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JOHNLOUCKS

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SPIROS VELIANITIS

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Philosophies and FrameworksPhilosophies and Frameworks Statistical Process ControlStatistical Process Control Acceptance SamplingAcceptance Sampling

Chapter 20Chapter 20Statistical Methods for Quality ControlStatistical Methods for Quality Control

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QualityQuality

QualityQuality is “the totality of features and is “the totality of features and characteristics of a product or service that characteristics of a product or service that bears on its ability to satisfy given needs.”bears on its ability to satisfy given needs.”

Organizations recognize that they must strive Organizations recognize that they must strive for high levels of qualityfor high levels of quality

They have increased the emphasis on methods They have increased the emphasis on methods for monitoring and maintaining quality.for monitoring and maintaining quality.

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Total QualityTotal Quality

Total Quality (TQ)Total Quality (TQ) is a people-focused management system that aims at is a people-focused management system that aims at continual increase in customer satisfaction at continually lower costcontinual increase in customer satisfaction at continually lower cost

TQ is a total system approach (not a separate work program) and an TQ is a total system approach (not a separate work program) and an integral part of high-level strategy.integral part of high-level strategy.

TQ works horizontally across functions, involves all employees, top to TQ works horizontally across functions, involves all employees, top to bottom, and extends backward and forward to include both the supply bottom, and extends backward and forward to include both the supply and customer chains.and customer chains.

TQ stresses learning and adaptation to continual change as keys to TQ stresses learning and adaptation to continual change as keys to organization success.organization success.

Regardless of how it is implemented in different organizations, Total Regardless of how it is implemented in different organizations, Total Quality is based on Quality is based on three fundamental principlesthree fundamental principles::

• a focus on customers and stakeholdersa focus on customers and stakeholders

• participation and teamwork throughout the organizationparticipation and teamwork throughout the organization

• a focus on continuous improvement and learninga focus on continuous improvement and learning

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Quality PhilosophiesQuality Philosophies

Dr. W. Edwards DemingDr. W. Edwards Deming• One of Deming’s major contributions was to direct One of Deming’s major contributions was to direct

attention away from inspection of the final product or attention away from inspection of the final product or service towards monitoring the process that produces service towards monitoring the process that produces the final product or service with emphasis of statistical the final product or service with emphasis of statistical quality control techniques. In particular, Deming stressed quality control techniques. In particular, Deming stressed that in order to improve a process one needs to reduce that in order to improve a process one needs to reduce the variation in the process.the variation in the process.

Joseph JuranJoseph Juran

• Proposed a simple definition of quality: Proposed a simple definition of quality: fitness fitness for usefor use

• His approach to quality focused on three quality His approach to quality focused on three quality processes: quality processes: quality planningplanning, quality , quality controlcontrol, and , and quality quality improvementimprovement

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Quality FrameworksQuality Frameworks

Malcolm Baldrige National Quality Award: Established in Malcolm Baldrige National Quality Award: Established in 1987 and given by the U.S. president to organizations that 1987 and given by the U.S. president to organizations that apply and are judged to be outstanding in seven areasapply and are judged to be outstanding in seven areas

ISO 9000: A series of five standards published in 1987 by ISO 9000: A series of five standards published in 1987 by the International Organization for Standardization in the International Organization for Standardization in Geneva, Switzerland. Geneva, Switzerland.

Six Sigma: Six Sigma: Six sigma level of qualitySix sigma level of quality means that for every means that for every million opportunities no more than 3.4 defects will occur.million opportunities no more than 3.4 defects will occur.

• The methodology created to reach this quality goal is The methodology created to reach this quality goal is referred to as referred to as Six SigmaSix Sigma..

• Six Sigma is a major tool in helping organizations achieve Six Sigma is a major tool in helping organizations achieve Baldrige levels of business performance and process Baldrige levels of business performance and process quality.quality.

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Quality TerminologyQuality Terminology

Quality AssuranceQuality Assurance: QA refers to the entire : QA refers to the entire system of policies, system of policies, procedures, and guidelines established by an procedures, and guidelines established by an organization to achieve and maintain quality. organization to achieve and maintain quality. QA consists of two functions:QA consists of two functions:

• Quality Engineering Quality Engineering - Its objective is to - Its objective is to include quality in the design of products and include quality in the design of products and processes and to identify potential quality processes and to identify potential quality problems prior to production.problems prior to production.

• Quality Control Quality Control - QC consists of making a - QC consists of making a series of inspections and measurements to series of inspections and measurements to determine whether quality standards are determine whether quality standards are being met.being met.

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Statistical Process Control (SPC)Statistical Process Control (SPC) In order to reduce the variation of a process, one needs to recognize In order to reduce the variation of a process, one needs to recognize

that the total variation is comprised of that the total variation is comprised of common causescommon causes and and specific causesspecific causes. . At any time there are numerous factors which At any time there are numerous factors which individually and in interaction with each other cause detectable individually and in interaction with each other cause detectable variability in a process and its output. Those factors that are not variability in a process and its output. Those factors that are not readily identifiable and occur randomly are referred to as the readily identifiable and occur randomly are referred to as the common causescommon causes, , while those that have large impact and can be while those that have large impact and can be associated with specialassociated with special circumstances or factors are referred to as circumstances or factors are referred to as specific causesspecific causes..

When a process has variation made up of only common causes then When a process has variation made up of only common causes then the process is said to be a the process is said to be a stable processstable process, which means that the , which means that the process is in process is in statistical control statistical control and remains relatively the same over and remains relatively the same over time. This implies that the process is predictable, but does not time. This implies that the process is predictable, but does not necessarily suggest that the process is producing outputs that are necessarily suggest that the process is producing outputs that are acceptable as the amount of common variation may exceed the acceptable as the amount of common variation may exceed the amount of acceptable variation. If a process has variation that is amount of acceptable variation. If a process has variation that is comprised of both common causes and specific causes then it is said comprised of both common causes and specific causes then it is said to be an to be an unstable processunstable process, which means that the process is not in , which means that the process is not in statistical control. An unstable process does not necessarily mean statistical control. An unstable process does not necessarily mean that the process is producing unacceptable products since the total that the process is producing unacceptable products since the total variation (common variation + specific variation) may still be less variation (common variation + specific variation) may still be less than the acceptable level of variation.than the acceptable level of variation.

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Statistical Process Control (SPC) ExampleStatistical Process Control (SPC) Example

Consider a manufacturing situation where a hole needs to be drilled Consider a manufacturing situation where a hole needs to be drilled into a piece of steel. We are concerned with the size of the hole, in into a piece of steel. We are concerned with the size of the hole, in particular the diameter, since the performance of the final product is a particular the diameter, since the performance of the final product is a function of the precision of the hole. As we measure consecutively function of the precision of the hole. As we measure consecutively drilled holes, with very fine instruments, we will notice that there is drilled holes, with very fine instruments, we will notice that there is variation from one hole to the next. Some of the possible common variation from one hole to the next. Some of the possible common sources can be associated with the density of the steel, air sources can be associated with the density of the steel, air temperature, and machine operator. As long as these sources do not temperature, and machine operator. As long as these sources do not produce significant swings in the variation they can be considered produce significant swings in the variation they can be considered common sources. On the other hand, the changing of a drill bit could common sources. On the other hand, the changing of a drill bit could be a specific source provided it produces a significant change in the be a specific source provided it produces a significant change in the variation, especially if a wrong sized bit is used!variation, especially if a wrong sized bit is used!

SPC Determination Steps:SPC Determination Steps:

• Sample and inspect the output of the production process.Sample and inspect the output of the production process.

• Using SPC methods, determined whether variations in output are Using SPC methods, determined whether variations in output are due to common causes or assignable causes.due to common causes or assignable causes.

• Decide whether the process can be continued or should be Decide whether the process can be continued or should be adjusted to achieve a desired quality level. adjusted to achieve a desired quality level.

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HHaa is formulated in terms of the production is formulated in terms of the production process being process being out of controlout of control..

Null HypothesisNull Hypothesis

Alternative HypothesisAlternative Hypothesis

SPC HypothesesSPC Hypotheses

SPC procedures are based on hypothesis-SPC procedures are based on hypothesis-testing methodology.testing methodology.

HH00 is formulated in terms of the production is formulated in terms of the production process being process being in controlin control..

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Identification ToolsIdentification Tools There are a number of tools used in practice to There are a number of tools used in practice to

determine whether specific causes of variation exist determine whether specific causes of variation exist within a process. In the remaining part of this chapter within a process. In the remaining part of this chapter we will discuss how time we will discuss how time series plotsseries plots, , the runs testthe runs test, , a a test for normality test for normality and and control charts control charts are used to identify are used to identify specific sources of variation. As will become evident specific sources of variation. As will become evident there is a great deal of similarity between time series there is a great deal of similarity between time series plots and control charts. In particular, the control charts plots and control charts. In particular, the control charts are time series plots of statistics calculated from are time series plots of statistics calculated from subgroups of observations, whereas when we speak of subgroups of observations, whereas when we speak of time series plots we are referring to plots of time series plots we are referring to plots of consecutive observations.consecutive observations.

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Time Series PlotsTime Series Plots A time series plot is a graph where the horizontal axis represents time A time series plot is a graph where the horizontal axis represents time

and the vertical axis represents the units in which the variable of concern and the vertical axis represents the units in which the variable of concern is measured. is measured. For example, consider the following series where the variable of For example, consider the following series where the variable of concern is the price of Anheuser Busch Co. stock on the last trading day for each concern is the price of Anheuser Busch Co. stock on the last trading day for each month from June 1995 to June 2000 inclusive. Using the computer we are able to month from June 1995 to June 2000 inclusive. Using the computer we are able to generate the following time series plot. Note that the horizontal axis represents generate the following time series plot. Note that the horizontal axis represents time and the vertical axis represents the price of the stock, measured in dollars.time and the vertical axis represents the price of the stock, measured in dollars.

When using a time series plot to determine whether a process is stable, what one When using a time series plot to determine whether a process is stable, what one is seeking is the answer to the following questions:is seeking is the answer to the following questions:

• 1. Is the mean constant?1. Is the mean constant?

• 2. Is the variance constant?2. Is the variance constant?

• 3. Is the series random (i.e. no pattern)?3. Is the series random (i.e. no pattern)? Rather than initially showing the reader time series plots of stable processes, we Rather than initially showing the reader time series plots of stable processes, we

show examples of non stable processes commonly experienced in practice.show examples of non stable processes commonly experienced in practice.

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Runs TestRuns Test Frequently non-stable processes can be detected by Frequently non-stable processes can be detected by

visually examining their time series plots. However, visually examining their time series plots. However, there are times when patterns exist that are not easily there are times when patterns exist that are not easily detected. A tool that can be used to identify nonrandom detected. A tool that can be used to identify nonrandom data in these cases is the runs test.data in these cases is the runs test.

To determine if the observed number significantly To determine if the observed number significantly differs from the expected number, we encourage the differs from the expected number, we encourage the reader to rely on statistical software (StatGraphics) and reader to rely on statistical software (StatGraphics) and utilize the p-values that are generated.utilize the p-values that are generated.

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Test for NormalityTest for Normality

Another attribute of a stable process, which you may Another attribute of a stable process, which you may recall lacks specific causes of variation, is that the recall lacks specific causes of variation, is that the series follows a normal distribution. To determine series follows a normal distribution. To determine whether a variable follows a normal distribution one whether a variable follows a normal distribution one can examine the data via a graph, called a histogram, can examine the data via a graph, called a histogram, and/or utilize a test which incorporates a chi-square and/or utilize a test which incorporates a chi-square test statistic.test statistic.

StatGraphics, will overlay the observed data with a StatGraphics, will overlay the observed data with a theoretical distribution calculated from the sample theoretical distribution calculated from the sample mean and sample standard deviation in order to assist mean and sample standard deviation in order to assist in the evaluation.in the evaluation.

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ExampleExample Stationarity? Stationarity? From the visual inspection From the visual inspection

(top chart), one can tell the series is (top chart), one can tell the series is stationary.stationary.

Normality? Normality? From this, one can see that From this, one can see that the distribution of the middle chart the distribution of the middle chart appears somewhat like a normal appears somewhat like a normal distribution. Not exactly, but in order to distribution. Not exactly, but in order to see how closely it does relate to theoretical see how closely it does relate to theoretical normal distribution, we rely on the Chi-normal distribution, we rely on the Chi-square test. As we can see from the table, square test. As we can see from the table, the p-value (significance level) equals the p-value (significance level) equals 0.1356. Since the p-value is greater than 0.1356. Since the p-value is greater than alpha (0.05), we retain the null hypothesis.alpha (0.05), we retain the null hypothesis.

Random? Random? Relying upon the nonparametric Relying upon the nonparametric test for randomness. we can just look at test for randomness. we can just look at the p-value, which in this case is 0.086 the p-value, which in this case is 0.086 (rounded). So, since the p-value again is (rounded). So, since the p-value again is larger than our value of α = 0.05, we are larger than our value of α = 0.05, we are able to conclude that we cannot reject the able to conclude that we cannot reject the null hypothesis.null hypothesis.

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Control ChartsControl Charts

SPC uses graphical displays known as SPC uses graphical displays known as control charts control charts to monitor a production process.to monitor a production process.

Control charts provide a basis for deciding whether Control charts provide a basis for deciding whether the variation in the output is due to common causes the variation in the output is due to common causes (in control) or assignable causes (out of control).(in control) or assignable causes (out of control).

Two important lines on a control chart are the Two important lines on a control chart are the upper upper control limitcontrol limit (UCL) (UCL) and and lower control limitlower control limit (LCL) (LCL)..

These lines are chosen so that when the process is These lines are chosen so that when the process is inin control, there will be a high probability that the control, there will be a high probability that the sample finding will be between the two lines.sample finding will be between the two lines.

Values outside of the control limits provide strong Values outside of the control limits provide strong evidence that the process is evidence that the process is outout of control. of control.

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Control Chart TypesControl Chart Types Variables Control ChartsVariables Control Charts

• XX-bar Chart - -bar Chart - This chart is This chart is used if the quality of the output is used if the quality of the output is measured in terms of a variable such as length, weight, measured in terms of a variable such as length, weight, temperature, and so on. temperature, and so on. x x represents the represents the mean valuemean value found in found in a sample of the output.a sample of the output.

• RR Chart - Chart - This This chart is used to monitor the chart is used to monitor the rangerange of the of the measurements in the sample.measurements in the sample.

• The X-Bar and R Charts procedure creates control charts for a The X-Bar and R Charts procedure creates control charts for a single numeric variable where the data have been collected in single numeric variable where the data have been collected in subgroups. subgroups.

Attributes Control ChartsAttributes Control Charts

• pp Chart - Chart - This This chart is used to monitor the chart is used to monitor the proportion defective proportion defective in the sample.in the sample.

• npnp Chart - This chart is used to monitor the Chart - This chart is used to monitor the number of defective number of defective items in the sample. items in the sample.

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xx Chart Structure Chart Structure

UCLUCL

LCLLCL

Process MeanProcess MeanWhen in ControlWhen in Control

Center LineCenter Line

TimeTime

x Upper Control Upper Control LimitLimit

Lower Control Lower Control LimitLimit

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RR Chart Chart

In practice, the In practice, the RR chart is usually constructed chart is usually constructed before the before the xx chart. chart.

If the If the RR chart indicates that the process chart indicates that the process variability is in control, then the variability is in control, then the xx chart is chart is constructed.constructed.

Because the control limits for the Because the control limits for the xx chart chart depend on the value of the average range, depend on the value of the average range, these limits will not have much meaning these limits will not have much meaning unless the process variability is in control.unless the process variability is in control.

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Control Limits for an Control Limits for an RR Chart: Process Chart: ProcessMean and Standard Deviation Unknown Mean and Standard Deviation Unknown

When Granite Rock’s packaging process is in control, When Granite Rock’s packaging process is in control, tthe weight of bags of cement filled by the process is he weight of bags of cement filled by the process is normally distributed with a mean of 50 pounds and a normally distributed with a mean of 50 pounds and a standard deviation of 1.5 pounds. Suppose Granite standard deviation of 1.5 pounds. Suppose Granite does not know the true mean and standard deviation does not know the true mean and standard deviation for its bag filling process. It wants tofor its bag filling process. It wants todevelop develop xx and and R R charts based on twenty samples of 5 charts based on twenty samples of 5 bags each. bags each.

The twenty samples, collected when the process was in The twenty samples, collected when the process was in control, resulted in an overall sample mean of 50.01 control, resulted in an overall sample mean of 50.01 pounds and an average range of .322 pounds.pounds and an average range of .322 pounds.

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R Chart for Granite Rock Co.

0.00

0.10

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0.60

0.70

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Sa

mp

le R

ang

e R

LCL

UCL

Control Limits for an Control Limits for an RR Chart: Process Chart: ProcessMean and Standard Deviation UnknownMean and Standard Deviation Unknown

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xx = 50.01, = 50.01, RR = .322, = .322, nn = 5 = 5__

==

UCL = UCL = xx + + AA22RR = 50.01 + .577(.322) = 50.196 = 50.01 + .577(.322) = 50.196==__

LCL = LCL = xx AA22RR = 50.01 = 50.01 .577(.322) = 49.824 .577(.322) = 49.824==__

Control Limits for an Control Limits for an xx Chart: Process Chart: ProcessMean and Standard Deviation UnknownMean and Standard Deviation Unknown

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49.7

49.8

49.9

50.0

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50.3

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mp

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ean

UCL

LCL

x Chart for Granite Rock Co.

Control Limits for an Control Limits for an xx Chart: Process Chart: ProcessMean and Standard Deviation UnknownMean and Standard Deviation Unknown

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Control Limits For a Control Limits For a pp Chart Chart

Every check cashed or deposited atEvery check cashed or deposited at

Norwest Bank must be encoded withNorwest Bank must be encoded with

the amount of the check before it canthe amount of the check before it can

begin the Federal Reserve clearingbegin the Federal Reserve clearing

process. The accuracy of the checkprocess. The accuracy of the check

encoding process is of utmost encoding process is of utmost

importance. If there is any discrepancyimportance. If there is any discrepancy

between the amount a check is madebetween the amount a check is made

out for and the encoded amount, the check isout for and the encoded amount, the check is

defective.defective.

Example: Norwest BankExample: Norwest Bank

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TwTwenty samples, each consisting of 400enty samples, each consisting of 400

checks, were selected and examinedchecks, were selected and examined

when the encoding process was knownwhen the encoding process was known

to be operating correctly. The numberto be operating correctly. The number

of defective checks found in the 20of defective checks found in the 20

samples are listed below.samples are listed below.

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Control Limits For a Control Limits For a pp Chart Chart

Example: Norwest BankExample: Norwest Bank

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Suppose Norwest does not know the Suppose Norwest does not know the proportion of defective checks, proportion of defective checks, pp, for the , for the encoding process when it is in control.encoding process when it is in control.

Control Limits For a Control Limits For a pp Chart Chart

We will treat the data (20 samples) collected We will treat the data (20 samples) collected as one large sample and compute the average as one large sample and compute the average number of defective checks for all the data. number of defective checks for all the data. That value can then be used to estimate That value can then be used to estimate pp..

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Control Limits For a Control Limits For a pp Chart Chart

Encoded Checks Proportion Defective

0.000

0.005

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Sample Number

Sa

mp

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rop

ort

ion

p

UCL

LCL

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NP ChartsNP Charts

The NP Chart procedure creates a control chart for data The NP Chart procedure creates a control chart for data that describes the number of times an event occurs in that describes the number of times an event occurs in m samples taken from a product or process. The data m samples taken from a product or process. The data might represent the number might represent the number of defective items in a of defective items in a manufacturing process, the number of customers that manufacturing process, the number of customers that return a product, or any other attribute that can be return a product, or any other attribute that can be classified as acceptable or unacceptable.classified as acceptable or unacceptable.

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Interpretation of Control ChartsInterpretation of Control Charts

The location and pattern of points in a control The location and pattern of points in a control chart enable us to determine, with a small chart enable us to determine, with a small probability of error, whether a process is in probability of error, whether a process is in statistical control.statistical control.

A primary indication that a process may be out A primary indication that a process may be out of control is a of control is a data point outside the control data point outside the control limitslimits..

Certain patternsCertain patterns of points within the control of points within the control limits can be warning signals of quality limits can be warning signals of quality problems: a large number of points on one side problems: a large number of points on one side of center line OR six or seven points in a row of center line OR six or seven points in a row that indicate either an increasing or decreasing that indicate either an increasing or decreasing trend.trend.

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Acceptance SamplingAcceptance Sampling

Acceptance samplingAcceptance sampling is a statistical method that is a statistical method that enables us to base the accept-reject decision on enables us to base the accept-reject decision on the inspection of a sample of items from the lot.the inspection of a sample of items from the lot.

The items of interest can be incoming shipments The items of interest can be incoming shipments of raw materials or purchased parts as well as of raw materials or purchased parts as well as finished goods from final assembly.finished goods from final assembly.

Acceptance sampling has advantages over 100% Acceptance sampling has advantages over 100% inspection.inspection.

Acceptance sampling is based on hypothesis-Acceptance sampling is based on hypothesis-testing methodologytesting methodology

• HH00: Good-quality lot: Good-quality lot

• HHaa: Poor-quality lot: Poor-quality lot

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Acceptance Sampling ProcedureAcceptance Sampling Procedure

Lot receivedLot received

Sample selectedSample selected

Sampled itemsSampled itemsinspected for qualityinspected for quality

Results compared withResults compared withspecified quality characteristicsspecified quality characteristics

Accept the lotAccept the lot Reject the lotReject the lot

Send to productionSend to productionor customeror customer

Decide on dispositionDecide on dispositionof the lotof the lot

Quality is Quality is notnot satisfactorysatisfactory

Quality isQuality issatisfactorysatisfactory