cpk&ppm

标准正态分布下Cpk,yield,PPM的关系

Statistics 2009-11-25 18:17:19 阅读600 评论0字号：大中小订阅

在标准正态分布情况下, mean=0, sigma=1, 概率函数为:

f(x)=(1/sqrt(2*pi))*e^(-x^2/2))

累计概率密度函数:

P(x)=f(x)在-z到z的积分

计算结果如下:

z取相应的sigma值

p即P(z)

yield=p*100%

Cpk=min((USL-mean)/(3*s),(mean-LSL)/(3*s)) or = abs(SL-mean)/(3*s) = z/(3*s)=z/3 (其中z为相应的sigma值)

PPM=(1-p)*10^6

Part I: 短期sigma, 不考虑mean值的偏差.

sigma p yield Cpk PPM

1 0.682689492137086 68.2700000% 0.33 317310

2 0.954499736103642 95.4500000% 0.67 45501

3 0.997300203936740 99.7300000% 1.0 2700

4 0.999936657516334 99.9900000% 1.33 63

5 0.99999942669685

6 99.9900000% 1.6

7 0.6

6 0.999999998026825 99.9999998% 2.0 0.002

PartII 长期sigma, 考虑到1.5倍sigma的均值偏差(即mean=1.5或-1.5)

考虑到对称性, 以f(x)=(1/sqrt(2*pi))*e^(-(x-1.5)^2/2))来计算p值

p=f(x)在-6到+6的积分, 计算结果如下表(由于Cpk只是针对于短期的sigma, 在此不计算Cpk值, 只计算对应的yield和ppm)

sigma p yield Cpk PPM

1 0.302327873400211 30.2300000% -- 697672

2 0.691229832194978 69.1200000% -- 308770

3 0.933189401058017 93.3100000% -- 66811

4 0.993790315684661 99.3700000% -- 6210

5 0.999767370880804 99.9700000% -- 233

6 0.999996602326843 99.9900000% -- 3.4

PartIII CL/LSL/USL与mean, z的关系

***以下共两个表格, 分短期sigma和长期sigma

Part I : 短期sigma情况下, 即不考虑均值的偏差, mean=0, sigma=1时, Cpk,yield,ppm 的关系如下:

附表一: short term sigma, sigma=1~6

z p(即phai(z)-phai(-z))Yield Cpk PPM 10.68268949213708668.269%0.333317310 1.10.72866787810723572.867%0.367271332 1.20.76986065955658476.986%0.400230139 1.30.80639903082877980.640%0.433193600 1.40.83848668153245883.849%0.467161513 1.50.86638559746228486.639%0.500133614 1.60.89040141660088489.040%0.533109598 1.70.91086907448291491.087%0.56789130 1.80.92813936177414892.814%0.60071860 1.90.94256688036799694.257%0.63357433 20.95449973610364195.450%0.66745500 2.10.96427115887436796.427%0.70035728 2.20.97219310497300397.219%0.73327806 2.30.97855177995664897.855%0.76721448 2.40.98360492815080898.360%0.80016395 2.50.98758066934844898.758%0.83312419 2.60.99067762395256399.068%0.8679322 2.70.99306605239391999.307%0.9006933 2.80.99488973933914499.489%0.9335110 2.90.99626837339923399.627%0.9673731 30.99730020393674099.730% 1.0002699 3.10.99806479357356499.806% 1.0331935 3.20.99862572412416899.863% 1.0671374 3.30.99903315171523299.903% 1.100966

3.40.99932614146864799.933% 1.133673 3.50.99953474184193299.953% 1.167465 3.60.99968178281968499.968% 1.200318 3.70.99978440053304499.978% 1.233215 3.80.99985530391215199.986% 1.267144 3.90.99990380731197199.990% 1.30096.1 40.99993665751632699.994% 1.33363.3

4.10.99995868498616099.996% 1.36741.3 4.20.99997330850197899.997% 1.40026.6 4.30.99998292018904799.998% 1.43317 4.40.99998917491218499.999% 1.46710.8 4.50.99999320465373399.999% 1.500 6.79 4.60.999995775090593100.000% 1.533 4.22 4.70.999997398384935100.000% 1.567 2.6 4.80.999998413343529100.000% 1.600 1.58 4.90.999999041633524100.000% 1.6330.958 50.999999426696856100.000% 1.6670.573

5.10.999999660346519100.000% 1.7000.339 5.20.999999800711474100.000% 1.7330.199 5.30.999999884197319100.000% 1.7670.115 5.40.999999933359103100.000% 1.8000.066 5.50.999999962020875100.000% 1.8330.037 5.60.999999978564819100.000% 1.8670.021 5.70.999999988019257100.000% 1.9000.011 5.80.999999993368508100.000% 1.9330.0066

5.90.999999996364984100.000% 1.9670.0036

60.999999998026825100.000% 2.0000.0019

Part II: 长期sigma情况下, 即考虑均值的偏差, mean=1.5*sigma=1.5, sigma=1时, Cpk,yield,ppm的关系如下:

附表二: long term sigma, z=1~6

z p(即phai(z)-phai(-z))Yield Cpk PPM

10.30232787340021130.233%697672

1.10.33991707036595733.992%660082

1.20.37862160400800737.862%621378

1.30.41818516023046941.819%581814

1.40.45830634942258745.831%541693

1.50.49865010196837049.865%501349

1.60.53886023406381153.886%461139

1.70.57857257150118757.857%421427

1.80.61742799804656861.743%382572

1.90.65508481234464865.508%344915

20.69122983219497969.123%308770

2.10.72558777365976972.559%274412

2.20.75792854804344975.793%242071

2.30.78807225337267978.807%211927

2.40.81589177830922681.589%184108

2.50.84131307482670684.131%158686

2.60.86431328154669886.431%135686

2.70.88491698402928188.492%115083

2.80.90319097550891390.319%96809

2.90.91923792822232191.924%80762 30.9331894010580089

3.319%66810 3.10.9451985958457399

4.520%54801 3.20.9554332364339249

5.543%44566 3.30.9640688875588399

6.407%35931 3.40.9712829610007609

7.128%28717 3.50.97724958140024997.725%22750 3.60.9821354096104439

8.214%17864 3.70.98609645284223898.610%13903 3.80.98927583207698498.928%10724 3.90.9918024307549559

9.180%8198 40.99379031568466299.379%6210 4.10.99533880125869199.534%4661 4.20.99653302020658899.653%3467 4.30.99744486635382699.744%2555 4.40.99813418488210999.813%1866 4.50.99865010098178299.865%1350 4.60.99903239625643999.903%968 4.70.99931286177976899.931%687 4.80.99951657570879399.952%483 4.90.99966307065663499.966%337 50.99976737088080699.977%233 5.10.99984089138928399.984%159 5.20.99989220025610199.989%108 5.30.99992765195084599.993%72

5.40.99995190365338299.995%48

5.50.99996832875688399.997%32

5.60.99997934249245699.998%21

5.70.99998665425068899.999%13

5.80.99999146009438099.999%8.5

5.90.99999458745602499.999% 5.4

60.999996602326834100.000% 3.4

CL对应于mean, USL对应于z, LSL对应于-z; 一般而言, 由于LSL和USL为固定值, 要提高Cpk, 只能通过改进工艺, 减小制程波动, 以减小sigma, 使

sigma<(USL-LSL)/2/3/Cpk_expected

Cpk_expected为期望的Cpk值, 一般取1.67或2.0

ps: 1. 以上表格中相应sigma的P(x)值为MATLAB计算的结果, 也可以通过正态分布表查表计算(正态分布表精度不是很高,另外一般只有z=0.01~3.99的正态分布值)或excel计算

2. 更详细的数据见本空间的其他blog, 如需更多的数据, 可以发邮件给我.