Steven Walfish
Statistical Outsourcing Services
Olney, MD
301-325-3129 steven@statisticaloutsourcingservices.com
1

 The Paradox
 Different types of sampling plans.
 Types of Risk
 Statistical Distribution
•
Normal
•
Binomial
•
Poisson
 When to Audit.
April 21, 2010 ASQ Section 511 2

 During an audit you increase the sample size if you have a finding…
 But, no findings might be because your sample size is too small to find errors.
April 21, 2010 ASQ Section 511 3

Common Sampling Strategies
 Simple random sample.
 Stratified sample.
 Systematic sample.
 Haphazard
 Probability proportional to size
April 21, 2010 ASQ Section 511 4

Decision
April 21, 2010
Accept
Reject
Reality
Accept Reject
Correct Decision Type II Error ( b
)
Consumer Risk
Type I Error ( a
)
Producer Risk
Correct Decision
Power (1b)
ASQ Section 511 5

Distribution Plot
Normal, Mean=0, StDev=1
 Typical bell-shaped curve.
0.4
0.3
0.2
0.1
0.0
0.0228
-2 0
X
2
0.0228
 Z-scores determine how many standard deviations a value is from the mean.
April 21, 2010 ASQ Section 511 6


Continuous Data Sample Size n


Z a

Z
 2 b
)
2
S
2
As the effect size decreases, the sample size increases.
 As variability increases, sample size increases.
 Sample size is proportional to risks taken.
April 21, 2010 ASQ Section 511 7

Binomial Distribution
 Binomial Distribution a 

 n x


 p x 

1 p
)
(n
 x)
 where:
• n is the sample size
• x is the number of positives
• p is the probability
• a is the probability of the observing x in a sample of n.
April 21, 2010 ASQ Section 511 8

Binomial Confidence Intervals
 Binomial Distribution a 

 n x


 p x 

1 p
)
(n
 x)
 Solve the equation for p given a
, x and n.
 x=0, n=11 and a
=0.05 (95% confidence).
• p=0.28 (table shows 0.30ucl)
 x=2, n=27 and a
=0.01 (99% confidence).
• p=0.298 (table shows 0.30ucl)
April 21, 2010 ASQ Section 511 9

 Describes the number of times an event occurs in a finite observation space.
 For example, a Poisson distribution can describe the number audit findings.
 The Poisson distribution is defined by one parameter: lambda. This parameter equals the mean and variance. As lambda increases, the
Poisson distribution approaches a normal distribution.
April 21, 2010 ASQ Section 511 10

 P(x) = probability of exactly x occurrences.

 x e
  x !

 is the mean number of occurrences.
April 21, 2010 ASQ Section 511 11

 If the average number (

) of audit findings is 5.5.
 What is the probability of a sample with exactly 0 findings?
•
0.0041 (0.41%)
 What is the probability of having 4 or less findings in a sample
•
(x=0 + x=1 + x=2 + x=3 + x=4)
•
0.0041 + 0.0225 + 0.0618 + 0.1133 + 0.1558 = 0.358
(35.8%)
April 21, 2010 ASQ Section 511 12

 The central confidence interval approach can be approximated in two ways:
 95% CI for x=6 would be (2.2,13.1)
1
2

2
0.975;2 x

1
2

2
0.025;2( x

1)
1.965

2 x
2
1.965
 x

1
2
2
April 21, 2010 ASQ Section 511 13

 What is missing in ALL calculations for the Poisson?
 No reference to sample size.
 Assumes a large population (np>5)
April 21, 2010 ASQ Section 511 14

Sample Size
50
100
500
1,000
Poisson
Mean
5
UCL
10.51
0.1
Binomial
19.88 9.94
0.05 10.22 10.22
0.01 2.09 10.45
0.005 1.05 10.47
April 21, 2010 ASQ Section 511 15

N

1
 N

1 was an unpublished report by the
AOAC in 1927.
 It was intended to be a quick rule of thumb for inspection of foods.
 Since it was unpublished, there was not a description of the statistical basis of it.
April 21, 2010 ASQ Section 511 16

N

1
 There is no known statistical justification for the use of the square root of n plus one’ sampling plan.

“Despite the fact that there is no statistical basis for a ‘square root of n plus one’ sampling plan, most firms utilize this approach for incoming raw materials.”
•
Henson, E., A Pocket Guide to CGMP Sampling, IVT.
April 21, 2010 ASQ Section 511 17

ANSI/ASQ Z1.4
Square root N plus one
 Lot Size N=1000
 Sample size n=32
 Acceptance Ac=0
 Rejection Re=1
 AQL=0.160%
 LQ = 6.94%
April 21, 2010
 Lot Size N=1000
 Sample size n=33
 Acceptance Ac=0
 Rejection Re=1
 AQL=0.153%
 LQ = 6.63%
ASQ Section 511 18

 Yes, it meets the Z1.4 definition of a sampling plan.
 It is statistically valid in that it defines the lot size, N, the sample size, n, the accept number,
Ac, and the reject number, Re.
 The Operational Characteristic, OC, curve can be calculated for any square root N plus one plan.
April 21, 2010 ASQ Section 511 19

 It is very common to use Z1.4 General Level I as the plan for audits.
 The sample sizes for square root N plus one are very close to the sample sizes for Z1.4 GL I.
 Square root N plus one can be used any where that Z1.4 GL I is or could be used.
April 21, 2010 ASQ Section 511 20

Sqrt(N+1) versus Z1.4
1000
100
10
1
1 10 100 1000
Lot Size
10000 100000 1000000
Sqrt (N+1)
Z1.4
April 21, 2010 ASQ Section 511 21

 Like Z1.4 GL I it can be used for audits.
 Any plan is justified by AQL and LQ
 It is easy to use and calculate.
 Works best with an Ac=0.
April 21, 2010 ASQ Section 511 22

Lot Size Sample Size
April 21, 2010
4
10
25
50
100
250
500
1000
10000
Ac=0 Ac=1
11
17
23
6
8
3
4
33
101
ASQ Section 511
AQL LQ AQL LQ
1.69
54 13.50
80
1.27
44 9.78
68
0.85
32
0.64
25
6.30
4.60
51
41
0.46
19
0.30
13
3.30
2.10
0.22
9.5
1.57
31
21
16
0.16
6.7
1.09
11
0.05
2.3
0.35
3.8
23

 How do you determine when you have too many findings?
 How do you determine the correct sample size for an audit?
 Would a confidence interval approach work?
•
As long as the observed number is lower than the upper confidence interval, the system is in control.
April 21, 2010 ASQ Section 511 24

 Need to use risk or statistical probability to determine when to audit:
•
Critical components
•
Low rank
•
High Volume suppliers
•
No third party data available
April 21, 2010 ASQ Section 511 25

 The results of an audit can help to establish acceptance controls.
 Better audit results would have less risk, and require smaller sample sizes for incoming inspection.
 Can use AQL or LTPD type of acceptance plans based on audit results.
April 21, 2010 ASQ Section 511 26

 Using the correct sampling strategy helps to assure coverage during an audit.
 Using confidence intervals to determine if a system is in control.
 More compliant systems require larger sample sizes.
April 21, 2010 ASQ Section 511 27

Steven Walfish steven@statisticaloutsourcingservices.com
301-325-3129 (Phone)
240-559-0989 (Fax)
April 21, 2010 ASQ Section 511 28