Why Sampling Plan is SQRT n+1 or √n+1 for Pharmaceuticals?

• The sampling formula SQRT n+1 in pharmaceuticals and its recommendations in various guidelines.
  
• Quality of pharmaceutical products majorly depends upon the sampling of the excipients and the active pharmaceutical ingredients. Proper sampling can give us confidence in our analysis. In other words we can say – sampling is a starting process but is has its importance.
  
• Number of containers to be sampled is an interesting part of the raw material sampling because it we receive 5000 containers of an excipent then it shall be very difficult to sample all containers and it is difficult too to analyze the thousands of samples. In such cases sampling plans are used to reduce the sampling and analysis of large number of containers. Generally in pharmaceuticals,  SQRT (n+1) or √n+1 formula is used to determine the number of containers to be sampled. Where n is the number of containers received. This formula is used to reduce the sampling of the large number of containers of the excipents. Some companies have their own limitations as if containers are 10 or less, all containers shall be sampled.
  
• WHO suggests 3 formulae of sampling for pharmaceutical ingredients in Technical Report Series TRS-929 - Annex 4, “WHO Guidelines for sampling of pharmaceutical products and related materials”.
  
•  n-plan: This plan is used when the material is uniform and supplier is recognized and reliable. Sample can be taken from any part of the container. Samples are taken by using the formula n=1+√N. Sampling units are selected randomly and all containers shall be sampled if those are four or less in number.
  
• p-plan: Samples are taken using this plan only when material is received from the reliable sources and identification of material is being done. Sampling is done by using the formula p=0.4√N and samples are collected in separate sample containers.
  
• r-plan: r-plan is used when the material is suspicious and received from the unknown source. Sampling is done using the formula r=1.5√N. It gives the more number of samples than the n-plan to build the confidence level.
  
• All samples are collected separately and transferred to quality control laboratory for identification. If sample passes the identification test; sample is analyzed for the assay.
  
• Department of Human Health Services, Food and Drug Administration (FDA) clearly writes in 21 CFR Part 111 Docket No. 2007N–0186 that there are a lot of sampling plans but we use SQRT n+1 and also suggests to sample 4 from 10 containers, 11 from 100 containers and 32 from 1000 containers.
  
• Therapeutic Goods Administration, Australia (TGA) states in its guideline “Sampling and testing of complementary medicines” that formula √n+1 can be used for the sampling of the excipents and sampling of the active material can be reduced.
  

Therefore, √n+1 is a widely accepted formula for the determination of the containers to sampled in pharmaceuticals and all major regulatory agencies recommend the same.