Statistics xác suất thống kê trong sản xuất sữa

Designing a QC System• The AQL must be determined by the Management • Definitions of Defectives must be made • Limits for Raw Materials must be set calculated • Determine Evaluation meth

Statistics

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!

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.

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.

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).

Defect rates detected at a certain sample size

Calculating sample size depending on AQL

Calculating sample size depending on AQL

100 x z AQL (%) Formula: n =

Example: AQL = 1:10,000 = 0.01%

Defect detected (C) = 1 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.

100 x 4.6 0.01

= 46,000 From the formula we get:

If we take 46.000 samples from a lot with a assumed defect rate of

1 : 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 < 1 : 10.000

Calculating sample size depending on AQL

Calculating sample size depending on AQL

100 x z AQL (%) Formula: n =

Example: AQL = 1:10,000 = 0.01%

Defect detected (C) = 1 at a Probability of 90%

In the chart follow the 90-line to the C=1 curve,

then go to the y-axis to get the z-value In this case 2.3.

100 x 2.3 0.01

= 23,000 From the formula we get:

If we take 23.000 samples from a lot with a assumed defect rate of

1 : 10.000, we have 90% chance to find one defect If we find < 1, we know with 90% probability

that the actual defect rate is < 1 : 10.000

Probability of detecting defects (%)

Sample size

Designing a QC System

• The AQL must be determined by the Management

• Definitions of Defectives must be made

• Limits for Raw Materials must be set (calculated)

• Determine Evaluation methods based on defect type and accuracy

• Determine Sampling plans based on confidence and

verification frequency required

The confidence level

• The probability is an expression of the degree of likelihood that the conclusions drawn from the results obtained by testing a certain

number (quantity) of material are correct

• The risk is an expression of the degree of the likelihood that the conclusions drawn from the result of testing a certain sample are incorrect

• % Probabilty + % Risk = 100%

Probabilities are usually expressed as the ratio: % Probability/100

Simulation of uncertainty

Throw a dice 30 times and register the number of sixes On a average

we get 5 sixes, but there is a considerable 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 = n x p

Poisson distribution: examples

Example 1:

• 4% of the Dutch population is more than 70 years old.

• What is the chance to find three persons in a group of 100 persons that are older than 70 years?

– M = n x p = 100 x 0.04 = 4

– P(x=3) = 4 3 x e -4 = 64/6 x 0.0183 = 0.1952 = 19.52%

3!

Example 2:

• Assumed defect rate in production is 1:1000 = 0.1% (p=0.001)

• QC takes 100 samples from each production

• What is the chance to find 1 defect?

– M = n x p = 100 x 0.001 = 0.1

Poisson distribution: examples

Example 3:

• Assumed defect rate in production is 1:1000 = 0.1% (p=0.001)

• QC takes 200 samples from each production

• What is the chance to find 1 defect?

– M = n x p = 200 x 0.001 = 0.2

– P(x=1) = 0.2 1 x e -0.2 = 0.2 x 0.82 = 0.164 = 16.4%

1!

Example 4:

• Accepted defect rate in production is 1:1000 = 0.1% (p=0.001)

• QC takes 2303 samples from a commissioning run

• What is the chance to find zero defects?

– M = n x p = 2303 x 0.001 = 2.303

– P(x=0) = 2.303 0 x e -2.303 = 1 x 0.0999 = 0.0999 = 9.99% (10%)

0!

Example 4 (cont’d) :

• Accepted defect rate in production is 1:1000 = 0.1% (p=0.001)

• QC takes 2303 samples from a commissioning run

• What is the chance to find zero defects?

– M = n x p = 2303 x 0.001 = 2.303

– P(x=0) = 2.303 0 x e -2.303 = 0.1= 10%

0!

• The chance to find 1 or more defects is 100 - 10 = 90%

• If the outcome of the sterility test is that zero defects are found in 2303

samples, we know that with 90% probability the defect rate will be less than 1:1000

Prob(accept) = 10% The production will be rejected with 90% probability.

• Examine 2303 packs, accept if no defect is found, reject otherwise If the true defect rate is 0.05% (5:10.000), what is the probability to accept the lot?

– P(x=0) = 0.23 0 x e -2.3 = 0.794 = 79.4% (2/10 of these lots will be rejected!)

Acceptance sampling

If the true defect rate is 0.1% (1:1000), what is the probability to accept the lot?

– P(x=2) = 5.322 2 x e -5.322 = 0.1 = 10%

2!

Prob(accept) = 0.1 = 10% The production will be rejected with 90% probability.

what is the probability to accept the lot?

– P(x=2) = 2.661 2 x e -2.661 = 0.503 = 50.3% (1/2 of these lots will be rejected!)

2!

what is the probability to accept the lot?

Better discrimination between good and bad production

Consumer’s and Producer’s risk

• Define

– a limit where the production is considered good

• p1 = AQL = Acceptable Quality level – a limit where it is considered bad

• p2 = LTPD = Lot Tolerance Percent Defective

• p1 < p2 – C is acceptance number, i.e if max c defectives are found, we accept

– Producer’s risk = Prob(reject if good i.e p1) = 1 - AcceptanceProbability(p1) – Consumer’s risk =Prob(accept if bad i.e p2) = AcceptanceProbability(p2)

Consumer’s and Producer’s risk

• Examine 2303 packs, accept if 0 defects are found, reject otherwise

Consumer’s and Producer’s risk

• Examine 3890 packs, accept if max 1 defect is found, reject otherwise

Consumer’s and Producer’s risk

• Examine 5322 packs, accept if max 2 defect is found, reject otherwise

Producer’s risk: MIL-STD 105D

– Focus on AQL (p1 = specification of ‘good batch”)

– Used for inspection of received goods

– AQL from 0.01% to 100%

– Seven inspection levels:

– Simple, double and multiple sampling

• Routine to find a plan

– Define lot size + inspection level + type of plan

– AQL gives the acceptance number

Producer’s risk: MIL-STD 105D

Table 1 – Sample size code letters

Lot or batch size Special Inspection Levels General Inspection Levels

Producer’s risk: MIL-STD 105D

Table 2A – Single sampling plans for normal inspection

Sample Acceptable Quality Levels

size Sample 0.010 0.015 0.025 0.040 0.065 0.10 0.15 0.25 0.40 0.65 1.0 1.5 2.5 4.0 6.5 10 15 25 40 65 100code size Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re Ac Re

Producer’s risk: MIL-STD 105D

Producer’s risk: MIL-STD 105D

Producer’s risk: MIL-STD 105D

Calculation of Confidence Interval

Example:

–Take 7200 samples, 4 are found to be defective

–What can be said about p = defect rate?

Estimate:

–p = 4 / 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.

Sampling

Sampling Calculation of Confidence Interval

From the table draw the limits at 90% confidence

Use the formula to calculate the defect levels

Sampling Calculation of Confidence Interval

From the table draw the limits at 90% confidence

Use the formula to calculate the defect levels

Statement: The defect level in the lot is with 90% confidence between 0.00 and 0.19%

Sampling Calculation of Confidence Interval

From the table draw the limits at 95% confidence

Use the formula to calculate the defect levels

Tools : Calculating the confidence interval

CALCULATION CONFIDENCE INTERVAL

Tools : Table confidence interval

90% Confidence level 95% Confidence level 90% Confidence level 95% Confidence level 90% Confidence level 95% Confidence levelAccept at Defects Accep at Defects Accept at Defects Accep at Defects Accept at Defects Accep at DefectsAQL % 0 1 2 3 0 1 2 3 AQL % 0 1 2 3 0 1 2 3 AQL % 0 1 2 3 0 1 21,0 :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 77781,1 :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 76831,2 :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 75901,3 :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 75001,4 :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 74121,5 :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 73261,6 :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 72411,7 :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 71591,8 :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 70791,9 :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 70002,0 :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 69232,1 :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 68482,2 :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 67742,3 :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 67022,4 :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 66322,5 :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 65632,6 :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 64952,7 :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 64292,8 :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 63642,9 :10000 0,029 7966 13448 18379 23034 10345 16379 21724 26724 6,5 :10000 0,065 3554 6000 8200 10277 4615 7308 9692 11923