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Control Charts for Variables Control Charts for Variables

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Control Charts for Variables

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• Control Charts for VariablesControl Charts for Variables

• IntroductionIntroductionA quality characteristic that is measured on a A quality characteristic that is measured on a

numericalnumerical scale is called a scale is called a variable.variable. dimensiondimension

length, widthlength, width weightweight temperaturetemperature volumevolume

• A quality characteristic that is a variable, it isA quality characteristic that is a variable, it isusually necessary to monitor both the mean value ofusually necessary to monitor both the mean value ofthe quality characteristic and its variability. the quality characteristic and its variability.

Control of the process average or mean qualityControl of the process average or mean qualitylevel is usually done with the control chart for meanslevel is usually done with the control chart for meansor or x x chartchart..

Process variability can be monitor with either Process variability can be monitor with either a control chart for the standard deviation, called the a control chart for the standard deviation, called the s s chartchart,,or a control chart for the range, called an or a control chart for the range, called an R R chartchart..

• The need for controlling both process mean

and process variability

• CONTROL CHART FOR CONTROL CHART FOR xx ANDAND RRStatistical Basis of the ChartsStatistical Basis of the Charts

Suppose that a quality characteristic is normallySuppose that a quality characteristic is normallydistributed with mean distributed with mean and standard deviation and standard deviation ,,where both where both and and are known.are known.

If If xx11,, xx22, ......, , ......, xxnn is a sample of size is a sample of size nn, the average, the averageof this sample isof this sample is

• we know that x is normally distributed with mean and standard deviation x = / n .

The probability is 1- that any sample mean will fallbetween

• In practice, we usually will no know and .They must be estimated from preliminary samples orsubgroups taken when the process is thought to be incontrol.

These estimates should usually be based onat least 20 to 25 samples.

Suppose that m samples are available, eachcontaining n observations on the quality characteristic.Typically, n will be small, often either 4, 5, or 6.

• The best estimator of , the process average, is the grand average,

• We may estimate from either the standarddeviations or the ranges of the m samples.

Let R1, R2, ....., Rm be the ranges of the m samples.The average range is

• The formulas for constructing the control limits onthe x chart are as follows:

• The formulas for constructing the control limits onthe R chart are as follows:

• ExampleExample

Piston for automotive engine are produced by a forgingPiston for automotive engine are produced by a forgingprocess. We wish to establish statistical control of inside process. We wish to establish statistical control of inside diameter of the ring manufactured by this process usingdiameter of the ring manufactured by this process usingxx andand R R charts. charts.

TwentyTwenty--five samples, each of size five, have been takenfive samples, each of size five, have been takenwhen we think the process is in control. The inside diameterwhen we think the process is in control. The inside diametermeasurement data from these samples are shown in table.measurement data from these samples are shown in table.

• x chart

• R chart

• CONTROL CHART FOR CONTROL CHART FOR xx ANDAND SS

xx AND AND S S charts are preferable to their more familiarcharts are preferable to their more familiarCounterparts, Counterparts, x x and and RR charts, when eithercharts, when either

1.1. The sample size The sample size nn is moderately large, is moderately large, nn > 10 or 12> 10 or 122.2. The sample size The sample size nn is variable.is variable.

• The construction of The construction of xx and and SS chartscharts

Setting up and operating control charts for Setting up and operating control charts for xx and and SSrequires about the same sequence of step as those forrequires about the same sequence of step as those forthe the xx and and RR charts, except that for each sample we charts, except that for each sample we must calculate the sample average must calculate the sample average xx and sampleand samplestandard deviation standard deviation SS..

• The sample variance :The sample variance :

• The formulas for constructing the control limits on the The formulas for constructing the control limits on the SS chartchart are as follow :are as follow :

For For givengiven

• For For not givennot given

• ExampleExample Construction of Construction of xx and and SS charts using the pistoncharts using the piston--ring inside diameter measurements in table.ring inside diameter measurements in table.

• Revise the Control LimitsRevise the Control Limits

Suppose that one or more of the values of either Suppose that one or more of the values of either x x and and RR plot out of control.plot out of control.

It is necessary to revise the control limit by examiningIt is necessary to revise the control limit by examiningeach of the out of control points, looking for an each of the out of control points, looking for an assignable cause.assignable cause.

If an assignable cause is found, the point is discardedIf an assignable cause is found, the point is discardedand an the control limits are recalculated, using onlyand an the control limits are recalculated, using onlythe remaining points.the remaining points.