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.
ANSI/A I/ASQ SQ Use ANS Describe
Z1.9 ± 19 Z1.9 1993. 93.
the Shainin Lot Plot Plan.
Outline:
An Introduction. Types of Sampling Plan. - Shai Shainin nin Lot Lot Plot Plan Plan.. - ANS ANSI/A I/ASQ SQ Z1. Z1.9 9
Variability Unknown Standard Range
Variability known
Deviation Method
Method
An
Introduction
W hen
a quality characteristic is measurable on a continuous scale, 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
Compare
LSL
ACCEPT
USL
Compare to required Quality Level
REJECT
-g
pu
pL
+g
LSL
USL
Percent nonconforming below and above specifications.
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 High er inspection inspection cost ± Meas Measurem urement. ent.
Higher High er clerical clerical cost cost ± Calcu Calculati lation. on.
Possibility of no nonconformity in a rejected lot.
Dependence on assumption of distribution shape.
Types of Variable Sampling Plan Variable Sampling Plan
Percent nonconform nonconforming ing To determine the proportion of product that is outside specification s pecification
E.g.: Shainin lot plot ANSI/ASQ Z1.9-1993
Process Parameter To control the average av erage and standard deviation of the distribution of the product to specified level E.g.: Acceptance control chart Sequential sampling for variable Hypothesis testing
Shainin Lot Plot Plan
Developed by Dorian Shainin (Hamilton Standard Division of United Aircraft Corporation, 1950).
The plan uses a plotted f requency requency distribution (histogram) to evaluate a sample for decisions (accept/reject a lot).
It is a practical plan for in-house for in-house inspection inspection.. inspection
&
receiving
Advantages:
Applicable to both normal and non-normal frequency distributions. Simple to use.
Lot Plotting Method
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)
Q&A
X ! 3W control limits
R !
ULL ! X LLL ! X
§ X
!
g § R
g 3 R d 2 3 R d 2
!
976.8 10
13.7 10
! 97.7
! 97.7
! 1.37
(3)(1.37) 2.326
! 99.5
(3)(1.37 ) ! 97.7 ! 95.9 2.326
Chart f or or
Ranges:: Ranges Factor for Central Line d2
Fig 10-14: Lot plot histogram
Lot Plot Evaluation
Decision is based on a comparison of the lot plot with different types of lot plots. Type
Distribution
- accepted without calculate lot limits. -lot limits within spec. limits. normal
-percentage of product beyond spec is obtained.
4
6 7 8
-accept. -lot limits outside spec. limits.
3
5
Comments -within spec.limits
1 2
-Review board determines the final disposition of the stock. nonnormal
skrewed Lot was screened & sorted
Bimodal condition nonnormal
Lot was screened
&
9 10 11
11
Bimodal condition Stray values
sorted
Summary 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 improvement in quality. quality. Inspectors can accept lots; however, disposition of unsatisfactory lots is left to a material-revi material-review ew board board..
Many users of the lot plot method have modif ied ied the Shainin method for their own situation.
The major criticism major criticism of the plan is that the shape of the plot does not always always give an accurate indication of the true distribution.. Shainin states that the plot is close distribution c lose enough to have no practical effect on the final decision, or if there are any errors, they are in a safe direction.
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. normally Composition of the standard: 9 different procedures to evaluate a lot.
Variability Unknown (Standard Deviation Method)
Single Specification
Form 1 (k method)
Quality Index m k
Variability known
Variability Unknown (Range Method)
LSL or USL
Form 2 (m method)
Double Specification
Form 2 (m method)
LSL and USL 1AQL or 2AQL
Quality Index p0 m M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
It is divided into 4 sections:
Sect Se ctio ion n A:
Gen ener eral al de desc scri ript ptio ion n of of sam sampl plin ing g pl plan an,, sa samp mple le
size, code determination, OC curves.
Sect Se ctio ion n B:
Unkn Un know own n Var aria iabi bili lity ty ± St Stan anda darrd Dev Devia iati tion on Me Meth thod od..
Section C:
Unknown Variabi ariability lity ± Range Metho Method. d.
Section Secti on D:
Know Variabi ariability lity..
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 specif ication limits, specif ication ication limit; limit; or both upper and lower specification called herein a double specif ication ication limit. limit.
For single specification Form 1 and Form 2 give identical end results.
Form 1: Decision on Acceptability Acceptability is to compare calculated Quality Index to Acceptability Constant, k. Form 2: Decision on Acceptability 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 ssumption): ):
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.
Determination of Sample Size
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 A -2
Note:
Unless otherwise specified, inspection Level II shall be used.
Inspection Level I may be specified when less discrimination discriminatio n 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 Z1. 9 Preceding 10 lots Accepted with total nonconforming less than limit number, Steady production, and Approval from responsible authority.
START
2 out of 5 consecutive lots not accepted
TIGHTENED
NORMAL
REDUCED
Lot not accepted, or Lot accepted but nonconformities found lie between Ac and Re of the plan, or Irregular production, or Other conditions warrant.
5 consecutive lots accepted 10 consecutive lots Remain on Tightened
Discontinue inspection
Varia ariabilit bility y
Unknown ± Stand Standard ard Deviation Deviation Method Method
There are two parts in this method
Single Specification Method Form 1 and 2.
Double Specification Method Form 2. Single Sing le Spec. Spec. Meth Method od ± 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 u k lot accepted.
Varia ariabilit bility y
Unknown ± Stand Standard ard Deviation Deviation Method Method
Single Sing le Spec. Spec. Meth Method od ± Form 2
It converts the computed results into percent nonconf orming orming (p0) into lot through Table B-5 and compares it with Maximum Allowable Percent Nonconforming, M. 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. sample size, n, and M.
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, Quality Index, QU or QL.
Obtain
E) If lot percent nonconforming, p u or pL e M lot accepted.
Table B-5 B-5
Q&A
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 Table A-2, the code letter D, which gives a sample sam ple size n=5 (Table (T able B-1). The temperatures for 5 samples are 197, 188, 184, 205, and 201oC. X !
§ X
n § X
s !
! 2
197 188 184 205 201 5 2 § X
n
n 1
!
low quality ind ind ex ex, Q L !
190.435 190.125 5 1 X L
!
s Acceptabil ity Cons tan t , k ! 1.53
o
! 195 C
195 180 8.80
! 8.80 ! 1.70
QL u k Accepted
Estimated a lot percent nonconforming nonconf orming below L: pL
Maximum allowable percent nonconforming, M
From Table Table B-5, B -5, pL = 0.66% From Table B-3, M=3.33%
pLeM accepted
Q&A
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 Table A-2, the code letter D, which gives a sample sam ple size n=5 (Table (T able B-1). X !
§ X
n § X
s !
o
! 195 C 2
2 § X
n
n 1
!
190.435 190.125
low quality ind ind ex ex, QU !
5 1 U X
s Acceptabil ity Cons tan t , k ! 1.40
!
! 8.80
208 195 8.80
! 1.48
QU u k Accepted
Estimated a lot percent nonconforming nonconf orming above above U: pU
Maximum allowable percent nonconforming, M
From Table Table B-5, B -5, pU = 4.22% From Table B-3, M=5.83%
pUeM accepted
Varia ariabilit bility y
Unknown ± Stand Standard ard Deviation Deviation Method Method
Double Doub le Spec. Spec. Meth Method od ± Form 2
It can be either: One
AQL value for both Upper and Lower Specification Limit Combined, Combined, or Different AQL values for Upper and Lower Specification Limit.
Varia ariabilit bility y
Unknown ± Stand Standard ard Deviation Deviation Method Method
Double Doub le Spec. Spec. Meth Method od ± Form 2
One AQL
value for both Upper and Lower Specification Limit Combined 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, and M 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, Quality Index, QU or QL.
Determine the estimated lot percent noncon f orming, orming, p=pL+pU (Table B-5). E) p e M lot accepted.
Q&A
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.
X !
§ X 197 188 184 205 201 ! ! 195o n 5 § X
s ! lo
2
2 § X
n
n 1
!
quality index, Q L !
Estimated From
190.435 190.125 5 1 X L s
!
195 180 8.80
! 8.80 ! 1.70
a lot percent nonconforming below L: p L
Table B-5, Table B -5, pL = 0.66%
QU !
U X s
Estimated From The
209 195 8.80
! 1.59 ( say 1.60)
a lot percent nonconforming nonconf orming above above U= pU
Table B-5, pU = 2.03% Table
lot meets acceptance criteria if pL+pU e M
From
!
Table B-3, M=3.33%
(0.66+2.03)% e 3.32 accepted
Varia ariabilit bility y
Unknown ± Stan Standard dard Deviatio Deviation n Method
Double Specification Method Form 2.
Diff erent erent 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, M u (AQL for upper spec limit) and ML(AQL for lower spec limit).
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, Quality Index, QU or QL.
E) Determine the estimated lot percent nonconforming, pL and pU (Table B-5), and p (= p L + pU )
F) Accept the lot if the following 3 conditions are met: pU
e MU
pL
e ML
p
e (MU or ML, which ever is larger)
Q&A
The minimum and maximum temperatures of operation for a certain device is specified as 180 and 209 oC. A lot of o f 40 items is submitted for inspection where inspection level II, normal inspection,and AQL = 1.0% f or or upper and upper and AQL=2.5% f or or lower specif ication ication 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.
L in e
Information needed
Value Obtained
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, p U
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
2.19%<3.32%
b)
Compare pL with ML
0.66%<9.80%
c)
Compare p with ML
2.85%<9.80%
Explanation
ANSI/ASQ Variability Unknown (Standard Deviation Method)
Quality Index m k
Variability known
Variability Unknown (Range Method)
Single Specification
Form 1
Z1.9
Form 2
LSL or USL
Double Specification
LSL and USL 1AQL or 2AQL
Form 2
Quality Index p0 m M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
Varia ariabilit bility y
Unknown ± Range Meth Unknown Method od
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:
QU
U X ! ;Q R
L
X L ! R
As
calculation of Quality indices for Form 2, there is an additional factor c factor c included: QU !
U X c R
; Q L
!
X L c R
Q&A
A lower specification limit for electrical resistance of a certain electrical component is 620 ohm. A lot of 100 items is submitted subm itted for inspection. Inspection Level II, normal inspection, with AQL=0.4% is to be used. From Tables A-2 and C-1 it i t 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)
L in e
Information needed
Value Obtained
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
7
Quality Index: QL
1.76
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
2.45%>1.14%
a)
Compare pL with M
The lot does not meet the acceptability criterion.
Explanation
ANSI/ASQ Variability Unknown (Standard Deviation Method)
Quality Index m k
Variability known
Variability Unknown (Range Method)
Single Specification
Form 1
Z1.9
Form 2
LSL or USL
Double Specification
LSL and USL 1AQL or 2AQL
Form 2
Quality Index p0 m M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
Variability
known
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:
QU
U X ! ;Q W
L
X L ! W
As
calculation of Quality indices for Form 2, there is an additional factor v factor v included (Table D-3 and D-4): QU !
U X v W
; Q L
!
X L v W
Q&A
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% AQL=1.5% is to be used. The variability is known to be 300 psi. From Tables 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.
L in e
Information needed
Value Obtained
1
Sample size, n
10
2
Known Varibility:
2
Sum of measurement
630,000
3
Sample mean, X-bar
63,000
4
Spec limit (lower), L
58,000
5
Quality Index: QL
1.67
6
k
1.70
7
Acceptability Criteria
1.67%<1.70%
a)
W
Explanation
3,000
Table D-1
Compare QL with k
The lot does not meet the acceptability criterion.
Summary Variability Unknown (Standard Deviation Method)
LSL or USL
Form 2
Quality Index m k
Z1.9 Variability known
Variability Unknown (Range Method)
Single Specification
Form 1
ANSI/ASQ
Double Specification
LSL and USL 1AQL or 2AQL
Form 2
Quality Index p0 m M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
QL or QU u k p, pU, or pL e M or k
Accept
