# Statistics are very important in the field of quality

278
As we view the world about us, statistics are everywhere. We hear statistics about politics, health, inflation, and the economy on digital or social media on a daily basis. Yet many people view the topic of statistics with fear, loathing, and trembling. The purpose of statistics is clear. Statistics is a group of tools that allows us to analyze data, make summaries, draw
inferences, and generalize from data.
Statistics are very important in the field of quality. In fact, during the first half century of
the quality movement, nearly all the work done in the field of quality related to statistics. This
work resulted in a body of tools that are used worldwide in thousands of organizations.
This chapter focuses on the use of statistical tools—not as control mechanisms, but as
the foundation for continual improvement. We present many statistical techniques and different
types of control charts. These tools represent powerful techniques for monitoring and improving processes. We also discuss the behavioral aspects of statistical process improvement. It is
important to recognize that it is not enough to learn the different charts and statistical techniques.
We also must know how to apply these techniques in a way that will document and motivate
continual improvement in organizations.
These techniques can be enjoyable to use, and we present them in a way that is intuitive
and easy to understand. Where possible, we develop shortcuts and simple statistical techniques
instead of more complex models. The primary goal is that these tools be used.
Chapter Objectives
After completing this chapter, you should be able to:
1. Discuss the basics of process variation and applied statistical methods.
2. Demonstrate the differences between random and non-random variation.
3. Implement x, R, X, MR, median, and s charts.
4. Develop control charts using Excel.
5. Interpret control charts.
C H A P T E R 1 1
Statistically Based
Quality Improvement
for Variables
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 279
StatiStical FundamentalS
What is Statistical thinking?
Statistical thinking is a decision-making skill demonstrated by the ability to draw conclusions
based on data. We make a lot of decisions based on intuition and gut feelings. Often we choose
friends, homes, and even spouses based on feelings. Therefore, intuition and feelings are very
important in making good decisions in certain circumstances.
However, intuitive decisions are sometimes biased and wrong-headed. Consider the case
of government. Many times it is the most vocal groups that seem to control political agendas. It is
difficult for mayors, governors, or presidents to determine exactly what the voting public wants
on any issue. As a result, decisions are sometimes made that satisfy the few but irritate the many.
Statistical thinking is based on these three concepts:
š All work occurs in a system of interconnected processes.
š All processes have variation (the amount of variation tends to be underestimated).
š Understanding variation and reducing variation are important keys to success.
In business, decisions need to be made based on data. If you want to know how to satisfy
your customers, you need to gather data about the customers to understand their preferences.
It is one thing to watch a production process humming along. It is a completely different thing
to gather data about the process and make adjustments to the process based on data. Statistical
thinking guides us to make decisions based on the analysis of data (see Quality Highlight 11-1 ).
QUALITY HIGHLIGHT 11-1 Statistical Tools in Action
Statistical tools have long been staples of the quality professional. Around the world, many firms have
adopted statistical tools with good results. One of the appealing features of statistical tools is that they
can be adapted and used in a wide variety of situations. For example, Ore-Ida Corporation, a subsidiary
of Heinz Corporation and a nationally known producer of consumer food products, uses statistics to
ensure that its food meets weight and measure requirements. One of the products that Ore-Ida produces
is called a Pita Pocket sandwich. The problem with the Pita sandwich was that if the sandwiches were
too large, they would not fit into the formed plastic Pita-holding package, and excess costs would be
incurred. If the sandwiches were too small, customers would perceive them as having less value. As a
result, Ore-Ida used statistical process control, experimental design, and process capability studies to
ensure that the sandwiches met requirements.
Statistical process control is not always immediately successful. Simplot Corporation, a competitor of Ore-Ida’s, attempted statistical process control and other tools of quality in its Caldwell, Idaho, facility. According to Bob Romero, manager of total quality management services, the company had to do
an educational assessment of its employees, which resulted in a picture that was anything but flattering.
Many of Simplot’s employees had marginal literacy skills. As a result, the company undertook a lengthy
program of training and education in literacy, after which new standards were created for employees that
included overall standards for literacy and the ability to use word-processing software and spreadsheets.
After completing this program, management again implemented statistical process control and other
quality management tools. This time they were successful in improving processes and reducing costs.
Jaco Manufacturing Company, a producer of industrial components, tube fittings, and injectionmolding machines, implemented statistical process control, process capability studies, and quality
management tools as a means for improving customer service. G. K. Products, Inc., of Ann Arbor,
Michigan—a Jaco customer—asked Jaco to reduce its costs by improving its inspection of plastic float
bodies. These float bodies are used in gas tanks so that the flow of fuel to the car’s engine will shut off
in case of rollover. Benefits that were achieved through this program included a 14% reduction in cycle
time, a decrease in scrap, thousands of dollars in cost savings, improved morale, and improved customer
satisfaction.
(continued)
280 3DUW š ,PSOHPHQWLQJ4XDOLW\
Why do Statistics Sometimes Fail in the Workplace?
Before beginning a discussion of statistical quality improvement, we must remember that many
times statistical tools do not achieve the desired results. Why is this so? Many firms fail to implement quality control in a substantive way. That is, they prefer form over substance. We provide
several reasons as a guide, which you can use to assess whether your organization will be successful in using statistics to improve. Reasons for failure of statistical tools include the following:
š Lack of knowledge about the tools; therefore, tools are misapplied.
š General disdain for all things mathematical creates a natural barrier to the use of statistics.
When was the last time you heard someone proclaim a love for statistics?
š Cultural barriers in a company make the use of statistics for continual improvement difficult.
š Statistical specialists have trouble communicating with managerial generalists.
š Statistics generally are poorly taught, emphasizing mathematical development rather than
application.
š People have a poor understanding of the scientific method.
š Organizations lack patience in collecting data. All decisions have to be made “yesterday.”
š Statistics are viewed as something to buttress an already-held opinion rather than a method
for informing and improving decision making.
š People fear using statistics because they fear they may violate critical statistical assumptions.
Time-ordered data are messy and require advanced statistical techniques to be used effectively.
š Most people don’t understand random variation, resulting in too much process tampering.
š Statistical tools often are reactive and focus on effects rather than causes.
š Another reason why people make mistakes with statistics is founded in the notions of type
I and type II errors. In the study of quality, we call a type I error producer’s risk and a
type II error consumer’s risk. In this context, producer’s risk is the probability that a good
product will be rejected. Consumer’s risk is the probability that a nonconforming product
will be available for sale. Consumer’s risk happens when statistical quality analysis fails
to result in the scrapping or reworking of a defective product. When either type I or type II
errors occur, erroneous decisions are made relative to products that can result in high costs
or lost future sales. Given these problems, we adopt the approach that statistics should be
used, they should be used correctly, and they should be taught correctly.
understanding Process Variation
All processes exhibit variation. There is some variation that we can manage and other variation
that we cannot manage. If there is too much variation, parts will not fit correctly, products will
not function properly, and a firm will gain a reputation for poor quality.
Two types of process variation commonly occur: random and nonrandom variation. Random
variation is uncontrollable, and nonrandom variation has a cause that can be identified. The statistical tools discussed later in this chapter are useful for determining whether variation is random.
Random variation is centered around a mean and occurs with a somewhat consistent
amount of dispersion. This type of variation cannot be controlled, so we refer to it as “uncontrolled variation.” The amount of random variation in a process may be either large or small.
When the variation is large, processes may not meet specifications on a consistent basis.
The statistical tools discussed in this chapter are not designed to detect random variation.
Figure 11-1 shows normal distributions resulting from a variety of samples taken from the same
population over time. We find a consistency in the amount of dispersion and the mean of the
As we learn from these examples and countless others, companies around the world either now
have implemented statistical quality tools or are in the process of adopting these tools. All processes exhibit variability. This fact alone makes statistical process tools invaluable to manufacturing and services
companies alike.
Video Clip:
Behavioral Aspects of SPC
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 281
process. The fact that not all observations within the distributions fall exactly on the target line
shows that there is variation. However, the consistency of the variation shows that only random
causes of variation are present within the process. This means that in the future, when we gather
samples from the process, we can expect that the distributions associated with such samples also
will take the same form.
Nonrandom (or “special cause”) variation results from some event. The event may be a
shift in a process mean or some unexpected occurrence. For example, we might receive flawed
materials from a supplier. There might be a change in work shift. An employee might come to
work under the influence of drugs and make errors. The machine may break or not function
properly. Figure 11-2 shows distributions resulting from a number of samples taken from the
same population over time where nonrandom variation is exhibited. Notice that from one sample
to the next, the dispersion and average of the process are changing. When we compare this figure
to random variation, it is clear that nonrandom variation results in a process that is not repeatable.
Target
Sample 3
Sample 2
Sample 1
Consistent means and
consistent dispersion
Figure 11-1 Random Variation
Target
Sample 3
Sample 2
Sample 1
Initial process
state
Smaller variance
and shift in mean
Increased variance
and shift in mean
Figure 11-2 Nonrandom Variation
282 3DUW š ,PSOHPHQWLQJ4XDOLW\
Process Stability
Process stability means that the variation we observe in the process is random variation (common cause) and not nonrandom variation (special or assignable causes). To determine process
stability, we use process charts. Process charts are graphs designed to signal process workers
when nonrandom variation is occurring in a process.
Sampling methods
To ensure that processes are stable, data are gathered in samples. Process control requires that
data be gathered in samples. For the most part, sampling methods have been preferred to the
alternative of 100% inspection. The reasons for sampling are well established. Samples are
cheaper, take less time, are less intrusive, and allow the user to frame the sample. In cases where
quality testing is destructive, 100% inspection would be impossible and would literally drive the
company out of business. In some processes, chemicals are used in testing, or destructive pull
tests are applied to cables. These destructive tests ruin the sample, but are useful to show that a
good product is being made.
However, recent experience has shown that 100% inspection can be effective in certain
instances. One hundred percent samples are also known as screening samples, sorting samples,
rectifying samples, or detailing samples. They have been most common in acceptance sampling
(refer to Chapter 9 Appendix), where a lot of material has been rejected in the past and materials
must be sorted to keep good materials and return defective materials for a refund.
Another example of 100% inspection is used when performing in-process inspection.
Many companies have asked their employees to inspect their own work as the work is being
performed, which can result in 100% inspection at every stage of the process! We should clarify
that in-process inspection also can be performed on a sampling basis. Because sampling is so
important, let’s look at some different types of samples.
random Samples
Randomization is useful because it ensures independence among observations. To randomize
means to sample in such a way that every piece of product has an equal chance of being selected
for inspection. This means that if 1,000 products are produced in a single day, each product has
a 1/1,000 chance of being selected for inspection on that day. Random samples are often the
preferred form of sampling and are yet often the most difficult to achieve. This is especially true
in process industries, in which multiple products are made by the same machines, workers, and
processes in sequence. In this case, there is not independence among observations because the
process results in ordered products that can be subject to machine drift (going out of adjustment
slowly over time).
Systematic Samples
Systematic samples have some of the benefits of random samples without the difficulty of randomizing. Samples can be systematic according to time or according to sequence. If a sample
is systematic according to time, a product is inspected at regular intervals of time, say, every
15 minutes. If a systematic sample is performed according to sequence, one product is inspected
every tenth iteration. For example, every tenth product coming off the line is sampled.
Sampling by rational Subgroups
A rational subgroup is a group of data that is logically homogeneous; variation within these data
can provide a yardstick for computing limits on the standard variation between subgroups. For
example, in a hospital it may not make sense to combine measurements such as body temperature or medication levels taken in the morning with measurements taken in the evening. Morning
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 283
measurements occur before medications are given and before the first meal of the day and constitute a rational subgroup. Evening measurements, another such subgroup, are taken after treatment has been provided during the day, medications have been administered, and patients have
been nourished. If data are gathered that combine these two subgroups (e.g., the night and day
measurements), differences between morning results and evening results will not be detected.
If variation among different subgroups is not accounted for, an unwanted source of nonrandom
variation is being introduced.
Planning for inspection
As you can see, much planning must be performed in developing sampling plans. Questions
must be answered about what type of sampling plan will be used, who will perform the inspection, who will use in-process inspection, sample size, what the critical attributes to be inspected
are, and where the inspection should be performed. There are rules for inspection that help to
prioritize where the inspection should be performed. Many firms compute the ratio between the
cost of inspection and the cost of failure resulting from a particular step in the process, in order
to prioritize where inspection should occur first.
control Plans
Control plans are an important part of a quality control system and are a required part of an ISO
9000 quality management system (QMS). After performing process failure modes and effects
analysis (PFMEA), inspection is put in place at critical points in a process. Control plans provide a documented, proactive approach to defining how to respond when process control charts
show that a process is out of control.
Figure 11-3 shows a sample control plan. In the production of a sprinkler head, four machines are used. The control plan outlines critical product characteristics that can be observed
or measured at each stage of the process. It shows how the inspection is to be performed and
prescribes reactions when a problem is detected.
The far-right column is referred to as the reaction plan. Control plans are usually kept by
the quality assurance people involved with ISO 9000 and are made available to line workers who
are responsible for executing process controls.
ProceSS control chartS
Now that we have discussed variation, it is time to learn about the tools used to understand
random and nonrandom variation. Statistical process control charts (also referred to as process
charts or control charts) are tools for monitoring process variation. Figure 11-4 shows a control
chart that has an upper limit, a center line, and a lower limit. Several different types of control
charts are discussed later. In this chapter, we introduce different statistical charts one by one.
However, you should know there is a generalized process for implementing all types of process
charts that we introduce first. This is a useful approach to learning control charts because the
process for establishing different types of control charts is the same. Although the process for
establishing different control charts is the same, the formulas used to compute the upper limit,
center line, and lower limit are different. You will learn the formulas later.
Variables and attributes control charts
To select the proper process chart, we must differentiate between variables and attributes. As we
already stated, a variable is a continuous measurement such as weight, height, or volume. An
attribute is an either-or situation. Here are examples of attributes: The motor is either starting
or not starting, or the lens is scratched or it is not. We discuss measurements in this chapter and
attributes in Chapter 12. While discussing attributes, we will also introduce reliability theory.
284 3DUW š ,PSOHPHQWLQJ4XDOLW\
Part Number: Part Name Description: Plant:
Sprinkler Head
Characteristics Methods
Process
Name/
Operation
Machine/
Tools
Fixture Product Process
Specifications/
Tolerance
Measurement
Technique
Sample
Size
Sample
Freq.
Control
Method Reaction Plan
Plastic
Injection
Molding
Machine 1 Appearance Free of Blemishes Visual Inspection 100% Continuous 100%
Inspection
Notify
Supervisor
Flow Marks 1st Piece
Inspection
Check
Sheet
Adjust Machine
Pot Holes 1st Piece
Inspection
Check
Sheet
Adjust Machine
Machine 2 Mounting
Hole
Location
Nozzle Diameter Vernier Caliper Rectifying
Sample
Check
Sheet
Adjust Machine
15 ± 1 mm Every 1/2
Hour
x & R Chart Quarantine and
Cause-andEffect Analysis
Machine 3 Dimensional 6 ± 1 mm Fixture 2 Sample Check
Sheet
Adjust/Recheck
Fixture #1 Perimeter 6 ± 1 mm Check Gap to
Fixture and Datum
Every 1/2
Hour
x & R Chart Quarantine and
Cause-andEffect Analysis
Figure 11-3 Control Plan
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 285
7DEOH11-VKRZVWKHPRVWFRPPRQW\SHVRIYDULDEOHDQGDWWULEXWHFKDUWV7KHYDULDEOHV
FKDUWVDUHX, xEDU1x2, R, MRDQGsFKDUWV7KHDWWULEXWHVFKDUWVDUHWKHp, np, c,DQGuFKDUWV,Q
WKHIROORZLQJSDJHVZHLQWURGXFHDQGGHYHORSWKHVHEDVLFFKDUWV:HEHJLQE\LQWURGXFLQJWKH
JHQHULFSURFHVVIRUGHYHORSLQJDOOFKDUWVDQGWKHQGLVFXVVWKHFKDUWVLQGLYLGXDOO\7KHUHDUHILYH
FHQWUDOUHTXLUHPHQWVIRUSURSHUO\XVLQJSURFHVVFKDUWV
1. <RXPXVWXQGHUVWDQGWKLVJHQHULFSURFHVVIRULPSOHPHQWLQJSURFHVVFKDUWV
2. <RXPXVWNQRZKRZWRLQWHUSUHWSURFHVVFKDUWV
3. <RXQHHGWRNQRZZKHQGLIIHUHQWSURFHVVFKDUWVDUHXVHG
4. <RXQHHGWRNQRZKRZWRFRPSXWHOLPLWVIRUWKHGLIIHUHQWW\SHVRISURFHVVFKDUWV
5. :HWUHDWHDFKRIWKHVHWRSLFVVHSDUDWHO\
A Generalized Procedure for Developing Process Charts
7KHSURFHVVIRUGHYHORSLQJDSURFHVVFKDUWLVWKHVDPHIRUDOPRVWDOOFKDUWV7KHRQO\GLIIHUHQFHV
DUHLQWKHDFWXDOVWDWLVWLFDOFRPSXWDWLRQV)ROORZLQJDUHWKHVWHSVXVHGLQGHYHORSLQJSURFHVV
FRQWUROFKDUWV
1. ,GHQWLI\critical operationsLQWKHSURFHVVZKHUHLQVSHFWLRQPLJKWEHQHHGHG7KHVHDUH
RSHUDWLRQVLQZKLFKWKHSURGXFWZLOOEHQHJDWLYHO\DIIHFWHGLIWKHRSHUDWLRQLVSHUIRUPHG
LPSURSHUO\
2. ,GHQWLI\critical product characteristics7KHVHDUHWKHDVSHFWVRIWKHSURGXFWWKDWZLOOUHVXOWLQHLWKHUJRRGRUSRRUIXQFWLRQLQJRIWKHSURGXFW
3. ‘HWHUPLQHZKHWKHUWKHFULWLFDOSURGXFWFKDUDFWHULVWLFLVDYDULDEOHRUDQDWWULEXWH
4. 6HOHFWWKHDSSURSULDWHprocess control chartIURPDPRQJWKHPDQ\W\SHVRIFRQWUROFKDUWV
7KLVGHFLVLRQSURFHVVDQGWKHW\SHVRIFKDUWVDYDLODEOHDUHGLVFXVVHGODWHU
5. (VWDEOLVKWKHcontrol limitsDQGXVHWKHFKDUWWRcontinually monitor and improve
6. Update the limitsZKHQFKDQJHVKDYHEHHQPDGHWRWKHSURFHVV
Understanding Process Charts
%HIRUHVKRZLQJKRZWRHVWDEOLVKSURFHVVFKDUWV\RXQHHGWRXQGHUVWDQGZKDWSURFHVVFKDUWVDUH
DQGKRZWKH\ZRUN:HXVHWKHFKDUWWRLOOXVWUDWHWKHIDFWWKDWWKHSURFHVVFKDUWLVQRWKLQJPRUH
Upper Control
Limit (UCL)
Center Line
(CL)
Lower Control
Limit (LCL)
Each point
represents data
from a sample
that are plotted
sequentially
The UCL, CL,
and LCL are
computed
statistically
FiGUre 11-4 Control
Chart
TABLE 11-1 Variables and Attributes
Variables Attributes
X (process population average) p (proportion defective)
x (mean or average) np (number defective or number nonconforming)
R (range) c (number nonconforming in a consistent sample space)
MR (moving range)
s (standard deviation) u (number defects per unit)
286 3DUW š ,PSOHPHQWLQJ4XDOLW\
than an application of hypothesis testing where the null hypothesis is that the process is stable.
An x Q chart is a variables chart that monitors average measurements. For example, suppose you
were a producer of 8½-inch * 11-inch notebook paper. Because the length of paper is measured
in inches, a variables chart such as the x chart is appropriate. If the length of the paper is a key
critical characteristic, we might inspect a sample of sheets to see whether the sheets are indeed
11 inches long.
To demonstrate how a process control chart works, we could use a hypothesis test instead
of a process chart to determine whether the paper is really 11 inches long. Therefore, the null
hypothesis is
H
0:m = 11 inches
The alternative hypothesis is
H
1:m ≠ 11 inches
To perform the hypothesis test, we establish a distribution with the following 95% (Z = 1.96)
rejection limits. If the standard error of the sample distribution (n = 10) is .001 inch, the rejection
limits are 11 ± 1.96(.001) = {11.00196, 10.99804}. Figure 11-5 shows the distribution with its
rejection regions.
Next, to test this hypothesis, we draw a sample of n = 10 sheets of paper and measure the
sheets. The measurements are shown in the following table:
Sheet Number Measurement
1 11.0001
2 10.9999
3 10.9998
4 11.0002
5 11.0004
6 11.0020
7 10.9980
8 10.9999
9 10.9870
10 11.0004
Sum 109.9877
Sample mean 10.99877
11.00
m
11.00196
m 1 1.960 s
10.99804
m 2 1.960 s
Rejection
region
Rejection
region
10.99877
(sample mean)
Figure 11-5
Hypothesis Testing
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 287
Because 10.99877 does not fall within either of the rejection regions shown in Figure 11-5,
we fail to reject the null hypothesis and conclude that the sheets do not differ significantly from
an average of 11 inches.
This is a basic hypothesis test. Now we use a process control chart to monitor this paper
production process. With process charts, we place the distribution on its side, as shown in Figure 11-6.
We draw a center line and upper and lower rejection lines, which we call control limits. We then
plot the sample average (10.99877) on the control chart. Because the point falls between the control limits, we conclude that the process is in control. This means that the variation in the process
is random (common).
Notice that the preceding example was based on a sample of n = 10. This means that
the distribution we drew was a sampling distribution (not a population distribution). Therefore,
we can invoke the central limit theorem. The central limit theorem states that when we plot the
sample means, the sampling distribution approximates a normal distribution.
_x
and R charts
Now that we have developed a process chart, you need to understand the different types of charts.
First, we discuss two charts that go hand in hand: x and R charts. We developed an x chart in the
prior example. When we are interested in monitoring a measurement for a particular product in a
process, there are two primary variables of interest: the mean of the process and the dispersion of
the process. The x chart aids us in monitoring the process mean or average. The R chart is used
in monitoring process dispersion.
The x chart is a process chart used to monitor the average of the characteristic being measured. To set up an x chart, select samples from the process for the characteristic being measured
and then form the samples into rational subgroups. Next, find the average value of each sample
by dividing the sums of the measurements by the sample size and plot the value on the process
control x chart.
The R chart is used to monitor the dispersion of the process. It is used in conjunction with
the x chart when the process characteristic is a variable. To develop an R chart, collect samples
from the process and organize them into subgroups, usually of three to six items. Next, compute
the range R by taking the difference of the high value in the subgroup minus the low value. Then
plot the R values on the R charts.
A standard process chart form is shown in Figure 11-7. This form has spaces for measurements and totals. In the example in Figure 11-8, our control chart form is filled out with
measurements from a process. Notice that there are k = 25 samples of size n = 4. For each of
the samples, totals, ranges, and averages are computed. The range is the difference between the
largest measurement and the smallest measurement in a particular sample.
Now that we have measurements, we need to compute a center line and control limits for
our x and R charts. The center line is the process average. The upper and lower control limits
are usually located three standard deviations from the center line. The formulas for computing
these lines are given in Figure 11-9. Figure 11-10 shows the completed formulas for the example
in Figure 11-8. Notice that the A2 and D4 table values come from the Factor for Control Limits
Upper control limit
Center line
10.99804 Lower control limit
10.99877
11
11.00196
Figure 11-6 Process Chart
288 3DUW š ,PSOHPHQWLQJ4XDOLW\
table in the lower-right corner of Figure 11-10. These table values provide estimates for the three
standard deviation limits for the x and R charts (when combined with R-bar [the mean of the R
values]). You also may notice that no formula is given for the lower limit of the R chart. This is
because the lower limit of R is zero for sample sizes less than or equal to six. For sample sizes
greater than six, D3 values must be used from Table A-1 in the appendix (the formula for the
lower control limit is shown in Table A-1). Notice that we have superimposed the control limits
computed in Figure 11-10 on the charts in Figure 11-8.
interpreting control charts
Before introducing other types of process charts, we discuss the interpretation of the charts.
Figure 11-11 shows several different signals for concern that are sent by a control chart, as
shown in the second and third boxes. When a point is found to be outside of the control limits,
we call this an “out-of-control situation.” When a process is out of control, variation is probably no longer random. If there are three standard deviation limits, the chance of a sample
average or range being out of control when the process is stable is less than 1%. Because this
Date
Time
1 2 3 4 5
Sum
Average, x
Range, R
Notes
Part number Chart number
Part name (product) Operation (process) Specification limits
Operator Machine Gauge Unit of measure Zero equals
Sample
measurements
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Ranges Averages
Variables Control Chart (x and R)
Figure 11-7 x and R Chart
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 289
probability is so small, we conclude that this was a nonrandom event and search for an assignable cause of variability.
Figure 11-11 presents examples of where nonrandom situations occur. You need not only
have an out-of-control situation to signal that a process is no longer random. Two points in succession farther than two standard deviations from the mean likely will be a nonrandom event
because the chances of it happening at random are very low. Five points in succession (either all
above or below the center line) are called a process run, which means that the process has shifted.
Seven points that are all either increasing or decreasing result in process drift. Process drift usually means that either materials or machines are drifting out of alignment. An example might be
a saw blade that is wearing out rapidly in a furniture factory. Large jumps of more than three or
four standard deviations result in erratic behavior. In all these cases, process charts help us to
understand when the process is or is not in control.
If a process loses control and becomes nonrandom, the process should be stopped immediately. In many modern process industries where lean is used, this will result in the stoppage
of several workstations. The team of workers who are to address the problem should use a control planning structured problem-solving process using brainstorming and cause-and-effect tools
Date
Time
1 2 3 4 5
Sum
Average, x
Range, R
Notes
Sample
measurements
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Ranges Averages
Part number Chart number
Part name (product) Operation (process) Specification limits
Operator Machine Gauge Unit of measure Zero equals
Variables Control Chart (x and R)
Figure 11-8 Completed x and R Chart
290 3DUW š ,PSOHPHQWLQJ4XDOLW\
A
mR = =
___________
A
mR =
______
URL
x = US – AmR = LRLX = LS + AmR =
Control Limits
Subgroups included
Modified Control Limits f or Averages
Based on specification limits and process capability
Applicable only if US – LS > 6s
Factors f or Control Limits
Limits f or Individuals
Compare with specification or
tolerance limits
R = S —– R = ———————— = ——–
k
US =
US – LS =
LS =
———
X =
x (Midspec or std) =
A2R = = ———
UCL
x = X + A2R =
LCL
x = X A2R =
UCLR = D4R = =
X = S —– x = ———————— = ——–
k
or
US = LS =
A2
1.880
1.023
0.729
0.577
0.483
D4
3.268
2.574
2.282
2.114
2.004
d2
1.128
1.693
2.059
2.326
2.534
d2
2.659
1.722
1.457
1.290
1.184
Am
0.779
0.749
0.728
0.713
0.701
n 23456
3—
R = =
———
3d2
6d2
UL
x = x + 3 R =
d2
LLx = x – — 3 R =
d2
6 = — s R =
Figure 11-9 x and R Chart Calculation Work Sheet
A
mR = =
___________
A
mR =
______
URL
x = US – AmR = LRLX = LS + AmR =
Control Limits
Subgroups included
Modified Control Limits f or Averages
Based on specification limits and process capability
Applicable only if US – LS > 6s
Factors f or Control Limits
Limits f or Individuals
Compare with specification or
tolerance limits
R = S —– R = =
k
US =
US – LS =
LS =
X =
x (Midspec or std) =
A2R = =
––––––
UCL
x = X + A2R =
LCL
x = X A2R =
UCLR = D4R = =
X = S —– x = =
k
or
US = LS =
A2
1.880
1.023
0.729
0.577
0.483
D4
3.268
2.574
2.282
2.114
2.004
d2
1.128
1.693
2.059
2.326
2.534
d2
2.659
1.722
1.457
1.290
1.184
Am
0.779
0.749
0.728
0.713
0.701
n 23456
3—
— 3 R = =
d
2
6 = — 6 R =
d
2
UL
x = x + 3 R =
d2
LLx = x – — 3 R =
d2
s
Figure 11-10 Calculation Work Sheet for Figure 11-8 Data
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 291
such as those discussed in Chapter 10 to identify the root cause of the out-of-control situation.
Typically, the cause is somewhere in the interaction among processes, materials, machinery, or
labor. Once the assignable cause of variation has been discovered, corrective action can be taken
to eliminate the cause. The process is then restarted, and people return to work.
The cause of the problem should be documented and discussed later during the weekly
departmental meeting. All workers should know why a problem in the process occurred. They
should understand the causes and the corrective actions that were taken to solve the problem.
Production companies that embark on this level of delegation of authority and development of employees find the transition difficult because the process is often stopped, and work is
interrupted. However, as time passes, the processes become more stable as causes of errors are
detected and eliminated. One manufacturer regularly produced poor-quality material that needed
UCL = Upper control limit
CL = Center line
LCL = Lower control limit
UCL
CL
LCL
Normal behavior. One data point out, above.
Investigate for cause
of poor performance.
One data point out,
below. Investigate for
cause of improvement.
Erratic behavior.
Investigate.
Seven successive data
points on increasing
or decreasing line.
Investigate for cause
of progressive change.
UCL
CL
LCL
Two data points near
upper limit (beyond 2
standard deviations
from the mean).
Investigate for cause
of poor performance.
Two data points near
lower limit (beyond 2
standard deviations
from the mean).
Investigate for cause
of improvement.
Five successive data
points above central
line. Investigate for
cause of sustained
poor performance.
UCL
CL
LCL
Five successive data
points below central
line. Investigate for
cause of sustained
below-mean
performance.
UCL
CL
LCL
Sudden change in level.
Investigate.
Figure 11-11 Control Chart Evidence for Investigation Source: Hansen, Bertrand L. Quality Control:
Theory and Applications. Upper Saddle River, NJ: Pearson Education (1964). ISBN: 013745208X. ©1964, p.65. Reprinted
and Electronically reproduced by permission of Pearson Education, Inc., New York, NY.
292 3DUW š ,PSOHPHQWLQJ4XDOLW\
to be scrapped. As a result, it had increased its master production schedules by 20% to cover up
this problem. The company decided instead to embark on a lot-size reduction program coupled
with giving the workers line-stop authority. During the first shift, the company reduced the number of scrapped pieces from an average of more than 1,000 to 6! At first, production suffered.
However, within two weeks of implementation, output volume had increased by more than 30%.
This was the result of less rework, scrap, and other problems because of poor quality. It is interesting to note that staff and machinery were not changed during this period. At first, management
thought its workers were unmotivated, resulting in the poor work. It wasn’t the people; it was the
process and the management.
7.583
6.125
4.667
1 2 3 4
A
mR = =
___________
A
mR =
______
URL
x = US – AmR = LRLX = LS + AmR =
Control Limits
Subgroups included
Modified Control Limits f or Averages
Based on specification limits and process capability
Applicable only if US – LS > 6s
Factors f or Control Limits
Limits f or Individuals
Compare with specification or
tolerance limits
R = S —– R = ———————— = ——–
k
US =
US – LS =
LS =
———
X =
x (Midspec or std) =
A2R = =
———
UCL
x = X + A2R =
LCL
x = X A2R =
UCLR = D4R = =
X = S —– x = ———————— = ——–
k
or
US = LS =
A2
1.880
1.023
0.729
0.577
0.483
D4
3.268
2.574
2.282
2.114
2.004
d2
1.128
1.693
2.059
2.326
2.534
d2
2.659
1.722
1.457
1.290
1.184
Am
0.779
0.749
0.728
0.713
0.701
n 23456
3—
R = =
———
3d2 6s
= — 6 R =
d
2
UL
x = x + 3 R =
d2
LLx = x – — 3 R =
d2
Figure 11-12 Calculation Work Sheet and x Chart
Day x Means Ranges
1 6 6 5 7 6 2
2 8 6 6 7 6.75 2
3 7 6 6 6 6.25 1
4 6 7 5 4 5.5 3
Excel File: Example 11-1
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 293
ExamplE 11-1 Using x _ and R Charts
Problem: The Sampson Company produces high-tech radar that is used in top-secret weapons
by the Secret Service and the Green Berets. It has had trouble with a particular round component
with a target of 6 centimeters. Samples of size 4 were taken during four successive days. The
results are in the following table.
1
There are several software packages and Excel add-ins that create control charts. A good place to find free Excel control
chart templates is www.freequality.org or on www.pearsonhighered.com/foster.
Figure 11-13 Example
11-1 Using Excel.
Source: Microsoft Excel,
Microsoft Corporation.
Used by permission.
Solution: The grand mean is 6.125. R is 2.
Develop a process chart to determine whether the process is stable. Because these are measurements, use x and R charts. Using the calculation work sheet, Figure 11-12 shows the values
for the process control limits.
The x control chart for this problem is shown with the appropriate limits. The R chart is also
in control. The sample averages were placed on the control chart, and the process was found to be
historically in control. Because the averages and ranges fall within the control limits, and no other
signals of nonrandom activity are present, we conclude that the process variation is random. Note
that this example is very simple. Generally, you use 15 to 20 subgroups to establish control charts.
using excel to draw x – and R charts
The problem in EXAMPLE 11-1 can be solved easily using Excel. Although there are more
elegant ways to develop control charts in Excel,1 we will demonstrate a simple “brute force”
method for creating x and R charts in Excel.
As you can see in Figure 11-13, we place the data in rows. From this we compute averages (xs), Rs, and R. Using these data, the center line (CL), upper limits (UCL), and lower limits
(LCL) are computed. Figure 11-13 provides all the needed equations. Try doing this for yourself.
Active Model: Example 11-1
294 3DUW š ,PSOHPHQWLQJ4XDOLW\
X and moving range (MR) charts for Population data
At times it may not be possible to draw samples because a process is so slow that only one or two
units per day are produced. If you have a variable measurement that you want to monitor, the X
and MR charts might just be the thing for you.
Important caveats are associated with the X and MR charts. Because you will not be sampling, the central limit theorem does not apply, which may result in the data being non-normally
distributed and an increase in the likelihood that you will draw an erroneous conclusion using a
process chart. Therefore, it is best to first make sure that the data are normally distributed.
If data are not normally distributed, other charts are available. A g chart is used when data
are geometrically distributed, and h charts are useful when data are hypergeometrically distributed. In statistics, an X is an individual observation from a population. Therefore, the X chart
reflects a population distribution. We call the three standard deviation limits in an X chart the
natural variation in a process. This natural variation can be compared with specification limits.
So, strictly speaking, X chart limits are not control limits; they are natural limits.
The formula for the center line and the natural limits for an X chart is as follows:
x { E
2(MR) (11.1)
where
x =
ΣX
k
and
X = a population value
k = the number of values used to compute x
E2 = 2.66 (n = 2) (see Table A-1 in the Appendix)
The formula for theMRchart is similar to that for theRchart (wheren = 2), except that the ranges
are computed as the differences from one sample to the next [n = 2; UCL = D4(MR);LCL = 0].
ExamplE 11-2 X and MR Charts in action
Problem: The EA Trucking Company of Columbia, Missouri, hauls corn from local fields to
the SL Processing Plant in Lincoln, Nebraska. Although the trucks generally take 6.5 hours to
make the daily trip, recently there seems to be more variability in the arrival times. Mr. Everett,
the owner, suspects that one of his drivers, Paul, may be visiting his girlfriend Janice en route
in Kansas City. The driver claims that this is not the case and that the increase is simply random
variation because of variability in traffic flows. The drivers keep written logs of departure and arrival times. Mr. Everett has listed these times in the following table. You are chosen as the analyst
to investigate this situation. What do you think?
Date Travel Times (Hrs.) Moving Range
1 6.4 —
2 6.2 0.2
3 5.8 0.4
4 7.3 1.5
5 8.6 1.3
6 6.0 2.6
7 6.5 0.5
Active Model:
Example 11-2
Excel File:
Example 11-2
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 295
Date Travel Times (Hrs.) Moving Range
8 6.3 0.2
9 7.2 0.9
10 7.3 0.1
11 7.5 0.2
12 7.2 0.3
13 8.0 0.8
14 7.8 0.2
15 8.2 0.4
16 7.0 1.2
17 7.8 0.8
x = 7.1235 MR = .725
Solution: You decide to develop an X and MR process chart to test the hypothesis concerning the
change. You conclude that, in fact, a run (from point 9 to point 15) indicates that trip times may
be increasing. However, this does not imply that the girlfriend is the cause. Further investigation
may be needed. (Note that E2 = 2.66 and D4 = 3.268.)
using excel to draw X and MR charts
The problem in Example 11-2 is now solved using Excel. Again, we use the “brute force” method
for creating X and MR charts in Excel. The process is very similar to what we did before. Notice
that E2 and D4 are both constants.
Interpreting the charts, there is a run on the X chart and an out-of-control point on the sixth
(fifth observation in the graph because there was no moving range for the first). This was because
of the jump from 8.6 hours down to 6 hours. It might be that our hero thought he should be on
better behavior after the long day on the fifth (see Figure 11-14).
Figure 11-14 Example 11-2 Using Excel. Source: Microsoft Excel, Microsoft Corporation. Used by permission.
296 3DUW š ,PSOHPHQWLQJ4XDOLW\
median charts
Although x charts generally are preferred for variables data, sometimes it is too time consuming
or inconvenient to compute subgroup averages. Also, there may be concerns about the accuracy
of computed means. In these cases, a median chart may be used (aka an & x chart). The main limitation is that you will use an odd sample size to avoid calculating the median. Generally, sample
sizes are 3, 5, or 7. Like the x chart, small sample sizes generally are used, although the larger the
sample size, the better is the sensitivity of the chart as a tool to detect nonrandom (special cause)
events (this is also true for x charts).
To prepare median charts, determine your subgroup size and how often you will sample.
The rule of thumb to establish a median control chart is to use 20 to 25 subgroups and a total of
at least 100 individual measurements.
Equations for computing the control limits are
Mean of medians = sum of the medians>number of medians = ≂ x (11.2)
LCL &
x = ≂ x + A &2R (11.3)
UCL &
x = ≂ x A &2R (11.4)
&A
2 values are found in Table 11-2. Median charts are usually used with R charts.
ExamplE 11-3 median Charts in action
Problem: The Luftig food company has gathered the following data with weights of its new
health food product. Because the published weight on the package is 6 ounces, Mr. Luftig wants
to know if the company is complying with weight requirements. Twenty samples of size 5 were
drawn.
Solution: The data are given here. Twenty samples of size 5 were drawn. Results show that the
process is not in control, with an average median of 6.23. The median process chart (see Figure
11-15) does show that some product is being made that is below 6 ounces. It also shows that
points 4, 7, and 10 are out of control.
Sample Observation 1 Observation 2 Observation 3 Observation 4 Observation 5
1 6.2 6.1 6.3 6.5 6.4
2 6.2 6.2 6.2 6.3 6.4
3 6.3 5.9 6.2 6.4 6.3
4 5.3 5.1 5.3 5.1 5.3
5 6.1 6.6 6.3 6.2 6.4
6 6.2 6.2 6.2 6.2 6.2
7 5.8 5.7 5.9 7.2 5.2
TABLE 11-2 Median Chart Values
n A
&
2 D4
3 1.187 2.575
5 0.691 2.115
7 0.508 1.924
9 0.412 1.816
Active Model:
Example 11-3
Excel File:
Example 11-3
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 297
Sample Observation 1 Observation 2 Observation 3 Observation 4 Observation 5
8 6.3 5.9 6.2 6.4 6.3
9 6.3 5.9 6.2 6.4 6.3
10 7.4 7.4 7.1 7.3 7.1
11 6.2 6.3 6.2 6.3 6.2
12 6.4 6.3 6.2 6.1 6.1
13 6.3 6.4 6.2 6.3 6.1
14 6.1 6.1 6.1 6.1 6.1
15 6.3 6.4 6.1 6.3 6.1
16 6.4 6.2 6.4 6.2 6.2
17 6.2 6.4 6.3 6.4 6.2
18 6.1 6.2 6.3 6.4 6.5
19 6.2 6.1 6.1 6.1 6.1
20 6.4 6.3 6.2 6.5 6.3
Figure 11-15 Example 11-3 Using Excel. Source: Microsoft Excel, Microsoft Corporation. Used by
permission.
using excel to draw median charts
Figure 11-15 shows the results for Example 11-3. Again, the columns are ordered such that the
data can be grouped properly and drawn using Excel. Excel makes creation of the chart quick
and easy. You will have the best results if you start with Example 11-1 and work all the Excel
examples. By now, you should have the hang of it. A good shortcut is to highlight the data in
columns B through F prior to invoking the Chart function.
298 3DUW š ,PSOHPHQWLQJ4XDOLW\
_x
and s charts
When you are particularly concerned about the dispersion of the process, it might be that the R
chart is not sufficiently precise. In this case, the x chart is recommended in concert with the s
chart or standard deviation chart. The standard deviation chart is often used where variation
in a process is small. For example, s charts are often used in monitoring the production of silicon
chips for computers.
Unfortunately, when using the s chart, because we do not compute ranges, new formulas
are used to compute the x limits. We introduce the formulas for the x and s charts because of their
importance for high-tech production.
The control limits for the s chart are computed using these formulas:
UCL
s = B4 * s Q (11.5)
LCL
s = B3 * s Q (11.6)
where
B
3 and B4 come from Table 11-3;
and s Q = Σsi/k (11.7)
where
si is the standard deviation for sample i
k is the number of samples.
Note that it is easy to find the sample standard deviation in Excel. If you don’t have Excel,
use the usual formulas for computing the sample standard deviations. We will show you how to
do this in Excel in Example 11-4.
After computing the limits, plot your sample means to see if the process is in control. If
the s chart is not in control, determine the cause for the out-of-control point, eliminate the cause,
and then recompute your control limits by throwing out the out-of-control data point(s). Do not
eliminate samples with out-of-control points if a cause cannot be identified.
When your s chart is in statistical control, use the following formula to estimate the process standard deviation:
se
st = s Q * 3(1 – C4 2)>C4 (11.8)
where
C4 can be found in Table 11-3.
Formulas for the x chart can now be created using the following formulas:
UCL =
x = x + A3(s) (11.9)
LCL =
x = x A3(s) (11.10)
where
A
3 can be found in Table 11-3 and = x is the grand mean.
TABLE 11-3 Values for x – and s Charts
n B3 B4 C4 A3
2 0 3.267 0.7979 2.659
3 0 2.568 0.8862 1.954
4 0 2.266 0.9213 1.628
5 0 2.089 0.9400 1.427
6 0.030 1.970 0.9515 1.287
7 0.118 1.882 0.9594 1.182
8 0.185 1.815 0.9650 1.099
9 0.239 1.761 0.9693 1.032
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 299
ExamplE 11-4 x – and s Charts in action
Problem: Twenty samples were taken for a milled rod. The diameters are needed to determine
whether the process is in control. Because these milled rods must be measured within 1/10,000
of an inch, it is determined that the process dispersion is important. Therefore, you need to use
an s and x chart to monitor the process. The data are found in Figure 11-16. We have 20 samples
with n = 3.
Figure 11-16
Example 11-4 Using
Excel. Source:
Microsoft Excel, Microsoft
Corporation. Used by
permission.
Solution: The control charts in Figure 11-16 show that the process is in control. There is no need
for corrective action. The solution method is demonstrated in the next section.
using excel to draw x – and s charts
Figure 11-16 shows the solution method for Example 11-4. Using the preceding formulas, we
computed the CL, UCL, and LCL for each chart. Notice that the LCL for the s chart is zero. Also,
notice that we have taken some shortcuts (What short cuts?) here compared with some of the other
charts we have drawn.
other control charts
Table 11-4 shows all the formulas for the process charts we have discussed in this chapter. These
are the major charts that are used the vast majority of times. Some other charts that are used more
rarely should be mentioned.
moving average chart
The moving average chart is an interesting chart used for monitoring variables and measurement
on a continuous scale. This chart uses past information to predict what the next process outcome
will be. Using this chart, we can adjust a process in anticipation of its going out of control.
Excel File:
Example 11-4
Active Model:
Example 11-4
300 3DUW š ,PSOHPHQWLQJ4XDOLW\
cusum chart
The cumulative sum, or cusum, chart is used to identify slight but sustained shifts in a universe in
which there is no independence between observations. A cusum chart looks very different from a
Shewhart process chart, as shown in Figure 11-17.
Some control chart concePtS For VariableS
choosing the correct Variables control chart
Now that we have developed control charts, we are in a position to discuss briefly some important control chart concepts before moving to process capability. The first concept has to do with
choosing the correct chart. Obviously, it is key to choose the correct control chart. Figure 11-18
shows a decision tree for the basic control charts. This flowchart helps to show when certain
charts should be selected for use.
TABLE 11-4 Summary of Variables Chart Formulas
Chart LCL CL UCL Constant Values
x = x A
2R = x = x + A2R (Appendix Table A-1)
R D
3R R D4R (Appendix Table A-1)
X = x E
2(MR) = x = x + E2(MR) (Appendix Table A-1)
Median
x
A
&
2R x x + A &2R (Appendix Table A-4)
x (with s) = x A3s = x = x + A3s (Appendix Table A-3)
s B
3s s B4s (Appendix Table A-3)
40
35
30
25
20
15
10
50
–5
–10
–15
–20
–25
–30
–35
–40
–45
1 5 10 15 20 25 30
Subgroup Number i
Cumulative Sum
Lower control limit
Upper control limit
P d
A B
A’
B’
U O
U
Figure 11-17 Cusum Chart
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 301
Are the data
Variable or Attribute?
Are you interested in
defects per unit or defectiv e
units?
Attributes
Variables
Is the sample size
constant?
Defectiv e Units
Use p chart
Use np or
p charts
Yes
No
Is the sample
space constant?
Defects per unit
Use c
charts
Use u
charts
Yes
No
Can subgroup av erages be
computed?
Use
median
charts
No
Is the subgroup
size 9 or more?
Yes
x/R
charts
No
Is it possible to
compute s for ev ery
subgroup?
Yes
x/s
charts
x/R
charts
Yes
No
Figure 11-18 Process for Selecting the Right Chart
302 3DUW š ,PSOHPHQWLQJ4XDOLW\
corrective action
When a process is out of control, corrective action is needed. Corrective action steps are similar
to continuous improvement processes:
1. Carefully identify the quality problem.
2. Form the appropriate team to evaluate and solve the problem.
3. Use structured brainstorming along with fishbone diagrams or affinity diagrams to identify
causes of problems.
4. Brainstorm to identify potential solutions to problems.
5. Eliminate the cause.
6. Restart the process.
7. Document the problem, root causes, and solutions.
8. Communicate the results of the process to all personnel so this process becomes reinforced
and ingrained in the organization.
how do We use control charts to continuously improve?
One of the goals of the control chart user is to reduce variation. Over time, as processes are
improved, control limits are recomputed to show improvements in stability. As upper and lower
control limits get closer and closer together, the process is improving. There are two key concepts here:
š The focus of control charts should be on continuous improvement.
š Control chart limits should be updated only when there is a change to the process.
Otherwise, any changes are unexpected.2
tampering with the Process
One of the cardinal rules of process charts is that you should never tamper with the process. You
might wonder, “Why don’t we make adjustments to the process any time the process deviates
from the target?” The reason is that random effects are just that—random. This means that these
effects cannot be controlled. If we make adjustments to a random process, we actually inject
nonrandom activity into the process. Figure 11-19 shows a random process. Suppose that we had
decided to adjust the process after the fourth observation. We would have shifted the process—
signaled by out-of-control observations during samples 12 and 19.
ProceSS caPability For VariableS
Once a process is stable, the next emphasis is to ensure that the process is capable. Process capability refers to the capability of a process to produce a product that meets specification. A highly
capable process produces high volumes with few or no defects and is the result of optimizing the
2
Wheeler, D., “When Do I Recalculate My Limits?” Quality Progress (May 1996): 79–80.
1 3 5 7 9 11 13 15 17 19 21 23
UCL
Target
LCL
Figure 11-19 The Effects of Tampering
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 303
interactions between people, machines, raw materials, procedures, and measurement systems.
World-class levels of process capability are measured by parts per million (ppm) defect levels,
which means that for every million pieces produced, only a small number (fewer than 100) are
defective. A Closer Look at Quality 11-1 looks at the need for capability in software.
A CLOSER LOOK AT QUALITY 11-1 A Justification for Meeting
Standards in Software Quality3
In this chapter, we discuss the importance of meeting standards and having controlled processes. We see
all around us the results of poor quality and defects. Among the areas where this is important is software
quality. Consider the following examples.
Poor software design in a radiation machine, known as Therac-25, contributed to the deaths of
three cancer patients. The Therac-25 was built by Atomic Energy of Canada Ltd., which is a Crown corporation of the government of Canada. In 1988, the company incorporated and sold its radiation-systems
assets under the Theratronics brand. According to Nancy Leveson, now a professor at MIT, the design
flaws included the incapability of the software to handle some of the data it was given and the delivery
of hard-to-decipher user messages.
During Operation Desert Storm, an Iraqi Scud missile hit a U.S. Army barracks in Saudi Arabia,
killing 28 Americans. The approach of the Scud should have been noticed by a Patriot missile battery.
A subsequent government investigation found a flaw in the Patriot’s weapons-control software that prevented the system from properly tracking the incoming missile.
During Operation Iraqi Freedom, the Patriot missile system mistakenly downed a British Tornado
fighter and (according to the Los Angeles Times an American F/A-18c Hornet). Reports show that investigators were looking at a glitch in the missile’s radar system that made it incapable of properly distinguishing between a friendly aircraft and an enemy missile.
In 2002, the Food and Drug Administration (FDA), which oversees medical-device software, said
that of 3,140 medical-device recalls, 242 were attributed to software failures. The FDA also says that the
number of software-related recalls may be underreported because it is often hard to determine the exact
cause of a problem in the immediate aftermath of an accident.
It is expected that these types of losses are likely to mount as complex software programs are
tied across networks. Imagine all the various pieces of corporate data that come together in systems for
CRMSs, SCMs, or ERPs. “Software is the most complicated thing that the human mind can come up
with and build,” says Gary McGraw, the chief technology officer at Citigal, a consulting firm specializing in improving software quality. Tools introduced in this chapter will be key for detecting whether
future software is functioning properly.
3
Gage, D., and J. McCormick, “Why Software Quality Matters,” Baseline 28 (March 2004): 34–59.
Six Sigma programs, such as those pioneered by Motorola Corporation, result in highly
capable processes. Six Sigma is a design program that emphasizes engineering parts so that they
are highly capable. As shown in Figure 11-20, these processes are characterized by specifications
that are ±6 standard deviations from the process mean. This means that even large shifts in the
process mean and dispersion will not result in defective products being built. If a process average is on the center line, a Six Sigma process will result in an average of only 3.4 opportunities
for defects per million units produced. The Taguchi method is a valuable tool for achieving Six
Sigma quality by helping to develop robust designs that are insensitive to variation.
Population versus Sampling distributions
To understand process capability, we must first understand the differences between population and sampling distributions. Population distributions are distributions with all individual
Video Clip:
Process Capability
304 3DUW š ,PSOHPHQWLQJ4XDOLW\
responses from an entire population. A population is defined as a collection of all the items or
observations of interest to a decision maker. A sample is a subset of the population. Sampling
distributions are distributions that reflect the distribution of sample means. We can demonstrate
the difference between a sample and a population. Suppose that you want to understand whether
a product conforms to specifications. Over a month’s time, a firm produces 10,000 units of product to stock. Because the product is fragile, it is not feasible to inspect all 10,000 units and
risk damaging some of the product in the inspection process. Therefore, 500 units are randomly
selected from the 10,000 to inspect. In this example, the population size N is 10,000 and the
sample size n is 500.
We now demonstrate the difference between a sampling distribution and a population distribution. Understanding the differences between sampling and population distributions is important: Population distributions have much more dispersion than sampling distributions. Consider
a class of 40 students, in which the tallest student is 6 feet 4 inches, and the shortest is 5 feet in
height. As shown in Figure 11-21, student height for this population is normally distributed, with
a mean of 5 feet 8 inches and a distribution ranging from 5 feet to 6 feet 4 inches.
Now suppose that you draw samples of size five from the population (with replacement).
Notice in Figure 11-21 that the mean of the sample is still 5 feet 8 inches, but the distribution
ranges only from 5 feet 4 inches to 6 feet. This is so because it is difficult to randomly obtain a
sample average that is more than 6 feet or less than 5 feet 4 inches. As a result, we see that sampling distributions have much less dispersion than population distributions.
Process
mean
(m)
m 1 3 s Upper
specification
(m 1 6 s )
Lower m 2 3 s
specification
(m 2 6 s )
Population
distrib ution
Figure 11-20 Six Sigma Quality
5054586064
Sampling
distrib ution
(n5 5)
Population
distrib ution
(N 5 40)
Figure 11-21 Population and Sampling Distributions for Class Heights
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 305
In the context of quality, specifications and capability are associated with population
distributions. However, sample-based process charts and stability are computed statistically
and reflect sampling distributions. Therefore, quality practitioners should not compare process chart limits with product specifications. To compare process charts limits with specification limits is not so much like comparing apples to oranges as it is comparing apples to
watermelons. We later show that process chart limits are statistically computed from sample
data. Specification (or tolerance) limits are set by design engineers who establish limits based
on the design requirements for a product. These design requirements might have to do with
making parts fit together properly or with the properties of certain materials used in making
products.
capability Studies
Now that we have defined process capability, we can discuss how to determine whether a process
is capable. That is, we want to know if individual products meet specifications. There are two
purposes for performing process capability studies:
1. To determine whether a process consistently results in products that meet specifications.
2. To determine whether a process is in need of monitoring through the use of permanent
process charts.
Process capability studies help process managers understand whether the range over which
natural variation of a process occurs is the result of the system of common (or random) causes.
There are five steps to perform process capability studies:
1. Select a critical operation. These may be bottlenecks, costly steps of the process, or places
in the process in which problems have occurred in the past.
2. Take k samples of size n, where x is an individual observation.
š Where 19 < k < 26
š If x is an attribute, n > 50 (as in the case of a binomial)
š Or if x is a measurement, 1 < n < 11
(Note: Small sample sizes can lead to erroneous conclusions.)
3. Use a trial control chart to see whether the process is stable.
4. Compare process natural tolerance limits with specification limits. Note that natural tolerance limits are three standard deviation limits for the population distribution. This can be
compared with the specification limits.
5. Compute capability indexes: To compute capability indexes, you compute an upper capability index (Cpu), a lower capability index (Cpl), and a capability index (Cpk). The
formulas used to compute these are
Cpu = 1USL – m2 >3s n (11.11)
Cpl = 1 m – LSL2 >3s n (11.12)
Cpk = min5Cpu, Cpl6 (11.13)
where
USL = upper specification limit
LSL = lower specification limit
m = computed population process mean
s n = Estimated process standard deviation = s n = R/d2 (11.14)
Make a decision concerning whether the process is capable. Although different firms use
different benchmarks, the generally accepted benchmarks for process capability are 1.25, 1.33,
306 3DUW š ,PSOHPHQWLQJ4XDOLW\
and 2.0. We will say that processes that achieve capability indexes (Cpk) of 1.25 are capable,
1.33 are highly capable, and 2.0 are world-class capable (Six Sigma).
ExamplE 11-5 process Capability
Problem: For an overhead projector, the thickness of a component is specified to be between 30
and 40 millimeters. Thirty samples of components yielded a grand mean (x) of 34 millimeters
with a standard deviation (s n ) of 3.5. Calculate the process capability index by following the
steps previously outlined. If the process is not highly capable, what proportion of product will
not conform?
Solution:
Cpu = (40 – 34)>(3)(3.5) = .57
Cpl = (34 – 30)>(3)(3.5) = .38
Cpk = .38
The process capability in this case is poor. To compute the proportion of nonconforming product
being produced, we use a Z table (Appendix A-2) with a standardized distribution. The formula
is
Z = (x – m)>s n (11.15)
Thus, for the lower end of the distribution:
Z = (30 – 34)>3.5 = -1.14
For the upper end of the distribution:
Z = (40 – 34)>3.5 = 1.71
Using a Z table (Table A-2 from the Appendix), the probability of producing bad product is
.1271 + .0436 = .1707. This means that, on average, more than 17% of the product produced
does not meet specification. This is unacceptable in almost any circumstance.
m 5 34 USL 5 40
(m 1 1.710s)
LSL 5 30 m 1 3s 5 44.5
(m 2 1.140s)
m 2 3s 5 23.5
p 5 .3729 p 5 .4564
.5 2 .3729 5 .1271 .5 2 .4564 5 .0436
Figure 11-22 Proportion of Product Nonconforming for Example 11-5
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 307
Ppk
If your data are not arranged in subgroups, and you have only population data to compute your
capability, use Ppk to compute your capability. Ppk stands for population capability index.
Rather than using the within-groups variation to estimate the sigma that you used in Cpk, use the
population standard deviation to compute your capability. Otherwise, the computations are the
same as Cpk. Here are the formulas:
Ppk = min 5Ppu, Ppl6 (11.16)
Ppu = (USL – m)>3s (11.17)
Ppl = (m – LSL)>3s (11.18)
s = 3Σ(xi x)2/(n – 1) (11.19)
where
USL = upper specification limit
LSL = lower specification limit
μ = population mean
s = population process standard deviation
Interpretation for Ppk is the same as for Cpk. The only difference is the use of population
parameters when computing the indexes.
ExamplE 11-6 population process Capability
Problem: The upper and lower specification limits (tolerances) for a metal plate are 3 millimeters
;0.002 millimeters. A sample of 100 plates yielded a mean x of 3.001 millimeters. We know that
the population standard deviation is .0002. Compute the Ppk for this product.
Solution:
Ppu = (3.002 – 3.001)>(.0002 * 3) = 1.67
Ppl = (3.001 – 2.998)>(.0002 * 3) = 5
Ppk = 1.67
Therefore, the process is highly capable.
the difference between capability and Stability
Once again, a process is capable if individual products consistently meet specification. A process
is stable if only common variation is present in the process. This is an important distinction. It
is possible to have a process that is stable but not capable. This would happen where random
variation was very high. It is probably not so common that an incapable process would be stable.
other StatiStical techniqueS in quality management
Throughout this chapter, we have focused on hypothesis testing and process charts. In Chapter
13 we discuss experimental design and off-line experimentation. Correlation and regression also
can be useful tools for improving quality, particularly in services.
308 3DUW š ,PSOHPHQWLQJ4XDOLW\
Although it is almost never appropriate to use regression on process data used in developing control charts, other types of data can be correlated and regressed to understand the customer.
For example, Figure 11-23 shows where conformance rates and quality costs were correlated in
one company. As conformance increased, costs increased as well. Table 11-5 shows that these
variables were significantly and positively related. The R2 values show the strength of the relationships between the variables for linear and nonlinear (quadratic) models.4
Such correlation is called interlinking.5 Interlinking is useful in helping to identify causal
relationships between variables.
100
Prevention and appraisal costs
Conformance
80
60
40
20
0
50 60 70 80 90 100
3
Figure 11-23 Plot of Prevention and Appraisal Costs with Conformance Source: Based on
S. T. Foster, Quality Costs Working Paper.
4
To learn more about this, see Foster, S. T., “An Examination of the Relationship between Conformance and QualityRelated Costs.” International Journal of Quality and Reliability Management 13, 4 (1996): 50–63.
TABLE 11-5 Relationship between Conformance
and PA Costs
Source: Based on S. T. Foster, “Quality Costs Working Paper.”
Model R2 p
First order 0.4002 0.0001
Quadratic 0.4675 0.0001
5
Collier, D., The Service/Quality Solution (Homewood, IL: Irwin, 1994).
Summary
In this chapter we have introduced the basic process charts and the fundamentals of statistical
quality improvement. The process for developing process charts is the same regardless of chart.
Therefore, the things that are required are:
You need to know the generic process for developing charts.
You need to be able to interpret charts.
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 309
You need to be able to choose which chart to use.
You need the formulas to derive the charts.
You need to understand the purposes and assumptions underlying the charts.
We have given you all these things for variables in this short chapter. You have everything
you need to get started. Have fun and enjoy yourself. Remember that the purpose of process
charts is to help you continually improve.
Key Terms
Attribute
Capability
Consumer’s risk
Control chart
Control plan
Cpk
Median chart
Nonrandom variation
Ppk
Process charts
Producer’s risk
Quality management
system (QMS)
R chart
Random variation
Reaction plan
s chart or standard
deviation chart
Sample
Stability
Statistical thinking
Variable
X chart
x chart
Discussion Questions
1. Discuss the concept of control. Is control helpful? Isn’t being controlling a negative?
2. The concept of statistical thinking is an important theme in this chapter. What are some examples of
statistical thinking?
3. Sometimes you do well on exams. Sometimes you have bad days. What are the assignable causes
when you do poorly?
4. What is the relationship between statistical quality improvement and Deming’s 14 points?
5. What are some applications of process charts in services? Could demerits (points off for mistakes) be
charted? How?
6. What is random variation? Is it always uncontrollable?
7. When would you choose an np chart over a p chart? An X chart over an x chart? An s chart over an R
chart?
8. Design a control chart to monitor the gas mileage in your car. Collect the data over time. What did you
find?
9. What does “out-of-control” mean? Is it the same as a “bad hair day”?
10. Design a control chart to monitor the amounts of the most recently charged 50 debits from your debit
card. What did you find?
problems
1. Return to the chart in Figure 11-8. Is this process stable? Explain.
2. Return to the data in Figure 11-8. Is this process capable? Compute both Cpk and Ppk.
3. For the following product characteristics, choose where to inspect first:
Characteristic Cost of Inspection Cost of Failure
A \$2.50 \$20
B \$2.00 \$19
C \$4.00 \$37
D \$3.00 \$38
310 3DUW š ,PSOHPHQWLQJ4XDOLW\
4. For the following product characteristics, choose where to inspect first:
Characteristic Cost of Inspection Cost of Failure
A \$35 \$200
B \$37 \$225
C \$38 \$175
D \$40 \$182
5. Interpret the following charts to determine whether the processes are stable.
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
(a) (d)
(b) (e)
(c) (f)
6. Interpret the following charts to determine whether the processes are stable.
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
UCL
CL
LCL
(a) (d)
(b) (e)
(c) (f)
7. Tolerances for a new assembly call for weights between 32 and 33 pounds. The assembly is made
using a process that has a mean of 32.6 pounds with a population standard deviation of .22 pounds.
The process population is normally distributed.
a. Is the process capable?
b. If not, what proportion will meet tolerances?
c. Within what values will 99.5% of sample means of this process fall if the sample size is constant at
10 and the process is stable?
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 311
8. Specifications for a part are 62″ +/- .01.″ The part is constructed from a process with a mean of 62.01″
and a population standard deviation of .033. The process is normally distributed.
a. Is the process capable?
b. What proportion will meet specifications?
c. Within what values will 95% of sample means of the process fall if the sample size is constant at 5
and the process is stable?
9. Tolerances for a bicycle derailleur are 6 cm +/- .001 cm. The current process produces derailleurs with
a mean of 6.0001 with a population standard deviation of .0004. The process population is normally
distributed.
a. Is the process capable?
b. If not, what proportion will meet specs?
c. Within what values will 75% of sample means of this process fall if the sample size is 6 and the
process is stable?
10. A services process is monitored using x and R charts. Eight samples of n = 10 observations have been
gathered with the following results:
Sample Mean Range
1 4.2 .43
2 4.4 .52
3 3.6 .53
4 3.8 .20
5 4.9 .36
6 3.0 .42
7 4.2 .35
8 3.2 .42
a. Using the data in the table, compute the center line, the upper control limit, and the lower control
limit for the x and R charts.
b. Is the process in control? Interpret the charts.
c. If the next sample results in the following values (2.5, 5.5, 4.6, 3.2, 4.6, 3.2, 4.0, 4.0, 3.6, 4.2), will
the process be in control?
11. A production process for the JMF Semicon is monitored using x and R charts. Ten samples of n = 15
observations have been gathered with the following results:
Sample Mean Range
1 251 29
2 258 45
3 233 36
4 275 25
5 234 35
6 289 20
7 256 3
8 265 19
9 246 14
10 323 46
a. Develop a control chart and plot the means.
b. Is the process in control? Explain.
12. Experiment: Randomly select the heights of at least 15 of the students in your class.
a. Develop a control chart and plot the heights on the chart.
b. Which chart should you use?
c. Is this process in control?
13. A finishing process packages assemblies into boxes. You have noticed variability in the boxes and desire to improve the process to fix the problem because some products fit too tightly into the boxes and
others fit too loosely. Following are width measurements for the boxes.
312 3DUW š ,PSOHPHQWLQJ4XDOLW\
Sample
1 2 3 4 5 6 7 8
68.51 68.94 68.66 68.49 68.64 68.34 68.99 68.92
68.46 68.20 68.44 68.94 68.63 68.42 68.94 68.91
68.54 68.54 68.55 68.56 68.62 68.99 68.95 68.97
68.34 68.56 68.77 68.62 68.32 68.02 68.95 68.93
68.46 68.70 68.70 68.69 68.34 68.03 68.94 68.96
68.46 68.70 68.64 68.56 68.24 68.47 68.97 68.95
Using x and R charts, plot and interpret the process.
14. For the data in Problem 13, if the mean specification is 68.5 ; .25 and the estimated process standard
deviation is .10, is the process capable? Compute Cpu, Cpl, and Cpk.
15. For the data in Problem 13, treat the data as if they were population data and find the limits for an X
chart. Is the process in control? Compare your answer with the answers to Problem 14. Hint: Use the
formula CL
x = x { (3>d2)R (Figure 11-9).
16. A Rochester, New York, firm produces grommets that have to fit into a slot in an assembly. Following
are dimensions of grommets (in millimeters):
Sample x
1 46 33 54 46 64
2 52 45 54 75 64
3 34 64 36 46 63
4 34 45 47 37 62
5 46 64 75 55 16
a. Use x and R charts to determine whether the process is in control.
17. Using the data from Problem 13, compute the limits for x and s charts. Is the process still in control?
18. Using the data from Problem 16, compute the limits for x and s charts. Is the process still in control?
19. Use a median chart to determine whether the process for the following data is centered.
Sample Observation 1 Observation 2 Observation 3 Observation 4 Observation 5
1 8.06 7.93 8.19 8.45 8.32
2 8.06 8.06 8.06 8.19 8.32
3 8.19 7.67 8.06 8.32 8.19
4 6.89 6.63 6.89 6.63 6.89
5 7.93 8.58 8.19 8.06 8.32
6 8.06 8.06 8.06 8.06 8.06
7 7.54 7.41 7.67 9.36 6.76
8 8.19 7.67 8.06 8.32 8.19
9 8.19 7.67 8.06 8.32 8.19
10 9.62 9.62 9.23 9.49 9.23
11 8.06 8.19 8.06 8.19 8.06
12 8.32 8.19 8.06 7.93 7.93
13 8.19 8.32 8.06 8.19 7.93
14 7.93 7.93 7.93 7.93 7.93
15 8.19 8.32 7.93 8.19 7.93
16 8.32 8.06 8.32 8.06 8.06
17 8.06 8.32 8.19 8.32 8.06
18 7.93 8.06 8.19 8.32 8.45
19 8.06 7.93 7.93 7.93 7.93
20 8.32 8.19 8.06 8.45 8.19
&KDSWHU š 6WDWLVWLFDOO\%DVHG4XDOLW\,PSURYHPHQWIRU9DULDEOHV 313
20. Use an x chart to determine whether the data in Problem 19 are in control. Do you get the same
answer?
21. The following data are for a component used in the space shuttle. Because the process dispersion is
closely monitored, use an x and s chart to see whether the process is in control.
Sample Observation 1 Observation 2 Observation 3
1 4.8000 4.7995 4.8005
2 4.7995 4.8007 4.8005
3 4.7995 4.8002 4.8012
4 4.7993 4.8000 4.8010
5 4.8007 4.8007 4.8005
6 4.8010 4.8007 4.8000
7 4.7995 4.7995 4.7995
8 4.8000 4.8002 4.8002
9 4.8012 4.8000 4.7998
10 4.7988 4.7995 4.8002
11 4.8005 4.7998 4.8002
12 4.8005 4.7995 4.8012
13 4.8000 4.8002 4.7995
14 4.8000 4.8005 4.8010
15 4.7986 4.8002 4.7990
16 4.7998 4.8007 4.7983
17 4.8005 4.7995 4.8010
18 4.8000 4.8002 4.8002
19 4.7993 4.7986 4.7995
20 4.8007 4.8017 4.7998
22. Develop an R chart for the data in Problem 21. Do you get the same answer?
23. Using the data from Problem 21, compute limits for a median chart. Is the process in control?
24. Design a control plan for exam scores for your quality management class. Describe how you would
gather data, what type of chart is needed, how to gather data, how to interpret the data, how to identify
causes, and remedial action to be taken when out-of-control situations occur.
25. For the sampling plan from Problem 24, how would you measure process capability?
26. For the data in Problem 16, if the process target is 50.25 with spec limits +/-5, describe statistically
the problems that would occur if you used your spec limits on a control chart where n = 5. Discuss
type I and type II errors.
CASES
Case 11-1 Ore-Ida Fries
www.heinz.com
An innovation in the frozen french fry industry was the
upright bag. When new equipment was introduced to
produce the bags, the Heinz Frozen Food Corporation
facility in Ontario, Oregon, was selected to produce the
new bag type.
When the new bags were produced, there were
problems with consistency. It was unclear whether the
problem was with the machinery or the “film” (the material used in the bags). One of the key measurements
was the distance from the universal product code (UPC)
and a black mark on the bag. A number of rolls of film
were randomly selected, and this measurement was
taken. The result of this actual study was the data that
follow.
We need to know whether the film is consistent.
Take the data that follow and use control charts to determine whether the measurements are consistent. Report
your results to management.
(continued)
314 3DUW š ,PSOHPHQWLQJ4XDOLW\
Sample Millimeters from Code to UPC Box
1 7 7 8 6 7
2 6 5 6 5 7
3 7 7 8 6 8
4 6 8 8 7 7
5 6 7 6 6 7
6 6 6 5 6 5
7 5 6 4 4 4
8 4 5 5 5 6
9 5 6 5 5 5
10 5 5 5 5 5
11 6 6 7 7 7
12 7 7 6 7 7
13 6 7 7 7 7
14 6 7 7 7 7
15 6 6 6 6 6
16 6 6 6 6 6
17 6 7 7 6 7
18 6 7 6 7 7
19 6 6 6 6 6
20 5 6 5 6 6
21 9 12 10 10 10
22 10 10 9 10 10
23 10 10 10 9 10
24 10 10 10 10 10
25 10 10 10 10 10
26 10 10 10 11 10
27 11 12 10 11 11
28 11 12 10 11 12
29 10 11 11 11 11
30 10 11 12 10 10
31 10 11 11 11 11
32 11 11 11 12 12
33 11 11 0 0 5
34 6 4 4 5 7
35 7 6 6 0 1
36 6 7 6 7 6
37 6 6 5 6 7
38 10 9 10 10 9
39 10 9 8 8 11
40 10 10 10 10 10
Sample Millimeters from Code to UPC Box
41 10 10 10 11 10
42 11 11 11 10 10
43 10 10 10 10 10
44 11 10 10 10 10
45 10 10 10 10 10
46 10 10 10 10 10
47 10 10 10 10 10
48 10 10 10 11 12
49 10 11 10 11 11
50 12 12 11 11 11
51 12 11 11 10 10
52 12 12 11 11 10
53 10 11 11 11 11
54 11 10 11 12 11
55 11 10 12 11 11
56 11 11 12 11 11
57 10 10 12 12 11
58 10 11 11 11 11
59 11 11 16 16 17
60 18 17 17 16 16
61 18 17 16 16 16
62 17 17 17 17 16
63 16 16 16 15 16
64 16 17 18 16 16
65 16 17 17 17 16
66 16 17 17 17 17
67 15 15 17 16 17
68 16 15 16 17 17
69 16 16 16 18 16
70 16 15 17 16 16
71 16 15 16 15 16
72 16 16 16 16 16
73 15 15 15 16 16
74 16 15 16 16 16
75 16 16 16 16 15
76 16 16 15 16 17
77 16 16 16 16 16
78 17 16 15 16 16
79 17 17 17 16 16
80 16 16 16 16 16

Don't use plagiarized sources. Get Your Custom Essay on
Statistics are very important in the field of quality
Just from \$10/Page

## Get professional assignment help cheaply

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason may is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

Our essay writers are graduates with diplomas, bachelor, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college diploma. When assigning your order, we match the paper subject with the area of specialization of the writer.

## Why choose our academic writing service?

• Plagiarism free papers
• Timely delivery
• Any deadline
• Skilled, Experienced Native English Writers
• Subject-relevant academic writer
• Adherence to paper instructions
• Ability to tackle bulk assignments
• Reasonable prices
• 24/7 Customer Support
• Get superb grades consistently

Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
• 24/7 support
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
• Expert Proofreading
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

We value our customers and so we ensure that what we do is 100% original..
With us you are guaranteed of quality work done by our qualified experts.Your information and everything that you do with us is kept completely confidential.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

The Product ordered is guaranteed to be original. Orders are checked by the most advanced anti-plagiarism software in the market to assure that the Product is 100% original. The Company has a zero tolerance policy for plagiarism.

### Free-revision policy

The Free Revision policy is a courtesy service that the Company provides to help ensure Customer’s total satisfaction with the completed Order. To receive free revision the Company requires that the Customer provide the request within fourteen (14) days from the first completion date and within a period of thirty (30) days for dissertations.

### Privacy policy

The Company is committed to protect the privacy of the Customer and it will never resell or share any of Customer’s personal information, including credit card data, with any third party. All the online transactions are processed through the secure and reliable online payment systems.

### Fair-cooperation guarantee

By placing an order with us, you agree to the service we provide. We will endear to do all that it takes to deliver a comprehensive paper as per your requirements. We also count on your cooperation to ensure that we deliver on this mandate.

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
Academic level
Number of pages
Urgency
You can contact our live agent via WhatsApp! Via +1 817 953 0426

Feel free to ask questions, clarifications, or discounts available when placing an order.

Order your essay today and save 20% with the discount code VICTORY