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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

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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*)

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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

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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

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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

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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.

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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

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Understanding Process Charts

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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)

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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

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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 *A*2 and *D*4 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

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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, *D*3 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

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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

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*A*

m*R *= =

___________

*A*

m*R *=

______

URL

x = US – *A*m*R *= LRLX = LS + *A*m*R *=

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) =

*A*2*R *= = ———

UCL

x = *X *+ *A*2*R *=

LCL

x = *X *– *A*2*R *=

UCLR = *D*4*R *= =

*X *= S —– *x *= ———————— = ——–

*k*

or

US = LS =

*A*2

1.880

1.023

0.729

0.577

0.483

*D*4

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

*A*m

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*

m*R *= =

___________

*A*

m*R *=

______

URL

x = US – *A*m*R *= LRLX = LS + *A*m*R *=

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) =

*A*2*R *= =

––––––

UCL

x = *X *+ *A*2*R *=

LCL

x = *X *– *A*2*R *=

UCLR = *D*4*R *= =

*X *= S —– *x *= =

*k*

or

US = LS =

*A*2

1.880

1.023

0.729

0.577

0.483

*D*4

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

*A*m

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

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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*

m*R *= =

___________

*A*

m*R *=

______

URL

x = US – *A*m*R *= LRLX = LS + *A*m*R *=

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) =

*A*2*R *= =

———

UCL

x = *X *+ *A*2*R *=

LCL

x = *X *– *A*2*R *=

UCLR = *D*4*R *= =

*X *= S —– *x *= ———————— = ——–

*k*

or

US = LS =

*A*2

1.880

1.023

0.729

0.577

0.483

*D*4

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

*A*m

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

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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 (*x*s), *R*s, 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

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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 *=

Σ*X*

*k*

and

*X *= a population value

*k *= the number of values used to compute *x*

*E*2 = 2.66 (*n *= 2) (see Table A-1 in the Appendix)

The formula for the*MR*chart is similar to that for the*R*chart (where*n *= 2), except that the ranges

are computed as the differences from one sample to the next [*n *= 2; UCL = *D*4(*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

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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 *E*2 = 2.66 and *D*4 = 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 *E*2 and *D*4 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 *&2*R *(11.3)

UCL &

*x *= ≂ *x *– *A *&2*R *(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

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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 *= *B*4 * *s *Q (11.5)

LCL

*s *= *B*3 * *s *Q (11.6)

where

*B*

3 and *B*4 come from Table 11-3;

and *s *Q = Σ*s**i*/*k *(11.7)

where

*s*i 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 – *C*4 2)>*C*4 (11.8)

where

*C*4 can be found in Table 11-3.

Formulas for the *x *chart can now be created using the following formulas:

UCL =

*x *= *x *+ *A*3(*s*) (11.9)

LCL =

*x *= *x *– *A*3(*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

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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

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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*

2*R *= *x *= *x *+ *A*2*R *(Appendix Table A-1)

R *D*

3*R R D*4*R *(Appendix Table A-1)

X = *x *– *E*

2(*MR*) = *x *= *x *+ *E*2(*MR*) (Appendix Table A-1)

Median

≂*x*

– *A*

&

2*R *≂ *x *≂ *x *+ *A *&2*R *(Appendix Table A-4)

*x *(with *s*) = *x *– *A*3*s *= *x *= *x *+ *A*3*s *(Appendix Table A-3)

s *B*

3*s s B*4*s *(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

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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

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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

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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

5*‘ *0*” *5*‘ *4*” *5*‘ *8*” *6*‘ *0*” *6*‘ *4*“*

Sampling

distrib ution

(*n*5 5)

Population

distrib ution

(*N *5 40)

Figure 11-21 Population and Sampling Distributions for Class Heights

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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*/*d*2 (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,

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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

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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(*x**i *– *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.

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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 *R*2 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.

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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

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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?

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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.

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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>*d*2)*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

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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*)

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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

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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

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- 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)

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.

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.

Read moreThe 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.

Read moreThe 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.

Read moreThe 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.

Read moreBy 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.

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The price is based on these factors:

Academic level

Number of pages

Urgency