statistical quality control_final

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Statistical Quality

ControlBITS-Pilani, Hyderabad Campus


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Introduction

Have you bought a product andfound it to be defective ?

New backpack with broken zipper

Tablets that are crushed in thestrip/blisters before use

Difficulty in assembling products

requiring minor assembly

As consumers, you expect every

product you buy should perform to


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WHAT IS STATISTICALQUALITY CONTROL?

Statistical quality control (SQC)is the term used to describe the setof statistical tools used by quality

professionals.

It is classified as: Descriptive statistics

Statistical process control

Acceptance sampling


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Classification of SQC

Descriptive statistics

Are used to describe quality characteristicsand relationships.

Included are statistics such as the mean,

standard deviation, the range and ameasure of the distribution of data.


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Classification of SQC cont.

Statistical process control (SPC) A statistical tool that involves inspecting a

random sample of the output from a processand deciding whether the process is

producing products with characteristics thatfall within a pre-determined range.

Acceptance sampling The process of randomly inspecting a

sample of goods and deciding whether toaccept the entire lot based on the results.


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Descriptive statistics are used todescribe certain qualitycharacteristics, such as the centraltendency and variability of observed

data.

Although this is useful, it does notprovide information on whether thereis problem with quality


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Acceptance sampling helps us decidewhether desirable quality has beenachieved for a batch of products, andwhether to accept or reject the items

produced

Although this information is helpful inmaking the quality acceptancedecision after the product has beenproduced, it does not help us identify

and catch a quality problem during


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Statistical QC Tools - Trinity

All three of these statistical qualitycontrol categories are helpful inmeasuring and evaluating the quality

of products or services.

However, statistical process control(SPC) tools are used most frequentlybecause they identify qualityproblems during the production

process.


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:COMMON AND

ASSIGNABLE CAUSES Common, or random causes ofvariation

E.g.: Cola bottles in grocery stores careful observation shows that no twobottles are filled exactly to the samelevel

Common causes of variation arebased on random causes that we

cannot identify


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:COMMON AND

ASSIGNABLE CAUSES It is important to find a range forsuch natural random variation

E.g.: If the average bottle of a soft drinkcontains 16 ounces of liquid, we maydetermine that the amount of naturalvariation is between 15.8 and 16.2ounces.

If production goes out of this range

this would lead us to believe that


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:COMMON AND

ASSIGNABLE CAUSES Assignable causes of variation These are causes that can be identified and

eliminated.

E.g.: poor quality in raw materials, anemployee who needs more training, or amachine in need of repair

In each of these examples the problem canbe identified and corrected.


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Descriptive Statistics

The most important descriptivestatistics are :

Measures of central tendency : Mean(average)

Measures of variability : standarddeviation and range

Measures of the distribution of data


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The Mean (average)

A statistic that measures the centraltendency of a set of data.

To compute the mean we simply sum allthe observations and divide by the totalnumber of observations.

The equation is given by


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The Range The difference between the largest and

smallest

observations in a set of data.

Standard Deviation

A statistic that measures the amount ofdata dispersion around the mean.

The equation is given by


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Distribution of data

A third descriptive statistic used tomeasure quality characteristics is the

shape of the distribution of the observeddata.

When a distribution is symmetric, there

are thesame number of observations below and

above the mean.

When a disproportionate number of


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Statistical ProcessControl (SPC) Methods

Statistical process control methodsextend the use of descriptivestatistics to monitor the quality of

the product and process.

Using statistical process control wewant to determine the amount ofvariation that is common or normal.


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Developing Control Charts

A control chart (also called processchart or quality control chart) is agraph that shows whether a sample

of data falls within the common ornormal range of variation.

A control chart has upper and lowercontrol limits that separate commonfrom assignable causes of variation.


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The common range of variation isdefined by the use of control chartlimits.

We say that a process is out ofcontrol (out of specification OOS) when a plot of data revealsthat one or more samples fall outsidethe control limits.


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The upper and lower control limits ona control chart are usually set at 3standard deviations from the mean

If we assume that the data exhibit anormal distribution, these controllimits will capture 99.74 percent ofthe normal variation


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Percentage of values captured bydifferentranges of standard deviation


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From the figure, we can infer that,observations that fall outside the setrange represent assignable causes of

variation.

However, there is a small probabilitythat a value that falls outside thelimits is still due to normal variation.


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Another name for this is alpha risk( ), where alpha refers to the sum ofthe probabilities in both tails of the

distribution that falls outside theconfidence limits.

Chance of Type I error for

3(sigma-standarddeviations)


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For limits of 3 standard deviationsfrom the mean,

the probability of a Type I error is .26% (100% 99.74%),

Whereas,

for limits of 2 standard deviations itis 4.56% (100% 95.44%).

Chance of Type I error for

3(sigma-standarddeviations)


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Types of Quality ControlCharts

Control charts are one of the mostcommonly used tools in statisticalprocess control.

The different characteristics that canbe measured by control charts canbe divided into two groups:

1. Variables

1. Attributes


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Control chart for variables

A control chart for variables is usedto monitor

characteristics that can be measuredand have

a continuum of values, such asheight, weight,

or volume.


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Control chart for attributes

A control chart for attributes is usedto monitor a product characteristicthat has a discrete value and can becounted.

Often they can be evaluated with asimple

yes or no decision

Examples include color, taste, or smell.


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Control Charts forVariables

When an item is inspected, thevariable being monitored ismeasured and recorded

For example, if we were producingcandles, height might be animportant variable.

Two of the most commonly usedcontrol charts for variables monitorboth the central tendency of the data

(the mean) and the variability of the


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Mean (x-bar) Charts

A control chart used to monitor changes inthe mean value of a process.

To construct a mean chart we first need toconstruct the centre line of the chart.

To do this we take multiple samples andcompute their means.

Usually these samples are small, with aboutfour or five observations.

Each sample has its own mean,


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Mean (x-bar) Charts

The centre line of the chart is then computed as the mean of allsample means, where is the number of samples


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Mean (x-bar) Charts


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Mean (x-bar) Charts

Another way to construct the controllimits is to use the sample range asan estimate of the variability of the

process. The range is simply the difference

between the largest and smallest

values in the sample The spread of the range can tell us

about the variability of the data.


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Mean (x-bar) Charts

Factors for three-sigma control


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ac o s o ee s g a co olimits of and R-charts


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Constructing a Mean (x-Bar)Chart from the Sample Range


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Range (R) Charts

A control chart that monitorschanges in the dispersion orvariability of process.

While x-bar charts measure shift inthe central tendency of the process,range charts monitor the dispersionor variability of the process.


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Range (R) Charts

The centre line of the control chart isthe average range, and the upperand lower control limits are

computed as :


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Range (R) Charts Note thatA2 is a factor that includes three standard

deviations of ranges and is dependent on thesample size being considered.

( ) h


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Range (R) Charts Resulting chart is depicted below:

Using Mean and Range Charts


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Using Mean and Range ChartsTogether

The mean or x-bar chart measures the centraltendency of the process, whereas the range chartmeasures the dispersion or variance of the process.

Since both variables are important, it makes senseto monitor a process using both mean and rangecharts.

It is possible to have a shift in the mean of theproduct but not a change in the dispersion.

Using Mean and Range Charts


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Using Mean and Range ChartsTogether

Process shifts captured by chartsand -charts

CONTROL CHARTS FOR


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CONTROL CHARTS FORATTRIBUTES

P-chart

A control chart that monitors theproportion of defects in a sample.

C-chart

A control chart used to monitor thenumberof

defects per unit.

CONTROL CHARTS FOR


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The P-Chart (Proportion chart)

P-charts are used to measure the proportionof items in a sample that are defective.

E.g.: proportion of broken tablets in a batch

P-charts are appropriate when both thenumber of defectives measured and the sizeof the total sample can be counted.

A proportion can then be computed andused as the statistic of measurement.

CONTROL CHARTS FOR


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P-chart is a control chart that monitorstheproportion of defects in a sample.

The center line is computed as theaverage proportion defective in thepopulation.

This is obtained by taking a number ofsamples of observations at random andcomputing the average value of p acrossall sam les.

CONTROL CHARTS FOR


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To construct the upper and lower controllimits for a p-chart, we use the followingformula:

As with the other charts,zis selected to be either 2or 3 standard deviations, depending on the amountof data we wish to capture in our control limits

Usually z = 3

CONTROL CHARTS FOR


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In this example the lowercontrol limit is negative,

which sometimes occursbecause the computationis an approximation of thebinomial distribution. Whenthis occurs, the LCL isrounded up tozero because we cannot

have a negative controllimit.

CONTROL CHARTS FOR


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The C-Chart (Count chart)

C-charts count the actual number of defects.

E.g.: Number of cfu on a petri-dish

C-charts do not give the proportion (we

cannot compute proportion of bacteria onpetri dish).

In C-charts, the types of units ofmeasurement we consider are a eriod of

CONTROL CHARTS FOR


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The C-Chart (Count chart)

The average number of defects, is thecenter line of the control chart.

The upper and lower control limits arecomputed as follows:


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Process Capability

Process capability

The ability of a production process tomeet or exceed preset specifications.

Product specifications(tolerances)

They preset ranges of acceptable qualitycharacteristics, such as productdimensions.

For a product to be considered


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Process Capability

E.g. for product specification:

Thickness of a particular tablet set at3.3 0.2 mm

It means, the standard requiredthickness of the tablet must be 3.3 mm,though, it is acceptable if it falls

between 3.1 to 3.5 mm

Specifications for a product are

preset on the basis of how the


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Process Capability

Measuring Process Capability

Simply setting up control charts to monitorwhether a process is in control does not

guarantee process capability.

Process capability is measured by theprocess capability index, Cp,

Cp is computed as the ratio of thespecification width to the width of the processvariability:

P C bilit


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Process Capability

Measuring Process CapabilityWhere, the specification width is thedifference between the upper specificationlimit (USL) and the lower specification limit

(LSL) of the process

The process width computed as 6 standard

deviations of the process being monitored

The reason we use 6 is that most of theprocess measurement (99.74 percent) falls

within 3 SD, which is a total of 6 standard


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Cp = 1 Cp < 1

Cp > 1


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2

1

3A Cp value of 1 means that 99.74 percent of the productsproduced will fall within the specification limits.

This also means that .26 percent (100% 99.74%) of the

products will not be acceptable.

The number .26 percent corresponds to 2600 parts per million(ppm) defective(0.0026 1,000,000)

That number can seem very high if we think of it in terms of


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Computation ofCp when processvariability not centered across

specification width


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In the figure the specification limitsare set between 15.8 and 16.2

ounces, with a mean of 16.0 ounces.

However, the process variation is notcentered; it has a mean of 15.9ounces.

Because of this, a certain proportionof products will fall outside thespecification range.


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Computation for such samples aredone as follows:

This measure of process capability helps us address apossible lack of centering of the process over thespecification range

To use this measure, the process capability of each halfof the normal distribution is computed and theminimum of the two is used.


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Example:

The Cp value of 1.00 leads us to conclude that the

process is capable.

However, from the graph we can see that theprocess is notcentered on the specification rangeand is producing out-of-spec products.


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Example:

Using only the Cp measure would lead to an

incorrect conclusion in this case

Computing Cpkgives us a different answer and leads us to adifferent conclusion:


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60


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Six Sigma Principles

The term Six Sigma was coined bythe Motorola Corporation in the1980s to describe the high level ofquality the company was striving toachieve.

Sigma ( ) stands for the number ofstandard deviations of the process

3 sigma ( ) means that 2600 m

Six sigma principles


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Six sigma principles

PPM defective for 3 versus 6 quality (not toscale)


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Acceptance Sampling

Acceptance sampling, refers to theprocess of randomly inspecting acertain number of items from a lot or

batch in order to decide whether toaccept or reject the entire batch.

Acceptance sampling is performedeither before or afterthe process,rather than during the process.


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Acceptance sampling before theprocess involves sampling materialsreceived from a supplier (E.g.: Rawmaterials for tablet manufacture)

Sampling afterthe process involves

sampling finished items that are tobe shipped either to a customer or toa distribution center (E.g.: Packedtablet bottles in warehouse)


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Acceptance sampling is used wheninspecting every item is notphysically possible or would beoverly expensive, or when inspecting

a large number of items would leadto errors due to worker fatigue

Another example of whenacceptance sampling would be usedis in destructive testing (E.g.:

Measurin crushin stren th of

Sampling Plans in


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Sampling Plans inacceptance sampling

Sampling Plan

A plan for acceptance sampling thatprecisely specifies the parameters of thesampling process and theacceptance/rejection criteria.

The variables to be specified include the size

of the lot (N), the size of the sampleinspected from the lot (n), the number ofdefects above which a lot is rejected (c), andthe number of samples that will be taken.

Sampling Plans in


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Sampling Plan

In single sampling, in which a randomsample is drawn from every lot, each item isclassified as either good or bad

Depending on number of bad items found,the entire batch is rejected

Sampling Plans in


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Sampling Plan Another type of acceptance sampling is

called double sampling

This provides an opportunity to sample thelot a second time if the results of the firstsample are inconclusive.

In double sampling we first sample a lot of

goods according to preset criteria fordefinite acceptance or rejection.

However, if the results fall in the middlerange, they are considered inconclusive and

a second sample is taken

pera ng arac er s c(OC) C


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p g(OC) Curves

OC Curves

A graph that shows the probability or chanceof accepting a lot given various proportionsof defects in the lot.

QC C


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QC Curves

Acceptable quality level (AQL)

The small percentage of defects thatconsumers are willing to accept.

Lot tolerance percent defective

(LTPD) The upper limit of the percentage of

defective items consumers are willing

to tolerate.

OC


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OC curves

Consumers risk

The chance of accepting a lot thatcontains a greater number of defects

than the LTPD limit.

This is the probability of making aType II errorthat is, accepting a lotthat is truly bad.

An OC curve showing producersi k ( ) d


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risk ( ) andconsumers risk ()

OC


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OC curves

Producers risk

The chance that a lot containing anacceptable

quality level will be rejected.

This is the probability of making aType I errorthat is, rejecting a lotthat is good.

OC curves


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OC curves

We can determine from an OC curvewhat the consumers and producersrisks are.

However, these values should not beleft to chance.

Rather, sampling plans are usuallydesigned to meet specific levels of

consumers and producers risk.

D l i OC C


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Developing OC Curves

An OC curve graphically depicts thediscriminating power of a samplingplan.

To draw an OC curve, we typicallyuse a cumulative binomial

distribution to obtain probabilities ofaccepting a lot given varying levelsof lot defects.

D l i OC C


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Partial Cumulative Binomial Probability Table

The top of the table shows values ofp, whichrepresents the proportionof defective items in a lot (5 percent, 10 percent, 20percent, etc.).

The left-hand column shows values ofn, whichrepresent the sample size being considered



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Developing OC CurvesOC curve with n = 5and c = 1

Average Outgoing


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g g gQuality (AOQ)

Average outgoing quality (AOQ)

The expected proportion of defectiveitems that will be passed to the

customer under the sampling plan.

From the OC curves, higher thequality of the lot, the higher is thechance that it will be accepted

Average Outgoing


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g g gQuality (AOQ)

Given that some lots are acceptedand some rejected, it is useful tocompute the average outgoing

quality (AOQ) of lots to get a senseof the overall outgoing quality of theproduct.

Assuming that all lots have the sameproportion of defective items, the

average outgoing quality can be

Average Outgoing


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g g gQuality (AOQ)

Usually we assume the fraction in the previous

equation to equal 1 and simplify the equation to thefollowing form:


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How much and how often


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to inspect ? Consider Product Cost and

Product Volume

trade-off between the cost of inspection

and the cost of passing on a defectiveitem should be considered

The inspection process should be set up

to consider issues of product cost andvolume.

Historical data must be considered



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to inspect ? Consider Process Stability

Stable processes that do not changefrequently do not need to be inspected

often. On the other hand, processes that are

unstable and change often should beinspected frequently.

For example, if it has been observedthat a particular type of drilling machinein a machine shop often goes out of

tolerance, that machine should be



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to inspect ? Consider Lot Size

The size of the lot or batch beingproduced is another factor to consider in

determining the amount of inspection A company that produces a small

number of large lots will have a smallernumber of inspections than a companythat produces a large number of smalllots

The reason is that every lot should have

some inspection, and when lots are

Where to Inspect ?


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Where to Inspect ?

Inbound Materials

Finished Products

Prior to Costly Processing

Which tools to use ?


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Which tools to use ?

Tools such as control charts are bestused at various points in theproduction process.

Acceptance sampling is best used forinbound and outbound materials.

It is also the easiest method to use

for attribute measures, whereas

Conclusions


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Conclusions

Statistical quality control can bedivided into three broad categories:descriptive statistics, acceptance

sampling, and statistical processcontrol (SPC).

Descriptive statistics are used todescribe quality characteristics, suchas the mean, range, and variance.

Conclusions


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Conclusions

Statistical process control (SPC)involves inspecting a random sampleof output from a process and

deciding whether the process isproducing products withcharacteristics that fall within presetspecifications.

There are two causes of variation in

the quality of a product or process:

Conclusions


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Conclusions

Common causes of variation arerandom causes that we cannotidentify.

Assignable causes of variation arethose that can be identified andeliminated.

A control chart is a graph used instatistical process control that shows

whether a sam le of data falls within

Conclusions


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Conclusions

Control charts for variables include x-bar charts and R-charts.

X-bar charts monitor the mean oraverage value of a productcharacteristic

R-charts monitor the range or

dispersion of the values of a product

Conclusions


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Conclusions

P-charts are used to monitor theproportion of defects in a sample.

C-charts are used to monitor theactual number of defects in asample.

Process capability is the ability of theproduction process to meet or

exceed reset s ecifications

Conclusions


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Conclusions

The term Six Sigma indicates a levelof quality in which the number ofdefects is no more than 3.4 parts permillion.

The goal of acceptance sampling is

to determine criteria for acceptanceor rejection based on lot size, samplesize, and the desired level ofconfidence.

Conclusions


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Conclusions

Operating characteristic (OC) curvesare graphs that show thediscriminating power of a samplingplan.

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