Homogeneous means uniform in composition and appearance throughout. The opposite is heterogeneous, which means non-uniform, with visible, distinct parts or clumps. 

There are three types of homogeneity: 

• Homogeneity of variance, ensuring consistent variance across groups for reliable statistical testing
• Homogeneity of subjects, ensuring participants share similar characteristics relevant to the study for internal validity
• Homogeneity of measures, emphasising consistent measurement tools and techniques for accurate, comparable data

With pharmaceutical materials, we consider homogeneous as aligning with the below:

• Definition: components are evenly distributed, appearing as a single substance
• Composition: uniform throughout
• Phases: only one phase of matter (e.g., all liquid, all gas)

The number of containers to sample depends on the relative risk of the material. With higher risk materials, 100% sampling is required. However, for lower risk materials a statistically significant sampling plan can be developed. Risk can be assessed in different ways, typically including aspects like past performance, criticality of the product and potential failure modes.

The general formula¹ is:

This is commonly applied for established, uniform materials. This is a non-statistical sampling method to select the number of containers to inspect from a population of size N.

For example, applying the formula, if there are 100 units, then 11 would be tested. If there are 1000 units, 33 would be tested.

The technique originated in the 1920s for agricultural inspectors working in the U.S ². Today, the approach is recommended by bodies like the U.S. Food and Drug Administration ³.

Although this approach is widely used for convenience and efficiency, it lacks a rigorous theoretical basis compared to other standards. In particular, the approach provides a lower level of confidence as the batch size increases, compared with other statistically driven methods ⁴. The three common statistical methods for assessing homogeneity are:

• Bartlett’s Test: often employed to test the homogeneity of variance across samples, providing a check for the assumption of equal variances, which many parametric tests require ⁵
• Levene’s Test: serves a similar purpose, offering robustness against departures from normality, making it suitable for assessing variance homogeneity in a broader range of data distributions ⁶
• The Coefficient of Variation (CV): another measure, quantifying the ratio of the standard deviation to the mean, offering insight into the relative variability within a sample ⁷

These approaches are outlined in internationally accepted standards like ANSI/ASQ Z1.4-2003 (R2018) ⁸ and ANSI/ASQ Z1.9-2003 (R2018) ⁹. Those who favour statistical approaches maintain that lot size alone provides an incomplete basis for determining sample size, whether by ‘square root’ sampling or percentage sampling. In contrast, sampling plans based on quantitative statements provide assessments of greater accuracy and meaning ¹⁰.

Also of importance is the ISO 2859 standard series:

• ISO 2859-1: the core standard for lot-by-lot inspection, using AQL to determine acceptable quality. It includes switching rules (normal, tightened, reduced) based on supplier performance
• ISO 2859-2: focuses on sampling plans for isolated lots using Limiting Quality (LQ), which is more stringent than AQL
• ISO 2859-3: specifies skip-lot sampling procedures to reduce inspection effort for high-quality suppliers
• ISO 2859-4: provides procedures for assessing declared quality levels

Nonetheless, the ‘square root of N’ method remains in widespread use in the pharmaceutical sector.

Representative materials

Obtaining a representative sample from a container for inspection or analytical testing is important. Sampling rods can be designed to take samples from the top, middle and bottom of containers. These are often called thief samplers, zone samplers or multi-level samplers, which are specialised tools used to obtain representative samples from liquids, powders or granules.

These tools are essential for identifying stratification within containers, such as tanks, drums and silos. The devices are usually manufactured from 316L stainless steel or from chemically resistant plastics (PTFE/PP). A key design feature is the use of a closure mechanism that ensures that the sample taken at the bottom is not contaminated by material from the top as the rod is removed.

Typical sampling instructions are:

1.    Insert: the closed sampling rod is inserted vertically into the container
2.    Sample: the device is opened at the desired depth (e.g., bottom, then middle, then top) to collect the material
3.    Close and retrieve: the rod is closed to seal the sample inside, before being lifted out
4.    Discharge: the contents are transferred to a container for analysis

This approach helps to evaluate the quality of the product by taking representative samples of the bulk material.