Statistics Technical Service Field Service Quality Assurance Statistical Sampling Variable • Figures Attribute • Yes or No Technical Service Field Service Quality Assurance Sampling Samples are divided into two groups • Random Samples – Drawn from a lot giving every sample the same chance to get picked • Aimed Samples – Drawn in direct connection to an event NEVER MIX THE TWO! Technical Service Field Service Quality Assurance Sampling Aimed Samples, objective • Used to monitor the influence of an event in a production lot • Accumulated results can be used to improve specific production steps, operational and technical Technical Service Field Service Quality Assurance Sampling Random Samples, objective • Used to give an estimation of the average defect rate in a production lot • Accumulated results can be used to estimate the average performance level of a plant Technical Service Field Service Quality Assurance Sampling Random Sampling, • • • • The size of a production lot does not matter Only the sample size determines the accuracy Percentage is not a good measurement Applying statistics is necessary due to limited amount of sampling (cost reasons) • For rare events we use the Poisson Distribution Technical Service Field Service Quality Assurance Sampling Defect rates detected at a certain sample size Technical Service Field Service Quality Assurance Sampling Calculating sample size depending on AQL 4.6 Technical Service Field Service Quality Assurance Sampling Calculating sample size depending on AQL Formula: n = 100 x z AQL (%) Example: AQL = 1:10,000 = 0.01% Defect detected (C) = at a Probability of 99% In the chart follow the 99-line to the C=1 curve, then go to the y-axis to get the z-value In this case 4.6 From the formula we get: 100 x 4.6 = 46,000 0.01 If we take 46.000 samples from a lot with a assumed defect rate of : 10.000, we have 99% chance to find one defect If we find < 1, we know with 99% probability that the actual defect rate is < : 10.000 Technical Service Field Service Quality Assurance Sampling Calculating sample size depending on AQL 2.3 Technical Service Field Service Quality Assurance Consumer’s and Producer’s risk • • • Examine 3890 packs, accept if max defect is found, reject otherwise We accept at or defect! P1 = 0.03% (good), P2 = 0.1% (bad) M = n x p = 3890 x 0.0003 = 1.167 P(x=0) = 1.1670 x e-1.167 = 0.31 = 31% 1! P(x=1) = 1.1671 x e-1.167 = 0.36 = 36% 1! - Prob(accept p1) = – (0.36+0.31) = 33% Producer’s risk = 33% M = n x p = 3890 x 0.001 = 3.89 P(x=1) = 3.89 x e-3.89 = 0.08 = 8% 1! Prob(accept p2) = 0.08 = 8% Consumer’s risk = 8% Technical Service Field Service Quality Assurance Consumer’s and Producer’s risk • • • Examine 5322 packs, accept if max defect is found, reject otherwise We accept at 0,1 or defects! P1 = 0.03% (good), P2 = 0.1% (bad) M = n x p = 5322 x 0.0003 = 1.596 P(x=0) = 1.5960 x e-1.596 = 0.20 = 20% 0! P(x=1) = 1.5961 x e-1.596 = 0.32 = 32% 1! P(x=2) = 1.5962 x e-1.596 = 0.26 = 26% 2! - Prob(accept p1) = – (0.20+0.33+0,26) = 22% Producer’s risk = 22% M = n x p = 5322 x 0.001 = 5.322 P(x=2) = 5.322 x e-5.322 = 0.08 = 8% 2! Prob(accept p2) = 0.08 = 8% Consumer’s risk = 8% Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D • General – – – – Focus on AQL (p1 = specification of ‘good batch”) Used for inspection of received goods AQL from 0.01% to 100% Seven inspection levels: • • • • I = less ambitious II = normal III = more ambitious S1 – S4 = very extensive inspection – Simple, double and multiple sampling • Routine to find a plan – Define lot size + inspection level + type of plan – AQL gives the acceptance number Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D Table – Sample size code letters Lot or batch size Special Inspection Levels General Inspection Levels S-1 S-2 S-3 S-4 I II III to A A A A A A B to 15 A A A A A B C 16 to 25 A A B B B C D 26 to 50 A B B C C D E 51 to 90 B B C C C E F 91 to 150 B B C D D F G 151 to 280 B C D E E G H 281 to 500 B C D E F H J 501 to 1200 C C E F G J K 1201 to 3200 C D E G H K L 3201 to 10000 C D F G J L M 10001 to 35000 C D F H K M N 35001 to 150000 D E G J L N P 150001 to 500000 D E G J M P Q 500001 and over D E H K N Q R Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D Table 2A – Single sampling plans for normal inspection Sample size code A B C D E F G H J K L M N P Q Sample size 13 20 32 50 80 125 200 315 500 800 1250 Acceptable Quality Levels R 2000 0.010 Ac Re 0.015 Ac Re 0.025 Ac Re 0.040 Ac Re 0.065 Ac Re 0.10 Ac Re 0.15 Ac Re 0.25 Ac Re 0.40 Ac Re 0.65 Ac Re 1.0 Ac Re 1.5 Ac Re 2.5 Ac Re 4.0 Ac 0 0 0 0 0 0 0 1 1 1 1 1 2 2 10 Ac 15 Re 10 11 14 15 21 22 2 3 10 11 14 15 3 10 11 14 15 21 22 21 22 2 3 10 11 14 15 3 10 11 14 15 21 22 21 22 10 11 14 15 14 15 21 22 21 22 2 3 10 11 14 15 3 10 11 14 15 21 22 21 22 2 3 10 11 14 15 3 10 11 14 15 21 22 21 22 10 11 14 15 21 22 8 11 11 7 10 10 6 3 8 5 15 2 Re 7 14 4 11 6 10 3 5 3 4 Ac 2 Re 3 Ac 2 Re 100 Ac 65 Re 3 2 40 Ac 1 25 Re 2 Ac 1 1 Re 1 6.5 Ac 1 1 Re Ac = Acceptance number Re = Rejection number Use first sample plan below Use first sample plan above Technical Service Field Service Quality Assurance Producer’s risk : MIL-STD 105D • Example – – – • Batch size is 1000 packages AQL is 2.5 : 100 (2.5%) Single sampling, normal inspection Table Table 2A – – Code letter is ‘J’ AQL gives the acceptance number – – P1 = 2.5% (good) = 0.025 M = n x p = 80 x 0.025 = – P(x=0) = 20 x e-2 0! P(x=1) = 21 x e-2 1! P(x=2) = 22 x e-2 2! P(x=3) = 23 x e-0.5 3! P(x=4) = 24 x e-2 4! P(x=5) = 25 x e-0.5 5! – – – – – - Examine 80 samples, accept at 5, reject at - N = 80, c = = 0.135 = 13.5% = 0.27 = 27% = 0.27 = 27% = 0.18 = 18% = 0.09 = 9% - Prob(accept p1) = – (0.135+0.27+0.27+0.18+0.09+0.04) = – 0.985 = 0.015 Producer’s risk = 1.5% = 0.04 = 4% Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D • Example – – – • Table – – • • Batch size is 100.000 packages AQL is : 1000 (0.1%) Single sampling, normal inspection Code letter is ‘N’ AQL gives the acceptance number P1 = 0.1% (good) = 0.001 M = n x p = 500 x 0.001 = 0.5 – P(x=0) = 0.50 x e-0.5 = 0.60 = 60% 0! – P(x=1) = 0.51 x e-0.5 = 0.30 = 30% 1! Table 2A - Examine 500 samples, accept at 1, reject at - N = 500, c = 1 - Prob(accept p1) = – (0.60+0.30) = – 0.90 = 0.10 Producer’s risk = 10% Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D • Example – – – • Table – – • • Batch size is 200.000 packages AQL is : 1000 (0.1%) Single sampling, normal inspection Code letter is ‘P’ AQL gives the acceptance number P1 = 0.1% (good) = 0.001 M = n x p = 800 x 0.001 = 0.8 – P(x=0) = 0.80 x e-0.8 = 0.45 = 45% 0! – P(x=1) = 0.81 x e-0.8 = 0.36 = 36% 1! – P(x=2) = 0.82 x e-0.8 = 0.14 = 14% 2! Table 2A - Examine 800 samples, accept at 2, reject at - N = 800, c = - Prob(accept p1) = – (0.45+0.36+0.14) = – 0.95 = 0.05 Producer’s risk = 5% Technical Service Field Service Quality Assurance Producer’s risk: MIL-STD 105D • Example – – – • Table – – • • Batch size is 100.000 packages AQL is : 10.000 (0.01%) Single sampling, normal inspection Code letter is ‘N’ AQL gives the acceptance number P1 = 0.01% (good) = 0.0001 M = n x p = 1250 x 0.0001 = 0.125 – P(x=0) = 0.1250 x e-0.125 = 0.88 = 88% 0! Table 2A - Examine 1250 samples, accept at 0, reject at - N = 1250, c = - Prob(accept p1) = – (0.88) = 0.12 Producer’s risk = 12% Technical Service Field Service Quality Assurance Sampling Calculation of Confidence Interval Example: –Take 7200 samples, are found to be defective –What can be said about p = defect rate? Estimate: –p = / 7200 * 100% = 0.056% –How uncertain is p? Now: confidence intervals We use a table that gives two sided confidence intervals for the mean = m = n*p, with confidence interval 0.90 and 0.95 Technical Service Field Service Quality Assurance Sampling Calculation of Confidence Interval Formula: m(x) n x 100 = p(x) % n = sample size = 2,400 c = defects found = From the table draw the limits at 90% confidence m(l) = lower confidence limit = m(u) = upper confidence limit = 2.436 Use the formula to calculate the defect levels p(l) = lower defect level = 0.00% p(u) = upper defect level = 0.10% Statement: The defect level in the lot is with 90% confidence between 0.00 and 0.10% Technical Service Field Service Quality Assurance Sampling Calculation of Confidence Interval Formula: m(x) n x 100 = p(x) % n = sample size = 2,436 c = defects found = From the table draw the limits at 90% confidence m(l) = lower confidence limit = 0.108 m(u) = upper confidence limit = 4.532 Use the formula to calculate the defect levels p(l) = lower defect level = 0.00% p(u) = upper defect level = 0.19% Statement: The defect level in the lot is with 90% confidence between 0.00 and 0.19% Technical Service Field Service Quality Assurance Sampling Calculation of Confidence Interval Formula: m(x) n x 100 = p(x) % n = sample size = 200 c = defects found = From the table draw the limits at 95% confidence m(l) = lower confidence limit = 0.051 m(u) = upper confidence limit = 5.323 Use the formula to calculate the defect levels p(l) = lower defect level = 0.03% p(u) = upper defect level = 2.66% Statement: The defect level in the lot is with 90% confidence between 0.03 and 2.66% Technical Service Field Service Quality Assurance Tools : Calculating the confidence interval CALCULATION CONFIDENCE INTERVAL Samples Defects 2.400 Calculated result 95 % Probability 90 % Probability m-lower value : 0,00 0,00 m-higher value : 2,44 3,29 Defect Rate Low: 0,000% 0,000% Defect Rate High: 0,102% 0,137% The defect rate is with 90% probability 0,00 % and 0,10 % between The defect rate is with 95% probability 0,00 % and 0,14 % between Technical Service Field Service Quality Assurance Tools : Table confidence interval AQL % 90% Confidence level 95% Confidence level 90% Confidence level 95% Confidence level 90% Confidence level Accept at Defects Accep at Defects Accept at Defects Accep at Defects Accept at Defects 3 AQL % 3 AQL % 95% Confidence level Accep at Defects 1,0 :10000 0,010 23100 39000 53300 66800 30000 47500 63000 77500 4,6 :10000 0,046 5022 8478 11587 14522 6522 10326 13696 16848 8,1 :10000 0,081 2852 4815 6580 8247 3704 5864 7778 1,1 :10000 0,011 21000 35455 48455 60727 27273 43182 57273 70455 4,7 :10000 0,047 4915 8298 11340 14213 6383 10106 13404 16489 8,2 :10000 0,082 2817 4756 6500 8146 3659 5793 7683 1,2 :10000 0,012 19250 32500 44417 55667 25000 39583 52500 64583 4,8 :10000 0,048 4813 8125 11104 13917 6250 9896 13125 16146 8,3 :10000 0,083 2783 4699 6422 8048 3614 5723 7590 1,3 :10000 0,013 17769 30000 41000 51385 23077 36538 48462 59615 4,9 :10000 0,049 4714 7959 10878 13633 6122 9694 12857 15816 8,4 :10000 0,084 2750 4643 6345 7952 3571 5655 7500 1,4 :10000 0,014 16500 27857 38071 47714 21429 33929 45000 55357 5,0 :10000 0,050 4620 7800 10660 13360 6000 9500 12600 15500 8,5 :10000 0,085 2718 4588 6271 7859 3529 5588 7412 1,5 :10000 0,015 15400 26000 35533 44533 20000 31667 42000 51667 5,1 :10000 0,051 4529 7647 10451 13098 5882 9314 12353 15196 8,6 :10000 0,086 2686 4535 6198 7767 3488 5523 7326 1,6 :10000 0,016 14438 24375 33313 41750 18750 29688 39375 48438 5,2 :10000 0,052 4442 7500 10250 12846 5769 9135 12115 14904 8,7 :10000 0,087 2655 4483 6126 7678 3448 5460 7241 1,7 :10000 0,017 13588 22941 31353 39294 17647 27941 37059 45588 5,3 :10000 0,053 4358 7358 10057 12604 5660 8962 11887 14623 8,8 :10000 0,088 2625 4432 6057 7591 3409 5398 7159 1,8 :10000 0,018 12833 21667 29611 37111 16667 26389 35000 43056 5,4 :10000 0,054 4278 7222 9870 12370 5556 8796 11667 14352 8,9 :10000 0,089 2596 4382 5989 7506 3371 5337 7079 1,9 :10000 0,019 12158 20526 28053 35158 15789 25000 33158 40789 5,5 :10000 0,055 4200 7091 9691 12145 5455 8636 11455 14091 9,0 :10000 0,090 2567 4333 5922 7422 3333 5278 7000 2,0 :10000 0,020 11550 19500 26650 33400 15000 23750 31500 38750 5,6 :10000 0,056 4125 6964 9518 11929 5357 8482 11250 13839 9,1 :10000 0,091 2538 4286 5857 7341 3297 5220 6923 2,1 :10000 0,021 11000 18571 25381 31810 14286 22619 30000 36905 5,7 :10000 0,057 4053 6842 9351 11719 5263 8333 11053 13596 9,2 :10000 0,092 2511 4239 5793 7261 3261 5163 6848 2,2 :10000 0,022 10500 17727 24227 30364 13636 21591 28636 35227 5,8 :10000 0,058 3983 6724 9190 11517 5172 8190 10862 13362 9,3 :10000 0,093 2484 4194 5731 7183 3226 5108 6774 2,3 :10000 0,023 10043 16957 23174 29043 13043 20652 27391 33696 5,9 :10000 0,059 3915 6610 9034 11322 5085 8051 10678 13136 9,4 :10000 0,094 2457 4149 5670 7106 3191 5053 6702 2,4 :10000 0,024 9625 16250 22208 27833 12500 19792 26250 32292 6,0 :10000 0,060 3850 6500 8883 11133 5000 7917 10500 12917 9,5 :10000 0,095 2432 4105 5611 7032 3158 5000 6632 2,5 :10000 0,025 9240 15600 21320 26720 12000 19000 25200 31000 6,1 :10000 0,061 3787 6393 8738 10951 4918 7787 10328 12705 9,6 :10000 0,096 2406 4063 5552 6958 3125 4948 6563 2,6 :10000 0,026 8885 15000 20500 25692 11538 18269 24231 29808 6,2 :10000 0,062 3726 6290 8597 10774 4839 7661 10161 12500 9,7 :10000 0,097 2381 4021 5495 6887 3093 4897 6495 2,7 :10000 0,027 8556 14444 19741 24741 11111 17593 23333 28704 6,3 :10000 0,063 3667 6190 8460 10603 4762 7540 10000 12302 9,8 :10000 0,098 2357 3980 5439 6816 3061 4847 6429 2,8 :10000 0,028 8250 13929 19036 23857 10714 16964 22500 27679 6,4 :10000 0,064 3609 6094 8328 10438 4688 7422 9844 12109 9,9 :10000 0,099 2333 3939 5384 6747 3030 4798 6364 2,9 :10000 0,029 7966 13448 18379 23034 10345 16379 21724 26724 6,5 :10000 0,065 3554 6000 8200 10277 4615 7308 9692 11923 3,0 :10000 0,030 7700 13000 17767 22267 10000 15833 21000 25833 6,6 :10000 0,066 3500 5909 8076 10121 4545 7197 9545 11742 3,1 :10000 0,031 7452 12581 17194 21548 9677 15323 20323 25000 6,7 :10000 0,067 3448 5821 7955 9970 4478 7090 9403 11567 3,2 :10000 0,032 7219 12188 16656 20875 9375 14844 19688 24219 6,8 :10000 0,068 3397 5735 7838 9824 4412 6985 9265 11397 3,3 :10000 0,033 7000 11818 16152 20242 9091 14394 19091 23485 6,9 :10000 0,069 3348 5652 7725 9681 4348 6884 9130 11232 3,4 :10000 0,034 6794 11471 15676 19647 8824 13971 18529 22794 7,0 :10000 0,070 3300 5571 7614 9543 4286 6786 9000 11071 3,5 :10000 0,035 6600 11143 15229 19086 8571 13571 18000 22143 7,1 :10000 0,071 3254 5493 7507 9408 4225 6690 8873 10915 3,6 :10000 0,036 6417 10833 14806 18556 8333 13194 17500 21528 7,2 :10000 0,072 3208 5417 7403 9278 4167 6597 8750 10764 3,7 :10000 0,037 6243 10541 14405 18054 8108 12838 17027 20946 7,3 :10000 0,073 3164 5342 7301 9151 4110 6507 8630 10616 3,8 :10000 0,038 6079 10263 14026 17579 7895 12500 16579 20395 7,4 :10000 0,074 3122 5270 7203 9027 4054 6419 8514 10473 3,9 :10000 0,039 5923 10000 13667 17128 7692 12179 16154 19872 7,5 :10000 0,075 3080 5200 7107 8907 4000 6333 8400 10333 4,0 :10000 0,040 5775 9750 13325 16700 7500 11875 15750 19375 7,6 :10000 0,076 3039 5132 7013 8789 3947 6250 8289 10197 4,1 :10000 0,041 5634 9512 13000 16293 7317 11585 15366 18902 7,7 :10000 0,077 3000 5065 6922 8675 3896 6169 8182 10065 4,2 :10000 0,042 5500 9286 12690 15905 7143 11310 15000 18452 7,8 :10000 0,078 2962 5000 6833 8564 3846 6090 8077 9936 4,3 :10000 0,043 5372 9070 12395 15535 6977 11047 14651 18023 7,9 :10000 0,079 2924 4937 6747 8456 3797 6013 7975 9810 4,4 :10000 0,044 5250 8864 12114 15182 6818 10795 14318 17614 8,0 :10000 0,080 2888 4875 6663 8350 3750 5938 7875 9688 4,5 :10000 0,045 5133 8667 11844 14844 6667 10556 14000 17222 Technical Service Field Service Quality Assurance [...]... variation m =n x p = 30 x 1/6 = 5 Notation: n = number of times p = probability each time to get a six m = mean or average number of sixes among n: m=nxp Technical Service Field Service Quality Assurance Statistics • • • • Examine 2400 packages from a production Assume defect rate is 0.1% (1 : 1000) How many defects will be found? Repeat M = n x p = 2400 x 0.1% = 2400 * 0.001 = 2.4 On a average we will