1. The Odds Are Against Auditing Statistical Sampling Plans
Steven Walfish Statistical Outsourcing Services Olney, MD

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

3. The Paradox
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

4. Common Sampling Strategies
Simple random sample.
Stratified sample.
Systematic sample.
Haphazard
Probability proportional to size
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ASQ Section 511

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

6. Normal Distribution Typical bell-shaped curve.
Z-scores determine how many standard deviations a value is from the mean.
April 21, 2010
ASQ Section 511

7. Continuous Data Sample Size
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

8. Binomial Distribution
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

9. Binomial Confidence Intervals
Binomial Distribution
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)
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ASQ Section 511

10. Poisson Distribution
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

11. Hypothesis Testing - Poisson
P(x) = probability of exactly x occurrences.
l is the mean number of occurrences.
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ASQ Section 511

12. Example of Poisson If the average number (l) of audit findings is 5.5.
What is the probability of a sample with exactly 0 findings?
(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)
= (35.8%)
April 21, 2010
ASQ Section 511

13. Poisson Confidence Interval
The central confidence interval approach can be approximated in two ways:
95% CI for x=6 would be (2.2,13.1)
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ASQ Section 511

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

15. Comparison
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ASQ Section 511

16. 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

17. 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

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

19. Is it a Real Sampling Plan?
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

20. Sample Size Comparison
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

21. Sample Size Comparison
April 21, 2010
ASQ Section 511

22. Is it a Good Plan? 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

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

24. Using Statistics 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

25. Deciding to Audit
Need to use risk or statistical probability to determine when to audit:
Critical components
Low rank
High Volume suppliers
No third party data available
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ASQ Section 511

26. Results of an Audit
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

27. Conclusion
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

28. Questions Steven Walfish steven@statisticaloutsourcingservices.com
(Phone)
(Fax)
April 21, 2010
ASQ Section 511