# The Effect of Lot Size

Dr. Wayne A. Taylor

Mil-Std-105E indexes sampling plans by AQL, Levels of Inspection, and lot size.  It is a common misconception that 105E includes lot size because “larger lots require more samples to obtain the same level of protection.”  In actuality, 105E takes more samples from larger lots in order to get better protection.  Figure 1 shows the OC curves of several of the 1.0% AQL sampling plans.  Increasing lot size increases the sample size letter code which steepens the OC curve resulting in better protection.

Figure 1: AQL=1% OC Curves

Better protection for larger lots can be justified by the fact that for larger lots the costs of rejecting good lots and the costs of accepting bad lots are higher.  Since the consequences of making wrong decisions are higher, it is logical to take more samples to lower the risk of making wrong decisions.

While this justification has merit when considering a single product, 105E is used to inspect a variety of products.  Should more samples be selected from a large lot of pencils or from a small lot of pace makers?  To overcome this objection, different levels of inspection are provided.  105E states that these inspection levels are to be selected based on the “discrimination” required.

The “discrimination” or protection provided by a sampling plan depends primarily on the number of units inspected and the acceptance number.  Lot size has only a minor effect limited to the case when 10% or more of the lot is inspected.  As a result, the single sampling plan n=13 and a=0 provides the same protection regardless of whether the lot size is 50, 200, or 200,000.  Figure 2 shows OC curves for these different lot sizes.  OC curves based on lot size, are called Type-A OC curves (hypergeometric distribution).  They are closely approximated by the Type-B OC curve which assumes an infinite lot size (binomial distribution).  The Type-B OC Curve represents the worse case.  It has the greatest chance of both accepting bad lots and rejecting good lots.

Figure 2: OC Curves of n=13, a=0

Since Type-B OC curves represent the worse case, sampling plans selected based on Type-B OC curves can be used to inspect any lot regardless of size.  When selecting statistically valid sampling plans, it is not necessary to use different sampling plans for different lot sizes.  A better strategy is to select one sampling plan based on the protection it provides, i.e., its OC curve.  The OC curves of the 105E plans are given in Table X of 105E.  Tables of sampling plans indexed by their OC curves are given in my book.

Appeared in FDC Control, Food Drug & Cosmetic Division ASQC, No. 103, Sept. 1994, p. 6

#### Note:

• All the OC curves shown in Mil=Std-105E are Type-B, so do not depend on lot size.
• Another justification of selecting sampling plans independent of lot size is given on pages 2-3 and 188 of the book  Statistical Procedures for the Medical Device Industry.  When sampling plans are selected based on risk, the risk components Severity, Occurrence, Detection, P1, P2 do not depend on lot size so the resulting sampling plans should not depend on lot size.

### 4 thoughts on “The Effect of Lot Size”

1. Dear. Dr. Wayne A. Taylor
My name is Lee Seung Ryul and I am a pharmaceutical company QA.

I saw your article, and I am very interested.

I also do not apply sample quantity based on lot size.
Currently, we have a fixed quantity of 125 such that C=0 based on AQL Limit=0.1.
This basis was established using OC Curve, hypergeometric distribution, and customer risk.

However, we do not accept data on this during the audit.

You are the only person who can credibly support this.

1. The hypergeometric distribution is used for OC curves based on lot size. It is based on the lot percent defective (X/N, X = number of defects, N = lot size) The binomial distribution is used for OC curves independent of lot size. It is based on the process percent defective (p = probability of a defect). I assume your statement about the hypergeometric is in error. The OC curves in ANSI Z1.4 are based on the binomial distribution ass described in Variation.com/the-effect-of-lot-size/.

The reason ANSI Z1.4 includes lot size as an index is not due to the effect of lot size on OC curve. It is based on an economic model that states as lot size increases the costs of making incorrect decisions increase so the OC curve should be tightened for larger lots. While there is some validity to this statement, there is still the question of what should the relationship be between lot size and the OC curve. This leads to multiple levels of inspection to choose between affecting the RQL of the selected sampling plan.

An alternative is to select lot size independent sampling plans directly based on just the AQL and RQL. This simplifies the process of selecting a sampling plan and aligns it with using a risk-based approach. Pages 2, 133 and 188 of the book Statistical Procedures for the Medical Device Industry justify this approach based on the fact that for a risk-based approach for selecting sampling plans, the fact that risk is lot size independent means the resulting sampling plans will be lot size independent.

2. Dear. Dr. Taylor,

I follow the justification laid out in STAT-09 Appendix D, however I was wondering whether the approach may be modified when you are talking about small lot sizes – in the range of 10 parts.

An example AQL = 1.5 (moderate to low risk product feature), moderate risk process, RQL = 18, attribute sampling would require between 8 and 13 samples (a=0).

If you were to sample 2 parts or 20% per batch – does this change the what you can say about the level of risk/ protection provided by the sampling plan vs. for example sampling 8 parts from a lot of 100 at same AQL / RQL as above.

Thanks and regards,

G

1. The lot size does not affect the protection. This is good news for large lots but bad news for small lots. For a lot size of 10 your options are on page 170 of Statistical Procedures for the Medical Device Industry

The plans with n=8 and below can be used. None have an RQL = 18. With small lots, attribute sampling plans do not offer great protection. Variables data is always preferred. The emphasis must be on controls of the process and linking data across lots like in trending.

Scroll to Top