (hopefully near zero).
If the proportion of failed chips rises above a set limit, the process is
examined to find the source of the problem.
Lower control limits for a P-chart are often ignored since this condition
is usually desirable. See the
"Displaying P-Charts" later in this topic.
The
upper and lower limits placed on the various control charts can be devised by
whatever makes sense to the process being measured. However, there are some
common ways to choose these limits. Recall that a collection of quantitative
observations has a distribution, and if those numbers have a normal (gaussian,
bell-shaped) type distribution, a measure
for spread is the standard deviation. Thus, when data are normally distributed,
(or if the sample sizes are large enough so the X-bars are approximately normal)
you can calculate confidence intervals based on the standard error of the mean.
An interval that is one standard error above and below the mean includes the
true mean about 68.3% of the time. An interval of two standard errors includes the true mean
about 95.4% of the time, and an interval of three standard deviations includes
the true mean about 99.7% of the time. To
devise lower and upper limits, most programs (including WINKS) use the
"three-sigma" (three standard errors) criteria.
That is, the limits are chosen so that there is about a 99.7% chance that
the true mean is within the limits. If
a mean is observed outside these limits, it is a "rare" event, and
signals an out of control situation. Limits
for R-Charts, S-Charts and P-Charts are also devised with the 3-sigma criteria.
In the WINKS program, if you do not want to use the 3-sigma limits, you
are given the option to enter your own upper and lower limits for the charts.
Note:
Control limits for an X-Bar and P-Chart can change from sample to sample
depending on sample size.
Note:
The term "in control" means in "statistical control".
That is, a process should not be called in control or out of control
based on the statistical calculations unless the limits make sense for the item
being measured. Statistical control
relates to points occurring between some mathematically calculated limits, and
may or may not have a relationship to reality.
X-Bar
and R-Charts
The
data for X-Bar or R-Charts should be stored in a database in the following
format:
The
data that will be used to calculate the means to be plotted come from a sample
of observations, with each sample containing a number of replicates. For
example, you might take samples of jars filled with jelly 25 times during the
day. Each time you take a sample, it consists of 3 jars. Thus, you have 25
samples, each with a size of 3 (3 replicates). The data for this chart would be
stored in a database using the following setup:
Observation
Sample Value
1
1 15.9
2
1 16.1
3
1 16.0
:
: :
4
2 16.2
5
2 15.9
:
: :
6
3 15.6
Each
jar should contain 16 ounces of jelly. You do not want the jars too empty or too
full. Thus, you may want to see that the average amount of jelly does not go
under or over certain limits. Also, you do not want the range to be too wide --
which may mean that the "average" jar contains 16 ounces, but the
amount in different jars may vary widely.
The
database needed for this analysis would contain 2 fields, SAMPLE and VALUE. As
an example, consider another problem: A manufacturer of automobile piston rings
measures the diameter (in millimeters) of the piston rings to track the accuracy
of the process. From 3 to 5 pistons were taken from 25 samples. The database to
perform this analysis must contain at least two fields, SAMPLE and OBSERVED. The
data are from Montgomery (1991, p 237). The data are in the file PISTONS.DBF. A
partial listing of the data is:
SAMPLE
OBSERVED
1
74.030
1
74.002
1
73.992
1
74.008
2
73.995
2
73.995
etc.
To
analyze this data, follow these steps:
Step
1: Open the database named PISTON.
Step
2: From the Analyze menu, select "Quality Control Charts" then
"X-Bar and R-Chart)".
Step 3: Choose the variable SAMPLE as the group variable and OBSERVED as the observation variable. Once you specify what fields to use, WINKS will display a preliminary screen showing the grand mean (X double-bar) and grand range (R-bar) and containing an options menu. See Figure 3.
Figure 3
From this options menu you can select to display the control chart, specify a custom mean and limits, or view/print a listing of the computations for the control limits . When you continue, the results of the analysis are displayed in the viewer. Click on Graph to display the X-Bar control chart.
Click
the Option button to select options such as axes labels, whether or not to
display control limits and the R-Chart.
It
turns out that the X-Bar chart reveals that the process is in control since no
points cross the upper or lower limits. Notice the upper and lower limits differ
across the range of the samples. When the sample size is smaller, the limits are
wider. This makes sense because with more data, you have a better estimate of
the mean. When you choose the option to view the results of the calculations,
you can examine the values for the mean and range at each sample and the upper
and lower limits for the mean and range. See Figure 4.
Figure 4
X-Bar
and S-Charts
You
can use the same PISTON data to display an X-Bar and S-Chart. The difference is
that the S-Chart, which appears at the bottom of the screen where the R-Chart
was, shows a plot of the standard deviations instead of ranges.
Otherwise, the charts are the same.
To
display an S-Chart, choose the "Control Chart for Means (X-Bar and
S-Chart)" option from the main
Analyze/"Control Charts" menu.
Control
Charts for Individual Measurements
In
a situation where the sample size used in process control is 1, a Control Chart
for Individual Measurements is appropriate.
The
result is similar to the X-Bar chart described in the previous X-Bar and R-Chart
example. A database on disk named PAINT contains information about the viscosity
for an aircraft primer paint (Montgomery, 1991, p. 242).
The control Chart for PAINT is shown in figure 5.
Figure 5
P-Charts
P-Charts
plot a proportion of items observed from within a sample. For example, you might
take a sample of 25 items from a manufacturing process each hour.
Then you count the number of defects in that sample.
You are interested in plotting the proportion of defects across time to
observe if an unusually high number of defects begin to occur.
The
format for the database is:
Sample
Size Splits
Holes
1
30 1
5
2
30 4
1
3
30 2
2
etc.
For
example, a large lumber yard makes a great deal of its profits from the sales of
plywood. Recently, customers have
been returning plywood sheets and asking for refunds. The customers complain that the sheets contain too many
defects such as splits or holes. The
owner of the lumber yard decided to carry out an investigation by looking at 150
plywood sheets that are produced each day for 20 days and recording the number
of sheets that contained splits or holes. The
data is in the database called PLYWOOD.DBF. To perform this example, follow
these steps:
Step
1: Open the database named PLYWOOD.DBF.
Step
2: From the Analyze menu, select "Quality Control Charts" then
"Proportions Chart (P-Chart)".
Step
3: Choose SAMPSIZE as the sample size variable and SPLITS and HOLES as the
defect count fields.
Step
4: The results will be displayed in the viewer. Click on Graph to display the
p-chart. The p-Chart is shown in figure 6.
Figure 6
The
SPLITS field contains the number of splits observed and the HOLES field contains
the number of holes. Thus, for
sample 1 the proportion of total defects found is 6/30.
Using this same database, you can also create a P-Chart that only
considers defects of type SPLITS. In
this case, your would choose only the Sample Size and SPLITS fields for
analysis.
The
minimum number of fields in the WINKS database needed for this chart is two, a
Sample Size field and a Count field. If
there are more than one Count (defects) fields, the program will add up the
defect fields to calculate the proportion defects for that sample. You do not
need a SAMPLE field, but you may want one if the field will contain information
about the sample, such as the hour taken.
In
the plywood example, when the P-Chart for the total defects is plotted, the
chart is in control with an average defect rate of about 4%.
If this is satisfactory to the lumber yard then production may continue.
If not, the lumber yard owner must decide to take steps to reduce the
defect rate. Note that even though
the chart that contains both types of defects is in control, if you plot
separate charts for splits and holes, it is discovered that the splits chart is
in control but the one for holes is not. This might indicate to management that
attention should be given to reducing this particular type of defect (holes). Also, the manufacturer may want to set an upper limit for
defects (say 0.06). If this were
done, then the original p-Chart would show several instances of being out of
control.
Notes
on Analyzing Control Limits
The
following items are adapted from The General Electric Handbook (1956) that
contains suggestions for deciding when a process is out of control:
