14 Variable Sampling Plan

Sampling Techniques by

Variables

Topic Outcome: At the end of this topic, students will be able to:

Discuss the advantages and disadvantages of variable sampling.

Apply Process Fraction Nonconforming Type of variable sampling.

Use ANSI/ASQ Z1.9 – 1993.Describe the Shainin Lot Plot Plan.

Outline: An Introduction. Types of Sampling Plan.

- Shainin Lot Plot Plan.

- ANSI/ASQ Z1.9

Variability UnknownStandard Deviation MethodRange Method

Variability known

An Introduction When a quality characteristic is measurable on a

continuous scale, and is know to have a distribution of a specific type (e.g. normal distribution); it is possible to use a sampling based on sample measurements such as:

Mean and

Standard Deviation.

Variables Sampling Plans.

Population

Sample

LSL USL

LSL USL

Compare

Compare to required Quality LevelACCEPT REJECT

Percent nonconforming below and above specifications.

LSL USL

pu

+-pL

Advantages of Variable Sampling Plan:

Same protection with smaller sample size.

Feedback of data on Process.

Extent of conformity given weightage.

Errors in measurement more likely detected.

Disadvantages of Variable Sampling Plan:

Applicable to only one quality characteristic at a time.

Higher inspection cost – Measurement.

Higher clerical cost – Calculation.

Possibility of no nonconformity in a rejected lot.

Dependence on assumption of distribution shape.

Types of Variable Sampling PlanVariable Sampling Plan

Percent nonconforming Process Parameter

To determine the proportion of product that

is outside specification

To control the average and standard deviation of

the distribution of the product to specified level

E.g.:Shainin lot plotANSI/ASQ Z1.9-1993

E.g.:Shainin lot plotANSI/ASQ Z1.9-1993

E.g.:Acceptance control chartSequential sampling for variablesHypothesis testing

Shainin Lot Plot Plan Developed by Dorian Shainin (Hamilton Standard Division

of United Aircraft Corporation, 1950).

The plan uses a plotted frequency distribution (histogram) to evaluate a sample for decisions (accept/reject a lot).

It is a practical plan for in-house inspection & receiving inspection.

Advantages:

Applicable to both normal and non-normal frequency distributions.

Simple to use.

The method for obtaining the lot plots is as follows:

A random sample of 10 subgroups (g) of 5 each for a total of 50 items is obtained from the lot.

The average, X-bar and range, R, are calculated for each subgroup.

A histogram is constructed (number of cell: 7-16).

Calculate the average of averages, X-double bar, and average of ranges, R-bar.

Calculate the upper and lower lot limits (ULL & LLL)

Lot Plotting Method

9.957.973

5.99326.2

)37.1)(3(7.97

3

326.2)37.1)(3(

2

2

d

RXLLL

d

RXULL

37.110

7.13

7.9710

8.976

g

RR

g

XX

Chart for Ranges:Factor for Central Line d2

3 control limits

Fig 10-14: Lot plot histogram

Decision is based on a comparison of the lot plot with 11 different types of lot plots.

Lot Plot Evaluation

Type Distribution Comments

1

normal

-within spec.limits

- accepted without calculate lot limits.

2-lot limits within spec. limits.

-accept.

3 -lot limits outside spec. limits.

-percentage of product beyond spec is obtained.

-Review board determines the final disposition of the stock.4

5nonnormal

skrewed

6 Lot was screened & sorted

7 Bimodal condition

8nonnormal Lot was screened & sorted

9

10 Bimodal condition

11 Stray values

Once learned, the lot plot procedure is relatively simple and has resulted in improved quality and lower inspection cost.

Unacceptable lots are returned to the producer, and this action will cause a subsequent improvement in quality.

Inspectors can accept lots; however, disposition of unsatisfactory lots is left to a material-review board.

Many users of the lot plot method have modified the Shainin method for their own situation.

The major criticism of the plan is that the shape of the plot does not always give an accurate indication of the true distribution. Shainin states that the plot is close enough to have no practical effect on the final decision, or if there are any errors, they are in a safe direction.

Summary

ANSI/ASQ Z1.9 MIL-STD-414 ASQ closely match ANSI/ASQ Z1.4 and

ISO/DIS 3951. Indexed by AQL (0.10-10.0%). Assumption: normally distributed random variable. Composition of the standard: 9 different procedures to

evaluate a lot.Variability Unknown

(Range Method)Variability Unknown

(Standard Deviation Method)Variability known

Single Specification Double Specification

Form 1(k method)

Form 2(m method)

Form 2(m method)

LSL orUSL

•LSL andUSL•1AQL or2AQL

Quality Index k Quality Index p0 M

k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity

It is divided into 4 sections: Section A: General description of sampling plan, sample

size, code determination, OC curves.

Section B: Unknown Variability – Standard Deviation

Method.

Section C: Unknown Variability – Range Method.

Section D: Know Variability.

It could apply either with a single specification limit or two specification limits.

Specification limits: It is the requirement that a quality characteristic should meet.

This requirement may be expressed as an upper spec. limit

(USL); or a lower spec. limit (LSL), called herein a single

specification limit; or both upper and lower specification limits,

called herein a double specification limit.

For single specification Form 1 and Form 2 give identical end results.

Form 1: Decision on Acceptability is to compare calculated Quality Index to Acceptability Constant, k.

Form 2: Decision on Acceptability is to convert calculated Quality index to Percent Nonconformity in lot p, and then compared to Maximum Allowable Percent Nonconformity, M.

IMPORTANT (Normality Assumption): This standard assumes the underlying distribution of

individual measurements to be normal in shape. Failure of this assumption affect OC curves and probabilities based on these curves. affect the estimate of percent nonconforming calculated from mean and standard deviation. The assumption should be verified prior to use of the standard. A variety of statistical tests and graphical techniques are available.

Relative samples are designated by code letters. The sample sizes code letter depends on the inspection

level and lot size. There are 5 inspection levels: General Levels I, II, and III Special Levels: S3 and S4 The sample size code letter applicable to specified

inspection level and for lots of given size shall be obtained from Table A-2

Determination of Sample Size

Note: Unless otherwise specified, inspection Level II shall

be used. Inspection Level I may be specified when less

discrimination is needed. Inspection Level III may be specified for greater

discrimination. Level S3 and S4 may be used when relatively small

sample sizes are necessary and large sampling risks can be tolerated.

Switching Rules for ANSI/ASQ Z1.9

• Preceding 10 lots Accepted with total nonconforming less than limit number,

• Steady production, and• Approval from

responsible authority.

• Lot not accepted, or• Lot accepted but

nonconformities found lie between Ac and Re of the plan, or

• Irregular production, or• Other conditions

warrant.

• 2 out of 5 consecutive lots not accepted

• 5 consecutive lots accepted

• 10 consecutive lots Remain on Tightened

• Discontinue inspection

NORMAL TIGHTENEDREDUCED

START

There are two parts in this method Single Specification Method Form 1 and 2. Double Specification Method Form 2. Single Spec. Method – Form 1

A) Determine sample size code (Table A-2) by using lot size and inspection level.

B) Select plan from Master Table B-1 and B-2. Obtain sample size, n, and the acceptability constant, k.

C) Select at random the sample of n units from the lot; inspect and record the measurement of each unit.

D) Compute x-bar,sample standard deviation, s, upper/lower specification limit.

E) If the upper/lower spec limit k lot accepted.

Variability Unknown – Standard Deviation Method

Single Spec. Method – Form 2 It converts the computed results into percent

nonconforming (p0) into lot through Table B-5 and compares it with Maximum Allowable Percent Nonconforming, M.


B) Select plan from Master Table B-3 and B-4. Obtain sample size, n, and M.


D) Compute x-bar, sample standard deviation, s, Quality Index, QU or QL.

E) If lot percent nonconforming, pu or pL M lot accepted.


Table B-5

The minimum temperature of operation for a certain device is specified as

180oC. A lot of 40 items is submitted for inspection where inspection level

II, normal inspection,and AQL = 1.0% are the criteria. (Form 1 & 2)

From Table A-2, the code letter D, which gives a sample size n=5

(Table B-1). The temperatures for 5 samples are 197, 188, 184, 205, and

201oC.

Q & A

53.1,tan

70.180.8

180195,

80.815

125.190435.190

1

1955

201205184188197

22

ktConsityAcceptabils

LXQindexqualitylow

nnX

Xs

Cn

XX

L

o

QL k Accepted

Estimated a lot percent nonconforming below L: pL

From Table B-5, pL = 0.66%

Maximum allowable percent nonconforming, M

From Table B-3, M=3.33%

pLM accepted

The maximum temperature of operation for a certain device is specified

as 208oC. A lot of 40 items is submitted for inspection where inspection

level II, normal inspection,and AQL = 1.5% are the criteria. (Form 1 & 2)

From Table A-2, the code letter D, which gives a sample size n=5

(Table B-1).

Q & A

40.1,tan

48.180.8

195208,

80.815

125.190435.190

1

195

22

ktConsityAcceptabils

XUQindexqualitylow

nnX

Xs

Cn

XX

U

o

QU k Accepted

Estimated a lot percent nonconforming above U: pU

From Table B-5, pU = 4.22%

Maximum allowable percent nonconforming, M


pUM accepted

Double Spec. Method – Form 2 It can be either:

One AQL value for both Upper and Lower Specification Limit Combined, or

Different AQL values for Upper and Lower Specification Limit.


Double Spec. Method – Form 2 One AQL value for both Upper and Lower Specification

Limit Combined



B) Select plan from Master Table B-3 and B-4. Obtain sample size, n, and M



Determine the estimated lot percent nonconforming, p=pL+pU (Table B-5).

E) p M lot accepted.

The minimum and maximum temperatures of operation for a certain device is specified as 180oC and 209oC. A lot of 40 items is submitted for inspection where inspection level II, normal inspection,and AQL = 1.0% are the criteria.

From Table A-2, the code letter D, which gives a sample size n=5 (Table B-1). The temperatures for 5 samples are 197, 188, 184, 205, and 201oC.

Q & A

Estimated a lot percent nonconforming below L: pL

From Table B-5, pL = 0.66%

70.180.8

180195,

80.815

125.190435.190

1

1955

201205184188197

22

s

LXQindexqualitylow

nnX

Xs

Cn

XX

L

o

)60.1(59.180.8

195209say

s

XUQU

Estimated a lot percent nonconforming above U= pU

From Table B-5, pU = 2.03%

The lot meets acceptance criteria if pL+pU M


(0.66+2.03)% 3.32 accepted

Double Specification Method Form 2. Different AQL values for Upper and Lower Specification

Limit. A) Determine sample size code (Table A-2) by using

lot size and inspection level. B) Select plan from Master Table B-3 and B-4. Obtain

sample size, n, Mu (AQL for upper spec limit) and ML(AQL for lower spec limit).




E) Determine the estimated lot percent nonconforming, pL and pU (Table B-5), and p (= pL + pU )

F) Accept the lot if the following 3 conditions are met:pU MU

pL ML

p (MU or ML, which ever is larger)

The minimum and maximum temperatures of operation for

a certain device is specified as 180 and 209oC. A lot of 40

items is submitted for inspection where inspection level II,

normal inspection,and AQL = 1.0% for upper and

AQL=2.5% for lower specification limits are the criteria.

From Table A-2, the code letter D, which gives a sample

size n=5 (Table B-1). The temperatures for 5 samples are

197, 188, 184, 205, and 201oC.

Q & A

Line Information needed Value Obtained Explanation

1 Sample size, n 5

2 Estimated Lot Standard Deviation, s 8.81

3 Sample mean, X-bar 195

4 Upper Specification Limit, U 209

5 Lower Specification Limit, L 180

6 Quality Index: QU 1.59

7 Quality Index: QL 1.70

8 Est. Lot Percent Ncf above U, pU 2.19% Table B-5

9 Est. Lot Percent Ncf below L, pL 0.66% Table B-5

10 Total Est PercentNcf, p 2.85%

11 Max. Allowable Percent Ncf above U, MU 3.32% Table B-3

12 Max. Allowable Percent Ncf below L, ML 9.80% Table B-3

13 Acceptability Criteria:

a) Compare pu with MU

b) Compare pL with ML

c) Compare p with ML

2.19%<3.32%

0.66%<9.80%

2.85%<9.80%

ANSI/ASQ Z1.9

Variability Unknown(Range Method)

Variability Unknown(Standard Deviation Method)

Variability known


Form 1 Form 2 Form 2

LSL orUSL




Basic approach is similar to those of Standard Deviation Method.

Master Tables used are Tables C-1 to C-5, instead of B-1 to B-5.

Quality indices for Form 1 are calculated as follows:

Variability Unknown – Range Method

R

LXQ

R

XUQ LU

;

As calculation of Quality indices for Form 2, there is an additional factor c included:

R

cLXQ

R

cXUQ LU

;

A lower specification limit for electrical resistance of a certain electrical

component is 620 ohm. A lot of 100 items is submitted for inspection.

Inspection Level II, normal inspection, with AQL=0.4% is to be used.

From Tables A-2 and C-1 it is seen that a sample of size 10 is

required.Suppose the values of the sample resistance in the order

reading from left to right are as follows:

643, 651, 619, 627, 658 (R1=658-619=39)

670, 673, 641, 638, 650 (R2=673-638=35)

and compliance with the acceptability criterion is to be determined. (using

Form 2)

Q & A


1 Sample size, n 10

2 Sum of measurement 6470

3 Sample mean, X-bar 647

4 Average Range, R-bar 37 (39+35)/2

5 Factor c 2.405 Table C-3

6 Spec limit (lower), L 620


8 Est. Lot Percent Ncf below L, pL 2.54% Table C-5

9 Max. Allowable Percent Ncf, M 1.14% Table C-3

10 Acceptability Criteria

a) Compare pL with M

2.45%>1.14%

The lot does not meet the acceptability criterion.

ANSI/ASQ Z1.9



Variability known



LSL orUSL




Basic approach is similar to those of Standard Deviation Method.

Master Tables used are Tables D-1 to D-5, instead of B-1 to B-5.

Quality indices for Form 1 are calculated as follows:

Variability known

LX

QXU

Q LU

;

As calculation of Quality indices for Form 2, there is an additional factor v included (Table D-3 and D-4):

vLXQ

vXUQ LU

;

The specified minimum yield point for certain steel castings is 58,000 psi.

A lot of 500 items is submitted for inspection. Inspection Level II, normal

inspection, with AQL=1.5% is to be used. The variability is known to be

300 psi. From Tables A-2 and D-1 it is seen that a sample size of 10 is

required. Suppose the yield points of the sample specimens are:

62,500; 60,500; 68,000; 59,000; 65,500

62,000; 61,000; 69,000; 58,000; 64,500

and compliance with the acceptability criterion is to be determined.

Q & A


1 Sample size, n 10

2 Known Varibility: 3,000

2 Sum of measurement 630,000

3 Sample mean, X-bar 63,000

4 Spec limit (lower), L 58,000


6 k 1.70 Table D-1

7 Acceptability Criteria

a) Compare QL with k

1.67%<1.70%

The lot does not meet the acceptability criterion.

Summary - ANSI/ASQ Z1.9



Variability known



LSL orUSL




QL or QU kp, pU, or pL M or k

Accept

14 Variable Sampling Plan

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