The best that I can determine is that the rule probably had its origin in the USDA in the 1920-30’s. But no confirming documents exist. There are actually three versions of the rule:

- Take the square root of the lot size to get the sample size. Accept on zero defects. This version of the rules determines the sampling plan to use.
- Take the square root of the number of cartons, open that number of cartons and select the required number of samples from them. In this case, the sampling plan including the number of samples is already determined. For example, if the sample size is 20 and 50 cartons exist, SQRT(50)+1=8 cartons must be opened. Select 3 samples from 4 cartons and 2 from the remaining 4. The square root of n plus one rule is used to obtain a representative sample.
- Take the square root of the number of drums, sample from this number of drums and composite the samples together to run a single test. For example, if 50 drums are received, take samples from SQRT(50)+1=8 drums, composite them together and measure the characteristic of interest. The rule is used again to obtain a representative sample.

Much of the discussion seems to confuse these distinct uses. Rule 1 should never be used. Sampling plans should be selected based on operating characteristics such as AQL and LTPD using tables of sampling plans like those given in my book Guide to Acceptance Sampling or one of the many standards such as ANSI Z1.4. The operating characteristic does not depend on the lot size as explained in the article “The Effect of Lot Size”. Therefore, such plans can be selected independently of lot size. One reference that might be of interest is:

Keith Borland (1950), “The Fallacy of the Square Root Sampling Rule,” Journal of the American Pharmaceutical Association, 39, No. 7, p373-377.

This reference describes why the square root of n plus one rule should not be used to select a sampling plan.

Rule 2 and 3 represent a reasonable compromise in many cases balancing the cost of testing with the precision of the results. However, there are certainly situations rule 2 and 3 should not be used. For example, printing defects where the process could produce 100 consecutive bad units all packed in a single carton and then correct itself.

Despite the lack of justification and documentation, this rule is commonly used. For example:

**GUIDE TO INSPECTIONS OF MANUFACTURERS OF MISCELLANEOUS FOOD PRODUCTS – VOLUME 1**

“For microscopic filth, excess shell, etc., sample the square root of the number of bags in the lot. Collect a minimum of six and a maximum of eighteen subs each consisting of 900 grams (2 lbs) taken 340 grams (2/3 lb) from each of the three bags. Collect the subs in duplicate for the 702(b) portion.”

“For retail size containers, sample the square root of the number of containers in the lot with a minimum of six and a maximum of 18 – 900 gram (2 lb) subs.”

“Bulk containers – collect 1 pint in duplicate from each container in lot. Sample 55 gal drums on a square root basis, collecting 1 pint from a minimum of 6 and maximum of 24 in duplicate.”

**Investigators Operations Manual – FDA May 1996**

Subchapter 420, Section 427.2 on random sampling states: “Sample size is usually described in your assignment, IOM Sample Schedule, Compliance Program, or the applicable schedules. If none of these furnish the sample size, a general rule is to collect samples from the square root of the number of cases or shipping containers but not less than 12 or more than 36 subs in duplicate. If there are less than 12 containers, all should be sampled. Discuss sample size and 702(b) requirements with your supervisor. See IOM 422.1. “

All reference to this rule has been deleted from the 2005 version.

I would love to hear from anyone who has further references, information or examples.

Sohail FarooqDr. Taylor,

This is Sohail from Pakistan.

I have been working in garment industry from 1993 in quality department through out my career. I have been doing final inspection from 2002. During my whole inspection experience I have been experiencing different types of carton selection methods in my career.

1) Square root carton selection of presented cartons.

2) Double square root carton selection of presented cartons.

3) 10% carton selection of presented cartons.

As per AQL table we cannot change the sample size according to the consignment quantity but inspector can change carton selection method as per his intuitions and dauts.

As far as AQL table is concern and my own research on this AQL table is concern if we need best result of random final inspection then we go for 10% carton selection of presented cartons.

Gloriana AlvarezDear Dr. Taylor,

At the company I work for, we currently use the √N+1 rule to determine a representative number of cartons for product sampling. The actual product sampling is based on ANSI Z1.4. However, we are considering moving away from the √N+1 rule.

What would you recommend as an alternative approach? Additionally, can the lot size and sample size columns in the ANSI Z1.4 tables be used for this purpose?

Thank you for your time and advice.

Gloriana

Wayne TaylorMy comments assume a sampling plan has been selected and one is determining how many cartons to open to get a representative sample. Something to consider are whether defects occur randomly or do they tend to occur in runs or clusters. If the defects occur totally at random, a small number of cartons can be used. One might specify something like a minimum of 5 cartons.

If defects can occur in runs or clusters, a larger number of cartons is needed. In fact, if the sample size exceeds the number of cartons, anything short of sampling every carton can reduce the protection provided by the sampling plan. Missing a carton can mean missing the run of defects in that carton. If the sample size is less than the number of cartons, similarly taking more than 1 sample per carton can reduce the protection provided by the sampling plan. In these cases, the cartons should be spread out across the lot and include the first and last carton in the lot. Further, no more than 2-3 samples per carton should be selected.

Totally random and runs of defects represent the extremes. In many cases both can occur resulting in middle ground.