Acceptance SamplingAcceptance sampling is a method used to accept or reject product based on a random sample of the product.

The purpose of acceptance sampling is to sentence lots (accept or reject) rather than to estimate the quality of a lot.

Acceptance sampling plans do not improve quality. The nature of sampling is such that acceptance sampling will accept some lots and reject others even though they are of the same quality.

The most effective use of acceptance sampling is as an auditing tool to help ensure that the output of a process meets requirements.

As mentioned acceptance sampling can reject good lots and accept bad lots. More formally:

Producers risk refers to the probability of rejecting a good lot. In order to calculate this probability there must be a numerical definition as to what constitutes goodAQL (Acceptable Quality Level) - the numerical definition of a good lot. The ANSI/ASQC standard describes AQL as the maximum percentage or proportion of nonconforming items or number of nonconformities in a batch that can be considered satisfactory as a process average

Consumers Risk refers to the probability of accepting a bad lot where:LTPD (Lot Tolerance Percent Defective) - the numerical definition of a bad lot described by the ANSI/ASQC standard as the percentage or proportion of nonconforming items or noncomformities in a batch for which the customer wishes the probability of acceptance to be a specified low value.

The Operating Characteristic Curve is typically used to represent the four parameters (Producers Risk, Consumers Risk, AQL and LTPD) of the sampling plan as shown below where the P on the x axis represents the percent defective in the lot:

Note: if the sample is less than 20 units the binomial distribution is used to build the OC Curve otherwise the Poisson distribution is used.

Chart2

0.99999214940.9999925124

0.99988582070.9998907682

0.99947436050.9994956532

0.99848868210.9985457706

0.99664193110.9967598938

0.99365994930.993866448

0.98930092810.9896229806

0.98336695180.9838280083

0.97570956370.9763271527

0.96623103180.9670150878

0.95488261670.9558345015

0.94166084750.9427730128

0.92660257090.9278587724

0.90977934560.9111553032

0.89129160530.8927559995

0.87126289170.8727785903

0.8498343670.8513597874

0.82715974330.8286502684

0.80340071180.8048100911

0.7787229110.7800045934

0.75329244750.754400805

0.72727295520.7281643698

0.70082316870.701456964

0.67409497310.674434182

0.64723188880.6472438563

0.6203679450.6200247699

0.59362689660.5929057216

0.56712173910.5660049005

0.54095447760.5394295297

0.51521611050.5132757405

0.48998678970.4876286381

0.46533612520.4625625287

0.44132360130.4381412733

0.41799908030.4144187442

0.39540336960.3914393573

0.37356883130.3692386609

0.35252001840.3478439602

0.33227432250.3272749643

0.31284262130.3075444405

0.29422991650.2886588653

0.27643595340.2706190638

0.25945581730.2534208289

0.24328050090.2370555161

0.22789744060.2215106085

0.21329101840.2067702488

0.19944302920.1928157382

0.18633311110.179625998

0.17393914190.1671779959

0.16223759820.1554471359

0.15120388280.1444076134

0.1408126170.1340327345

0.13103790340.1242952044

0.12185355810.115167382

0.11323331570.1066215066

0.10515100780.0986298955

0.09758071820.0911651162

0.09049691520.0842001341

0.08387456470.0777084381

0.07768922370.0716641463

0.07191711770.0660420924

0.06653520320.0608178959

0.06152121590.0559680174

0.05685370790.0514698001

0.05251207320.0473014993

0.04847656480.0434423017

0.04472830260.0398723352

0.04124927480.0365726704

0.0380223330.0335253159

0.03503118220.0307132069

0.03226036580.0281201888

0.02969524760.0257309975

0.02732199050.0235312349

0.02512753250.0215073429

0.02309956110.0196465745

0.02122648630.0179369629

0.01949741210.016367291

0.01790210790.0149270578

0.01643097920.0136064466

0.01507503780.0123962917

0.01382587340.0112880458

0.01267562360.0102737475

0.01161694630.00934599

0.01064299160.0084978892

0.00974737410.0077230545

0.00892414750.0070155589

0.00816777850.0063699107

0.00747312250.005781027

0.00683540010.0052442071

0.00625017440.0047551079

0.00571332970.0043097201

0.00522105080.0039043458

0.00476980330.0035355771

0.00435631570.0032002757

0.00397756090.0028955543

0.00363074050.0026187579

0.00331326870.0023674476

0.00302275760.0021393841

0.00275700320.001932513

0.00251397250.0017449511

0.00229179120.0015749728

LTPD

AQL

Producers Risk

Consumers Risk

Poisson

Binomial

P

Probability of Acceptance

OC Curve

Sheet1

Sampling Plan

n120

c3first sample

0.2794317523

pMean number of defective unitsProbability of acceptance (Poisson)Probability of acceptance (Binomial)p(2)0.2611013704

0.0010.120.99999214940.9999925124p(3)0.2198748382

0.0020.240.99988582070.9998907682

0.0030.360.99947436050.9994956532

0.0040.480.99848868210.9985457706Second sample

0.0050.60.99664193110.9967598938

0.0060.720.99365994930.993866448p(1)0.0311606801

0.0070.840.98930092810.9896229806p(0)0.0059205292

0.0080.960.98336695180.9838280083

0.0091.080.97570956370.9763271527

0.011.20.96623103180.9670150878

0.0111.320.95488261670.9558345015

0.0121.440.94166084750.9427730128

0.0131.560.92660257090.9278587724

0.0141.680.90977934560.9111553032

0.0151.80.89129160530.8927559995

0.0161.920.87126289170.8727785903

0.0172.040.8498343670.8513597874

0.0182.160.82715974330.8286502684

0.0192.280.80340071180.8048100911

0.022.40.7787229110.7800045934

0.0212.520.75329244750.754400805

0.0222.640.72727295520.7281643698

0.0232.760.70082316870.701456964

0.0242.880.67409497310.674434182

0.02530.64723188880.6472438563

0.0263.120.6203679450.6200247699

0.0273.240.59362689660.5929057216

0.0283.360.56712173910.5660049005

0.0293.480.54095447760.5394295297

0.033.60.51521611050.5132757405

0.0313.720.48998678970.4876286381

0.0323.840.46533612520.4625625287

0.0333.960.44132360130.4381412733

0.0344.080.41799908030.4144187442

0.0354.20.39540336960.3914393573

0.0364.320.37356883130.3692386609

0.0374.440.35252001840.3478439602

0.0384.560.33227432250.3272749643

0.0394.680.31284262130.3075444405

0.044.80.29422991650.2886588653

0.0414.920.27643595340.2706190638

0.0425.040.25945581730.2534208289

0.0435.160.24328050090.2370555161

0.0445.280.22789744060.2215106085

0.0455.40.21329101840.2067702488

0.0465.520.19944302920.1928157382

0.0475.640.18633311110.179625998

0.0485.760.17393914190.1671779959

0.0495.880.16223759820.1554471359

0.0560.15120388280.1444076134

0.0516.120.1408126170.1340327345

0.0526.240.13103790340.1242952044

0.0536.360.12185355810.115167382

0.0546.480.11323331570.1066215066

0.0556.60.10515100780.0986298955

0.0566.720.09758071820.0911651162

0.0576.840.09049691520.0842001341

0.0586.960.08387456470.0777084381

0.0597.080.07768922370.0716641463

0.067.20.07191711770.0660420924

0.0617.320.06653520320.0608178959

0.0627.440.06152121590.0559680174

0.0637.560.05685370790.0514698001

0.0647.680.05251207320.0473014993

0.0657.80.04847656480.0434423017

0.0667.920.04472830260.0398723352

0.0678.040.04124927480.0365726704

0.0688.160.0380223330.0335253159

0.0698.280.03503118220.0307132069

0.078.40.03226036580.0281201888

0.0718.520.02969524760.0257309975

0.0728.640.02732199050.0235312349

0.0738.760.02512753250.0215073429

0.0748.880.02309956110.0196465745

0.07590.02122648630.0179369629

0.0769.120.01949741210.016367291

0.0779.240.01790210790.0149270578

0.0789.360.01643097920.0136064466

0.0799.480.01507503780.0123962917

0.089.60.01382587340.0112880458

0.0819.720.01267562360.0102737475

0.0829.840.01161694630.00934599

0.0839.960.01064299160.0084978892

0.08410.080.00974737410.0077230545

0.08510.20.00892414750.0070155589

0.08610.320.00816777850.0063699107

0.08710.440.00747312250.005781027

0.08810.560.00683540010.0052442071

0.08910.680.00625017440.0047551079

0.0910.80.00571332970.0043097201

0.09110.920.00522105080.0039043458

0.09211.040.00476980330.0035355771

0.09311.160.00435631570.0032002757

0.09411.280.00397756090.0028955543

0.09511.40.00363074050.0026187579

0.09611.520.00331326870.0023674476

0.09711.640.00302275760.0021393841

0.09811.760.00275700320.001932513

0.09911.880.00251397250.0017449511

0.1120.00229179120.0015749728

Sheet1

Poisson

Binomial

P

Probability of Acceptance

OC Curve

Sheet2

LTPD

AQL

Producers Risk

Consumers Risk

Poisson

Binomial

P

Probability of Acceptance

OC Curve

Sheet3

MBD00000020.unknown

MBD00134818.unknown

MBD00000054.unknown

Designing Sampling PlansOperationally, three values need to be determined before a sampling plan can be implemented (for single sampling plans):

N = the number of units in the lot n = the number of units in the samplec = the maximum number of nonconforming units in the sample for which the lot will be accepted.While there is not a straightforward way of determining these values directly given desired values of the parameters, tables have been developed. Below is an excerpt of one of these tables.

Excerpt From a Sampling Plan Table with Producers Risk = 0.05 and Consumers Risk = 0.10

C

LTPD/AQL

n(AQL)

n(LTPD)

0

44.89

0.052

2.334

1

10.946

.355

3.886

2

6.509

.818

5.324

3

4.89

1.366

6.68

4

4.057

1.97

7.992

5

3.549

2.613

9.274

6

3.206

3.286

10.535

7

2.957

3.981

11.772

8

2.768

4.695

12.996

9

2.618

5.426

14.205

Three alternatives for specifying sampling plansProducers Risk and AQL specifiedConsumers Risk and LTPD specifiedAll four parameters specified

For the first two cases we simply choose the acceptance number and divide the appropriate column by the associated parameter to get the sample size.Example 1: Given a producers risk of .05 and an AQL of .015 determine a sampling planc = 1: n = .355/.015 ~ 24c = 4: n = 1.97/.015 ~ 131Example 2: Given a consumers risk of .1 and a LTPD of .08 determine a sampling planc = 0: n = 2.334/.08 ~ 29c = 5: n = 9.274/.08 ~ 116

Note: you can check these with the OC curve posted on my web site

All four parameters specifiedWhen all four parameters are specified we must first find a value close to the ratio LTPD/AQL in the table. Then values of n and c are found.

Example: Given producers risk of .05, consumers risk of .1, LTPD 4.5%, and AQL of 1% find a sampling plan.Since 4.5/1 = 4.5 is between c= 3 and c = 4. Using the n(AQL) column the sample sizes suggested are 137 and 197 respectively. Note using this column will ensure a producers risk of .05. Using the n(LTPD) column will ensure a consumers risk of .1

Double SamplingIn an effort to reduce the amount of inspection double (or multiple) sampling is used. Whether or not the sampling effort will be reduced depends on the defective proportions of incoming lots. Typically, four parameters are specified:

n1=number of units in the first samplec1=acceptance number for the first samplen2=number of units in the second samplec2=acceptance number for both samples

ProcedureA double sampling plan proceeds as follows:

A random sample of size n1 is drawn from the lot.If the number of defective units (say d1 ) c1 the lot is accepted.If d1 c2 the lot is rejected.If neither of these conditions are satisfied a second sample of size n2 is drawn from the lot.If the number of defectives in the combined samples (d1 + d2) > c2 the lot is rejected. If not the lot is accepted.

ExampleConsider a double sampling plan with n1 = 50; c1 = 1; n2 = 100; c2 = 3. What is the probability of accepting a lot with 5% defective?

The probability of accepting the lot, say Pa, is the sum of the probabilities of accepting the lot after the first and second samples (say P1 and P2)

To calculate P1 we first must determine the expected number defective in the sample (50*.05)we can then use the Poisson function to determine the probability of finding 1 or fewer defects in the sample, e.g., Poisson(1,2.5,true) = .287

To calculate P2 we need to determine first the scenarios that would lead to requiring a second sample. Then for each scenario we must determine the probability of acceptance. P2 will then be the some of the probability of acceptance for each scenario.

Example continuedFor this example, there are only two possible scenarios that would lead to requiring a second sample: Scenario 1: 2 nonconforming items in the first sample Scenario2: 3 nonconforming items in the first sample. The probability associated with these two scenarios is found as follows:P(scenario 1) = Poisson(2,2.5,false) = .257P(scenario 2) = Poisson(3,2.5,false) = .214Under scenario 1 the lot will be accepted if there are either 1 or 0 nonconforming items in the second sample. The expected number of nonconforming items in the second sample .05(100) = 5the probability is then: Poisson(1,5,true) = .04therefore the probability of acceptance under scenario 1 is (.257)*(.04) = .01Under scenario 2 we will accept the lot only if there are 0 nonconforming items in the second sample, i.e., Poisson(0,5,false) = .007therefore the probability of acceptance under scenario 2 is (.214)*(.007) = .0015so the probability of acceptance on the second sample is .01 + .0015 = .0115 So the overall probability of acceptance is (probability of acceptance on first sample + probability of accepting after 2 samples) = .287 + .0115 = .299

OC curve for double sampling

Chart6

0.99879089570.99999230060.0000002502

0.99532115980.99988953890.0000038468

0.98981417290.99949815040.0000187141

0.98247690370.99857537840.0000568402

0.97350097880.99687328220.0001333697

0.96306368690.99416626590.0002658112

0.95132892110.99026678610.0004733497

0.93844806440.98503276920.0007762514

0.92456081990.97836937650.0011953522

0.90979598960.97022707460.0017516226

0.89427220610.96059744220.0024657976

0.87809861780.94950773750.0033580689

0.86137553170.9370149490.0044478278

0.84419501640.92319982210.0057534576

0.82664146730.908161180.0072921665

0.80879213540.89201073790.0090798578

0.79071762410.87486851290.0111310325

0.77248235350.85685887480.0134587206

0.75414499570.83810723260.0160744366

0.73575888230.81873732960.0189881569

0.71737238570.79886909620.0222083157

0.69902927580.77861700380.0257418165

0.68076905420.75808885840.0295940576

0.66262726620.73738497080.0337689682

0.64463579290.71659764550.0382690543

0.6268231240.69581093180.0430954527

0.60921461250.67510058890.0482479906

0.59183271350.65453421870.0537252504

0.57469720580.63417152930.0595246381

0.55782540040.61406469530.0656424544

0.54123233330.59425878640.0720739676

0.52493094680.57479224090.0788134872

0.50893225780.55569736630.0858544381

0.49324551490.53700084890.0931894339

0.47787834450.51872426240.1008103499

0.4628368870.50088456490.1087083947

0.4481259240.48349457690.1168741796

0.43374899570.46656343660.1252977867

0.41970851120.45009702710.1339688341

0.40600584970.43409837440.1428765395

0.39264145590.41856801530.1520097802

0.37961492760.40350433460.1613571511

0.3669250970.38890387120.1709070198

0.35457010680.37476159610.1806475783

0.34254747980.36107116130.1905668926

0.33085418430.34782512290.2006529488

0.31948669340.33501513890.2108936964

0.30844104120.32263214420.221277089

0.29771287340.31066650460.2317911218

0.28729749520.2991081510.2424238669

0.27718991430.28794669670.253163505

0.26738488160.27717153810.2639983556

0.25787692770.26677194180.2749169037

0.24866039710.25673711830.2859078244

0.23972947950.24705628530.2969600055

0.2310782380.237718720.3080625674

0.22270063540.22871380340.3192048807

0.21459055820.22003105650.3303765825

0.20674183850.21166016960.3415675896

0.19914827350.20359102640.3527681112

0.19180364380.19581372240.3639686588

0.18470172980.18831857910.3751600546

0.17783632650.18109615490.3863334389

0.17120125670.17413725230.3974802756

0.16479038330.16743292290.4085923561

0.15859761980.16097446980.4196618025

0.15261694030.15475344820.4306810695

0.14684238780.14876166480.4416429447

0.14126808190.14299117550.4525405484

0.13588822540.13743428230.4633673321

0.13069711040.13208352920.4741170768

0.12568912330.12693169780.4847838895

0.12085874880.12197180230.4953621999

0.11620057440.11719708360.5058467558

0.11170929280.1126010040.5162326184

0.10737970490.10817724140.5265151567

0.10320672170.10391968270.5366900417

0.09918536610.09982241850.5467532399

0.09531077380.09587973650.5567010067

0.09157819440.0920861160.5665298796

0.08798299190.08843622170.5762366705

0.08452064450.08492489790.5858184585

0.08118674490.08154716310.5952725818

0.07797699950.07829820410.6045966304

0.0748872280.07517337110.6137884376

0.07191336280.07216817180.6228460723

0.06905144730.06927826680.6317678304

0.06629763550.06649946450.6405522272

0.063648190.06382771630.6491979888

0.0610994810.0612591120.6577040442

0.05864798430.05878987530.6660695168

0.05629028020.05641635930.674293717

0.05402305090.05413504270.6823761335

0.05184307950.05194252530.6903164258

0.04974724740.04983552440.6981144166

0.04773253290.04781087080.7057700835

0.04579600870.0458655050.7132835521

0.04393484010.04399647420.7206550882

0.04214628310.04220092820.7278850907

0.0404276820.04047611650.7349740847

0.03877646730.0388193850.7419227141

0.0371901540.03722817250.7487317354

0.03566633910.03570000840.7554020106

0.03420269940.03423250910.7619345013

0.032796990.03282337560.7683302619

0.03144704160.03147039030.7745904343

0.03015075880.03017141490.7807162413

0.0289061180.02892438750.7867089816

0.02771116530.02772731990.7925700237

0.02656401440.02657829550.7983008013

0.02546284490.02547546690.8039028076

0.02440590050.02441705350.8093775907

0.02339148660.02340133920.8147267488

0.02241796860.02242667060.8199519258

0.02148377040.02149145440.8250548066

0.02058737230.0205941560.8300371133

0.01972730920.01973329680.8349006011

0.01890216920.0189074530.8396470541

0.01811059160.01811525330.8442782822

0.01735126520.01735537730.8487961172

P1

Pa = P1 + P2

P(rejection 1st)

p

Pa

OC Chart - Double Sampling Plan

Sheet1

n150

c11

n2100

c23

Sample 1Sample 2

pMean number of defective unitsProbability of acceptance (Poisson)Probability of c1+1 defects in sample 1Probability of c1+2 defects in sample 1Probability of acceptance after sample 2Probability of acceptance on combined sampleProbability of rejecting the lot after the first sampleProbability a lot disposition decision is made after the first sampleAverage sample number

0.0010.050.99879089570.00118903680.00001981730.00120140490.99999230060.00000025020.998791145950.120885406

0.0020.10.99532115980.00452418710.00015080620.0045683790.99988953890.00000384680.995325006750.4674993327

0.0030.150.98981417290.00968296470.00048414820.00968397760.99949815040.00001871410.98983288751.0167112972

0.0040.20.98247690370.01637461510.0010916410.01609847470.99857537840.00005684020.982533743951.7466256066

0.0050.250.97350097880.02433752450.0020281270.02337230340.99687328220.00013336970.973634348552.636565151

0.0060.30.96306368690.03333681990.0033336820.0311025790.99416626590.00026581120.963329498153.6670501924

0.0070.350.95132892110.04316214550.00503558360.0389378650.99026678610.00047334970.951802270954.8197729136

0.0080.40.93844806440.05362560370.00715008050.04658470480.98503276920.00077625140.939224315856.0775684174

0.0090.450.92456081990.06455985040.00968397760.05380855660.97836937650.00119535220.925756172157.4243827904

0.010.50.90979598960.07581633250.01263605540.0604310850.97022707460.00175162260.911547612158.8452387875

0.0110.550.89427220610.08726365880.01599833750.06632523620.96059744220.00246579760.896738003760.326199627

0.0120.60.87809861780.09878609450.01975721890.07140911970.94950773750.00335806890.881456686661.8543313396

0.0130.650.86137553170.11028217030.02389447020.07563941740.9370149490.00444782780.865823359463.4176640581

0.0140.70.84419501640.12166339940.02838812650.07900480560.92319982210.00575345760.84994847465.0051525962

0.0150.750.82664146730.1328530930.03321327320.08151971270.908161180.00729216650.833933633866.6066366198

0.0160.80.80879213540.14378526850.03834273830.08321860250.89201073790.00907985780.817871993268.2128006789

0.0170.850.79071762410.15440364420.04374769920.08415088880.87486851290.01113103250.801848656769.8151343347

0.0180.90.77248235350.16466071220.04939821370.08437652120.85685887480.01345872060.785941074171.4058925853

0.0190.950.75414499570.17451688680.05526368080.08396223680.83810723260.01607443660.770219432372.9780567665

0.0210.73575888230.18393972060.06131324020.08297844730.81873732960.01898815690.754747039274.5252960781

0.0211.050.71737238570.19290318420.06751611450.08149671050.79886909620.02220831570.739580701376.0419298667

0.0221.10.69902927580.20138700560.07384190210.07958772810.77861700380.02574181650.724771092377.5228907704

0.0231.150.68076905420.20937606380.08026082440.07731980420.75808885840.02959405760.710363111878.963688819

0.0241.20.66262726620.21685983260.0867439330.07475770460.73738497080.03376896820.696396234480.3603765608

0.0251.250.64463579290.22383187250.09326328020.07196185260.71659764550.03826905430.682904847281.7095152775

0.0261.30.6268231240.23028936510.09979205820.06898780790.69581093180.04309545270.669918576783.008142333

0.0271.350.60921461250.23623268750.10630470940.06588597640.67510058890.04824799060.657462603184.2537396895

0.0281.40.59183271350.24166502470.11277701150.06270150520.65453421870.05372525040.645557963885.4442036172

0.0291.450.57469720580.24659201540.11918614080.05947432350.63417152930.05952463810.634221843986.5778156115

0.031.50.55782540040.25102143020.12551071510.0562392950.61406469530.06564245440.623467854787.653214525

0.0311.550.54123233330.25496287860.13173082060.05302645310.59425878640.07207396760.613306300988.6693699148

0.0321.60.52493094680.2584275430.1378280230.04986129410.57479224090.07881348720.60374443489.6255565984

0.0331.650.50893225780.26142793810.1437853660.04676510850.55569736630.08585443810.594786695990.5213304071

0.0341.70.49324551490.26397769230.14958735890.0437553340.53700084890.09318943390.586434948891.3565051201

0.0351.750.47787834450.26609135090.15521995470.0408459180.51872426240.10081034990.578688694492.1311305605

0.0361.80.4628368870.26778419890.16067051940.03804767790.50088456490.10870839470.571545281792.845471827

0.0371.850.4481259240.26907210090.16592779550.03536865290.48349457690.11687417960.565000103693.4999896381

0.0381.90.43374899570.26997135770.17098185990.03281444090.46656343660.12529778670.559046782494.0953217572

0.0391.950.41970851120.27049857860.17582407610.03038851590.45009702710.13396883410.553677345394.6322654696

0.0420.40600584970.27067056650.18044704430.02809252470.43409837440.14287653950.548882389295.1117610789

0.0412.050.39264145590.27050421620.18484454770.02592655940.41856801530.15200978020.544651236195.5348763876

0.0422.10.37961492760.27001642430.1890114970.0238894070.40350433460.16135715110.540972078795.9027921306

0.0432.150.3669250970.26922400970.19294387360.02197877420.38890387120.17090701980.537832116896.2167883239

0.0442.20.35457010680.26814364320.19663867170.02019148930.37476159610.18064757830.535217685196.4782314944

0.0452.250.34254747980.26679178720.20009384040.01852368150.36107116130.19056689260.533114372496.6885627551

0.0462.30.33085418430.26518464160.20330822530.01697093870.34782512290.20065294880.531507133196.8492866909

0.0472.350.31948669340.26333809920.2062815110.01552844550.33501513890.21089369640.530380389896.9619610182

0.0482.40.30844104120.26126770550.20901416440.0141911030.32263214420.2212770890.529718130197.0281869852

0.0492.450.29771287340.25898862650.21150737830.01295363120.31066650460.23179112180.529503995297.0496004774

0.052.50.28729749520.25651562070.21376301720.01181065580.2991081510.24242386690.529721362197.0278637949

0.0512.550.27718991430.25386301660.21578356410.01075678240.28794669670.2531635050.530353419396.9646580685

0.0522.60.26738488160.25104469440.21757206840.00978665660.27717153810.26399835560.531383237296.8616762814

0.0532.650.25787692770.24807407190.21913209680.00889501410.26677194180.27491690370.532793831396.7206168667

0.0542.70.24866039710.24496409390.22046768450.00807672120.25673711830.28590782440.534568221596.54317784790.040427682

0.0552.750.23972947950.24172722520.22158328980.00732680570.24705628530.29696000550.536689485196.3310514943

0.0562.80.2310782380.23837544550.22248374910.0066404820.237718720.30806256740.539140805496.0859194616

0.0572.850.22270063540.23492024820.22317423570.0060131680.22871380340.31920488070.541905516195.8094483898

0.0582.90.21459055820.23137264030.2236602190.00544049820.22003105650.33037658250.544967140795.503285933

0.0592.950.20674183850.22774314550.22394742640.00491833110.21166016960.34156758960.548309428195.1690571924

0.0630.19914827350.22404180770.22404180770.00444275290.20359102640.35276811120.551916384794.8083615311

0.0613.050.19180364380.22027819710.22394950040.00401007860.19581372240.36396865880.555772302694.4227697433

0.0623.10.18470172980.21646141750.22367679810.00361684930.18831857910.37516005460.559861784494.0138215585

0.0633.150.17783632650.21260011440.22323012010.00325982840.18109615490.38633343890.564169765493.5830234559

0.0643.20.17120125670.20870248440.22261598330.00293599560.17413725230.39748027560.568681532393.1318467696

0.0653.250.16479038330.20477628510.22184097550.00264253960.16743292290.40859235610.573382739492.6617260649

0.0663.30.15859761980.20082884650.22091173110.002376850.16097446980.41966180250.578259422492.1740577649

0.0673.350.15261694030.19686708190.21983490820.00213650790.15475344820.43068106950.583298009991.6701990124

0.0683.40.14684238780.19289750040.21861716710.0019192770.14876166480.44164294470.588485332591.1514667457

0.0693.450.14126808190.18892621850.21726515130.00172309370.14299117550.45254054840.593808630290.6191369764

0.073.50.13588822540.18495897350.2157854690.00154605690.13743428230.46336733210.599255557590.07444425

0.0713.550.13069711040.18100113560.21418467720.00138641870.13208352920.47411707680.604814187289.5185812773

0.0723.60.12568912330.17705772150.21246926580.00124257460.12693169780.48478388950.610473012888.9526987209

0.0733.650.12085874880.17313340660.21064564470.00111305340.12197180230.49536219990.616220948888.3779051247

0.0743.70.11620057440.16923253870.20872013110.00099650920.11719708360.50584675580.622047330387.795266974

0.0753.750.11170929280.16535915060.20669893820.00089171120.1126010040.51623261840.627941911387.2058088738

0.0763.80.10737970490.16151697280.20458816550.00079753650.10817724140.52651515670.633894861686.610513835

0.0773.850.10320672170.15770944670.20239378990.0007129610.10391968270.53669004170.639896763486.0103236584

0.0783.90.09918536610.15393973650.20012165750.00063705240.09982241850.54675323990.645938605985.4061394054

0.0793.950.09531077380.15021074220.19777747730.00056896270.09587973650.55670100670.652011780584.7988219488

0.0840.09157819440.14652511110.19536681480.00050792160.0920861160.56652987960.658108074184.1891925923

0.0814.050.08798299190.142885250.19289508750.00045322970.08843622170.57623667050.664219662583.5780337529

0.0824.10.08452064450.13929333680.19036756020.00040425330.08492489790.58581845850.67033910382.9660896979

0.0834.150.08118674490.13575133150.18778934180.00036041820.08154716310.59527258180.676459326782.3540673302

0.0844.20.07797699950.13226098760.18516538260.00032120470.07829820410.60459663040.682573629981.7426370136

0.0854.250.0748872280.12882386250.18250047190.00028614310.07517337110.61378843760.688675665681.1324334352

0.0864.30.07191336280.12544132810.17979923690.0002548090.07216817180.62284607230.69475943580.5240564965

0.0874.350.06905144730.12211458050.17706614180.00022681950.06927826680.63176783040.700819277779.9180722291

0.0884.40.06629763550.11884465030.17430548710.0002018290.06649946450.64055222720.706849862779.3150137312

0.0894.450.063648190.11563241120.171521410.00017952630.06382771630.64919798880.712846178878.7153821185

0.094.50.0610994810.11247858990.16871788490.0001596310.0612591120.65770404420.718803525178.1196474874

0.0914.550.05864798430.10938377440.16589872450.00014189090.05878987530.66606951680.724717501177.5282498852

0.0924.60.05629028020.10634842220.16306758070.00012607910.05641635930.6742937170.730583997176.9416002851

0.0934.650.05402305090.10337286890.16022794680.00011199180.05413504270.68237613350.736399184476.3600815623

0.0944.70.05184307950.10045733560.15738315910.00009944590.05194252530.69031642580.742159505375.7840494676

0.0954.750.04974724740.09760193650.15453639950.0000882770.04983552440.69811441660.74786166475.2138335985

0.0964.80.04773253290.0948066860.15169069760.00007833790.04781087080.70577008350.753502616474.6497383612

0.0974.850.04579600870.09207150550.14884893380.00006949630.0458655050.71328355210.759079560774.0920439263

0.0984.90.04393484010.08939622980.1460138420.00006163410.04399647420.72065508820.764589928373.5410071718

0.0994.950.04214628310.08678061360.14318801250.00005464510.04220092820.72788509070.770031373872.9968626151

0.150.0404276820.08422433750.14037389580.00004843450.04047611650.73497408470.775401766772.4598233303

0.1015.050.03877646730.08172701310.13757380540.00004291760.0388193850.74192271410.780699181571.9300818503

0.1025.10.0371901540.07928818910.13478992140.00003801850.03722817250.74873173540.785921889571.4078110528

0.1035.150.03566633910.07690735590.13202429440.00003366930.03570000840.75540201060.791068349770.8931650287

0.1045.20.03420269940.0745839510.12927884830.00002980970.03423250910.76193450130.796137200770.3862799314

0.1055.250.032796990.07231736290.12655538510.00002638560.03282337560.76833026190.801127251969.8872748081

0.1065.30.03144704160.07010693640.12385558770.00002334870.03147039030.77459043430.806037475969.3962524102

0.1075.350.03015075880.0679519760.12118102380.00002065610.03017141490.78071624130.810867000268.913299984

0.1085.40.0289061180.06585175010.11853315030.00001826950.02892438750.78670898160.815615099668.4384900401

0.1095.450.02771116530.0638054950.1159133160.00001615460.02772731990.79257002370.82028118967.9718811025

0.115.50.02656401440.0618124180.11332276630.00001428120.02657829550.79830080130.824864815667.5135184353

0.1115.550.02546284490.05987170090.11076264660.0000126220.02547546690.80390280760.829365652567.0634347483

0.1125.60.02440590050.05798250310.10823400570.0000111530.02441705350.80937759070.833783491266.6216508813

0.1135.650.02339148660.05614396460.10573780.00000985270.02340133920.81472674880.838118235366.1881764665

0.1145.70.02241796860.05435520890.10327489680.0000087020.02242667060.81995192580.842369894365.7630105684

0.1155.750.02148377040.0526153450.1008460780.00000768410.02149145440.82505480660.84653857765.346142304

0.1165.80.02058737230.05092347080.09845204360.00000678370.0205941560.83003711330.850624485664.9375514397

0.1175.850.01972730920.04927867450.09609341520.00000598760.01973329680.83490060110.854627910364.5372089683

0.1185.90.01890216920.04768003710.09377073960.00000528380.0189074530.83964705410.858549223464.1450776643

0.1195.950.01811059160.04612663450.09148449170.00000466170.01811525330.84427828220.862388873863.7611126188

0.1260.01735126520.04461753920.08923507840.00000411210.01735537730.84879611720.866147382563.385261754

Sheet1

&A

Page &P

Pa1

Pa = Pa1 + Pa2

P(rejection 1st)

p

Pa

OC Chart - Double Sampling Plan

Sheet2

probability of rejecting on first sample

probability of acceptance

probability of acceptance on first sample

Average sample number

p

Average sample number

ASN Curve

P1

Pa = P1 + P2

P(rejection 1st)

p

Pa

OC Chart - Double Sampling Plan

Sheet3

Average Sample NumberSince the goal of double sampling is to reduce the inspection effort, the average number of units inspected is of interest. This is easily calculated as followsLet PI = the probability that a lot disposition decision is made on the first sample, i.e., (probability that the lot is rejected on the first sample + probability that the lot is accepted on the first sample)

Then the average number of items is:

n1(PI) +(n1 + n2)(1 - PI)= n1 + n2(1 - PI)

The average sample number curve can be derived from the data on the OC chart

Chart5

50.120885406

50.4674993327

51.0167112972

51.7466256066

52.636565151

53.6670501924

54.8197729136

56.0775684174

57.4243827904

58.8452387875

60.326199627

61.8543313396

63.4176640581

65.0051525962

66.6066366198

68.2128006789

69.8151343347

71.4058925853

72.9780567665

74.5252960781

76.0419298667

77.5228907704

78.963688819

80.3603765608

81.7095152775

83.008142333

84.2537396895

85.4442036172

86.5778156115

87.653214525

88.6693699148

89.6255565984

90.5213304071

91.3565051201

92.1311305605

92.845471827

93.4999896381

94.0953217572

94.6322654696

95.1117610789

95.5348763876

95.9027921306

96.2167883239

96.4782314944

96.6885627551

96.8492866909

96.9619610182

97.0281869852

97.0496004774

97.0278637949

96.9646580685

96.8616762814

96.7206168667

96.5431778479

96.3310514943

96.0859194616

95.8094483898

95.503285933

95.1690571924

94.8083615311

94.4227697433

94.0138215585

93.5830234559

93.1318467696

92.6617260649

92.1740577649

91.6701990124

91.1514667457

90.6191369764

90.07444425

89.5185812773

88.9526987209

88.3779051247

87.795266974

87.2058088738

86.610513835

86.0103236584

85.4061394054

84.7988219488

84.1891925923

83.5780337529

82.9660896979

82.3540673302

81.7426370136

81.1324334352

80.5240564965

79.9180722291

79.3150137312

78.7153821185

78.1196474874

77.5282498852

76.9416002851

76.3600815623

75.7840494676

75.2138335985

74.6497383612

74.0920439263

73.5410071718

72.9968626151

72.4598233303

71.9300818503

71.4078110528

70.8931650287

70.3862799314

69.8872748081

69.3962524102

68.913299984

68.4384900401

67.9718811025

67.5135184353

67.0634347483

66.6216508813

66.1881764665

65.7630105684

65.346142304

64.9375514397

64.5372089683

64.1450776643

63.7611126188

63.385261754

Average sample number

p

Average sample number

ASN Curve

Sheet1

n150

c11

n2100

c23

Sample 1Sample 2

pMean number of defective unitsProbability of acceptance (Poisson)Probability of c1+1 defects in sample 1Probability of c1+2 defects in sample 1Probability of acceptance after sample 2Probability of acceptance on combined sampleProbability of rejecting the lot after the first sampleProbability a lot disposition decision is made after the first sampleAverage sample number

0.0010.050.99879089570.00118903680.00001981730.00120140490.99999230060.00000025020.998791145950.120885406

0.0020.10.99532115980.00452418710.00015080620.0045683790.99988953890.00000384680.995325006750.4674993327

0.0030.150.98981417290.00968296470.00048414820.00968397760.99949815040.00001871410.98983288751.0167112972

0.0040.20.98247690370.01637461510.0010916410.01609847470.99857537840.00005684020.982533743951.7466256066

0.0050.250.97350097880.02433752450.0020281270.02337230340.99687328220.00013336970.973634348552.636565151

0.0060.30.96306368690.03333681990.0033336820.0311025790.99416626590.00026581120.963329498153.6670501924

0.0070.350.95132892110.04316214550.00503558360.0389378650.99026678610.00047334970.951802270954.8197729136

0.0080.40.93844806440.05362560370.00715008050.04658470480.98503276920.00077625140.939224315856.0775684174

0.0090.450.92456081990.06455985040.00968397760.05380855660.97836937650.00119535220.925756172157.4243827904

0.010.50.90979598960.07581633250.01263605540.0604310850.97022707460.00175162260.911547612158.8452387875

0.0110.550.89427220610.08726365880.01599833750.06632523620.96059744220.00246579760.896738003760.326199627

0.0120.60.87809861780.09878609450.01975721890.07140911970.94950773750.00335806890.881456686661.8543313396

0.0130.650.86137553170.11028217030.02389447020.07563941740.9370149490.00444782780.865823359463.4176640581

0.0140.70.84419501640.12166339940.02838812650.07900480560.92319982210.00575345760.84994847465.0051525962

0.0150.750.82664146730.1328530930.03321327320.08151971270.908161180.00729216650.833933633866.6066366198

0.0160.80.80879213540.14378526850.03834273830.08321860250.89201073790.00907985780.817871993268.2128006789

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0.0733.650.12085874880.17313340660.21064564470.00111305340.12197180230.49536219990.616220948888.3779051247

0.0743.70.11620057440.16923253870.20872013110.00099650920.11719708360.50584675580.622047330387.795266974

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0.0763.80.10737970490.16151697280.20458816550.00079753650.10817724140.52651515670.633894861686.610513835

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0.0834.150.08118674490.13575133150.18778934180.00036041820.08154716310.59527258180.676459326782.3540673302

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0.0874.350.06905144730.12211458050.17706614180.00022681950.06927826680.63176783040.700819277779.9180722291

0.0884.40.06629763550.11884465030.17430548710.0002018290.06649946450.64055222720.706849862779.3150137312

0.0894.450.063648190.11563241120.171521410.00017952630.06382771630.64919798880.712846178878.7153821185

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Sheet1

&A

Page &P

Pa1

Pa = Pa1 + Pa2

P(rejection 1st)

p

Pa

OC Chart - Double Sampling Plan

Sheet2

probability of rejecting on first sample

probability of acceptance

probability of acceptance on first sample

Average sample number

p

Average sample number

ASN Curve

Pa1

Pa = Pa1 + Pa2

P(rejection 1st)

p

Pa

OC Chart - Double Sampling Plan

Sheet3