How to Calculate Sampling Size of Raw Materials | n, p and r Plans

When raw materials are delivered to a manufacturing plant, they pose a threat to the quality of finished goods. The quality department cannot presume that every box, even though it comes from a certified supplier, is uniform and pure. This is why sampling plays such a crucial part in GMP compliance.

One of the most common questions I get asked in my internal GMP training is how many containers need to be sampled. Unfortunately, there is no simple answer. There are companies which still insist on sampling each and every box they receive while some others create less detailed sampling plan based on the qualification results of their suppliers.

Knowledge of n, p and r sampling plans provides useful information for quality specialists who are supposed to create scientifically based sampling plans in accordance with regulatory requirements.

Importance of Raw Material Sampling

Raw material testing starts long before the sample gets to the laboratory. If the sampling fails to obtain a representative sample, the most advanced analytical method won’t yield reliable results.

Bad sampling practices can lead to:

• Acceptance of contaminated material
• Failure to detect mixing or switching of containers
• Wrong batch releasing decision
• Regulatory remarks
• Product recalls

This is why sampling is considered one of the critical quality control processes in pharmaceutical manufacturing.

Regulatory Perspective

Application of law forces the producers to write down procedures indicating the way in which raw materials would be sampled from, identified, tested and released. Checks by inspectors include:

• Procedures for sampling used
• Plans for sampling
• Documents related to the qualification of suppliers
• Places in which the products become sampled
• Equipment for sampling
• Document stating the identity of the sample
• Environment in which the sample is taken
• Training of personnel

The regulations of the FDA and GMP within the EU introduce no specific formula governing the sampling of any product but hope that the manufacturers can justify their sampling approach based on scientific principles, supplier qualification, historical records of quality and risk assessment.

Factors that Influence Sampling Size

There are numerous factors that are to be taken into consideration before the selection of the sampling plan. These are as follows:

• Supplier qualification level
• Criticality of the material
• Total number of containers
• Manufacturing procedure
• Variability of the material
• Past cases of non-conformance
• Risk of cross-contamination
• Packaging configuration

The sampling strategy of a high-risk API is usually different from that of low-risk excipients received from a well-qualified supplier.

Understanding the Three Sampling Plans

In the realm of manufacturing pharmaceuticals, there exist three sampling techniques called the n, p and r methods. Each method has its own utility and ought to be selected based on risk rather than ease of implementation.

1. The n Plan (100% Container Sampling)

The n plan is viewed as the most conservative of the three plans, where every container in a batch is taken for sampling.

For example, in a shipment containing 50 drums, all the drums are sampled.
This technique ensures a high degree of certainty in detection of variability from container to container, however, a considerable amount of time and resources is needed. The n method is used in the following cases:

• Active pharmaceutical ingredients
• Sterile raw materials
• Highly potent materials
• Unused suppliers
• Materials with previous quality problems
• Starting materials that are considered to be critical

Although, this plan consumes many resources, it considerably lowers the risk of accepting uneven or incorrectly identified materials.

2. The p Plan (Square Root Plus One)

The p plan is one of the most favorably accepted sampling methods for parties doing qualification of suppliers. To calculate the number of sampling units, the formula given below is used:

Total number of sampling units = √N + 1
Where:
N = Total number of items to be sampled
The value is rounded to the nearest whole number.

Example 1
Let us assume that there are 100 drums in one shipment.
Calculations:
Total number of sampling units = √100 + 1 = 10 + 1 = 11
Thus, only 11 drums are sampled.

Example 2
A batch of goods contains 49 bags.
Calculations:
Total number of sampling units = √49 + 1 = 7 + 1 = 8
Therefore, in this case, samples are taken from 8 bags.

Therefore, the p plan forms a very easy instrument in the sampling of goods while ensuring buyers that the product satisfies certain quality requirements provided that suppliers keep demonstrating good quality records during their performance.

3. The r Sampling Plan

The reduced sampling plan is a professional risk-based sampling plan that applies to materials from approved suppliers whose performance is monitored continuously. The r plan is different from n and p plans because it does not have a mathematical model. The decision on how many containers to sample is based on documented risk assessment and supplier performance data.

The r plan may involve the following:

• Supplier audit data
• Number of lots accepted in a row
• Criticality
• Process capability
• History of complaints
• Analysis of trends
• Transportation conditions
• Previous laboratory performance

For instance, a pharmaceutical manufacturer may choose to take only a limited number of samples if the same pharmaceutical grade lactose is received from suppliers with ten years of compliance history provided that identity testing and supplier approval procedures are observed.

The basis for the Reduced Sampling Plan must always be documented and reviewed from time to time, as it must not be used only for the sake of reducing the amount of work in the laboratory.

Comparing the Sampling Plans

Sampling Plan	Containers Sampled	Typical Use
n Plan	Every container	High-risk materials, new suppliers, API containers
p Plan	√N + 1	Qualified suppliers, routine raw materials, Excipients
r Plan	Risk-based reduced sampling	Highly reliable suppliers with documented performance

Example in Real Life

Let’s say you get 225 packs of microcrystalline cellulose from a certified vendor.
Use various sampling schemes:
n Plan – Testing 225 packs.
p Plan – √225 + 1 = 15 + 1 = 16 packs.
r Plan – Sampling amount is defined according to certified risk assessment, for instance, it could be 8 or 10 packs.
The choice should be based on risk management specifics rather than personal opinion.

Common Mistakes During Raw Material Sampling

Deficiencies in sampling are often caused by poor procedures in the audits of GMP rather than wrong calculations. The following mistakes are frequently made:

• Minimizing sampling for unapproved vendors.
• Having a common sampling scheme for all materials.
• Failure to control vendor performance.
• Choosing most available containers instead of random sampling.
• Failure to change the sampling scheme according to vendor or process updates.
• Failure to provide justification for reduced sampling.
• Thinking that less sampling means no testing of identity.

Creating a Risk-Based Sampling Strategy

The successful strategy for sampling must be based on statistical guidelines and scientific knowledge. While considering a sampling SOP, it is necessary to analyze the following questions:

• Is the supplier completely approved and regularly inspected?
• Has the material passed many tests and has it been of suitable quality?
• What is the danger of defective containers for the product?
• Is there any proof of variability of different containers?
• Does packing and delivery create a risk of contamination?
• Does the testing meet the demands of regulations and standards?

These questions can help select n, p or r plan, according to the answers.

Documentation Requirements

Any sampling event must be completely traceable. The sampling documentation must involve:

• Material name
• Batch number
• Supplier name
• Total number of containers received
• Sampling plan (n, p or r)
• Formula for calculating the size of the sample
• Container numbers
• Time of sampling
• Sampler name and its signature
• Equipment used for sampling
• Environmental conditions where applicable

Complete documentation is important not only for the compliance with GMP regulations but for providing evidence during inspections.

Sampling raw materials is much more complex than simply selecting several samples for laboratory testing because it has to be performed according to science-based principles. Such a process affects the reliability of the research results as well as the quality of the pharmaceutical product. While sampling according to the n scheme ensures maximum confidence through 100% container sampling, the p scheme provides effective compromise between confidence and efficiency by means of √N + 1 principle. Sampling according to r scheme, when reinforced by supplier qualification and risk assessment, allows optimization of resources while maintaining quality.

According to my personal experience, the most effective sampling plans are those that are always kept dynamic. They change owing to the evolution of manufacturing process, supplier performance and regulatory requirements. Sampling plan should never be chosen simply on the ground of "this is how we have always done it." Instead, it should be developed on the basis of scientific rationale, risk management principles and strong commitment to quality preservation.

Regulatory References

1. EU GMP Guidelines, Volume 4, Chapter 5: Production
https://health.ec.europa.eu/medicinal-products/eudralex/eudralex-volume-4_en
2. FDA Guidance for Industry: CGMP for Finished Pharmaceuticals (21 CFR Part 211)
https://www.ecfr.gov/current/title-21/chapter-I/subchapter-C/part-211
3. WHO Technical Report Series No. 1025, Annex 4: WHO Good Manufacturing Practices for Pharmaceutical Products
https://www.who.int/publications/i/item/9789240001824

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