But the minimal number that guarantees 3 high-clay samples is when you pick enough to exceed the maximum number that can avoid high-clay. - Appcentric

Understanding the Context

Determining clay content often relies on laboratory testing, a time-consuming and costly process. Direct testing of every sample is impractical, especially in large-scale surveys. Instead, we rely on sampling strategies grounded in statistical confidence. Suppose you aim to infer that at least three out of a larger batch are high-clay samples. A key insight is: you must sample enough to exceed the maximum number of samples that could possibly avoid high-clay content.

Understanding the Statistical Guardrail

Let’s define high-clay as soil samples exhibiting clay content above a predefined threshold (say, 40% or more by volume). The challenge is distinguishing true high-clay samples from noise. Because clay content fluctuates within a site, sampling fewer than necessary risks missing the three high-clay samples entirely.

Here’s the core principle:

Key Insights

> To guarantee detection of at least three high-clay samples with high statistical confidence, your sample size must exceed the maximum number of possible samples that could contain fewer than three high-clay specimens.

Mathematically, if p is the proportion of soil likely to be high-clay (a guess based on prior data), the worst-case scenario assumes sampling only low-clay soils. The maximum number of samples that can avoid high-clay depends on how small “high-clay” samples are.

Suppose you assume high-clay samples represent a minority (say, ≤25% of total). Then in a sample of size n, the maximum number of samples that could be non-high-clay (0 to 2 high-clay samples) depends on distribution. But to guarantee at least three high-clay samples, the number tested must exceed the sum of all combinations that could contain up to two such samples.

The Minimal Guaranteeing Number

A practical rule of thumb, supported by sample size theory and statistical power analysis, is:

Final Thoughts

> Test at least 7 samples to guarantee detection of at least three high-clay samples when high-clay soils are relatively rare or clustered but still present.

Why 7? Because with only 6 samples, it’s statistically possible (under certain distributions) to include up to two high-clay samples—and miss the third. Sample size 7 shifts this risk decisively beyond that threshold. Specifically:

• Testing 6 samples: maximum 2 high-clay samples possible at low density.
  
• Testing 7 samples: regardless of clustering, at least 3 are confirmed high-clay with high confidence.

This number balances practicality with statistical rigor, ensuring you avoid Type II errors (failing to detect actual high-clay content).

Real-World Application

In field surveys, this principle guides adaptive sampling:

• Start with small batches; if fewer than three high-clay samples are found, expand size.
  
• When forced to draw 7 representative cores, you are statistically locking in detection of at least three high-clay samples.