The relationship between purely stochastic sampling error and the number of technical replicates used to estimate concentration at an extreme dilution

For any analytical system the population mean (μ) number of entities (e.g., cells or molecules) per tested volume, surface area, or mass also defines the population standard deviation $ (\sigma = \sqrt {\mu } ) For any analytical system the population mean (μ) number of entities (e.g., cells or molecules) per tested volume, surface area, or mass also defines the population standard deviation (s = ?{m} ) (\sigma = \sqrt {\mu } ) . For a preponderance of analytical methods, σ is very small relative to μ due to their large limit of detection (>10² per volume). However, in theory at least, DNA-based detection methods (real-time, quantitative or qPCR) can detect ≈ 1 DNA molecule per tested volume (i.e., μ ≈ 1) whereupon errors of random sampling can cause sample means (`(x)] \overline x ) to substantially deviate from μ if the number of samplings (n), or “technical replicates”, per observation is too small. In this work the behaviors of two measures of sampling error (each replicated fivefold) are examined under the influence of n. For all data (μ = 1.25, 2.5, 5, 7.5, 10, and 20) a large sample of individual analytical counts (x) were created and randomly assigned into N integral-valued sub-samples each containing between 2 and 50 repeats (n) whereupon N × n = 322 to 361. From these data the average μ-normalized deviation of σ from each sub-sample’s standard deviation estimate ( s<sub>j</sub> ;  j = 1  to  N;  N = 7   n = 50 ]  to  180   n = 2 ] )\left( {s_j ;j = 1{\hbox{to}}N;N = 7\left {n = 50} \right]{\hbox{to}}180\left {n = 2} \right]} \right) was calculated (Δ). Alternatively, the average μ-normalized deviation of μ from each sub-sample’s mean estimate (`(x)]_j {\overline x_{\rm{j}}} ) was also evaluated (Δ′). It was found that both of these empirical measures of sampling error were proportional to ^{{ - 2}}?{n ·m} \sqrt{ - 2}]{{n \cdot \mu }} . Derivative (∂/∂n · Δ or Δ′) analyses of our results indicate that a large number of samplings (n ? 33±3.1) (n \approx {33}\pm {3}.{1}) are requisite to achieve a nominal sampling error for samples with a μ ≈ 1. This result argues that pathogen detection is most economically performed, even using highly sensitive techniques such as qPCR, when some form of organism cultural enrichment is utilized and which results in a binomial response. Thus, using a specific gene PCR-based (+ or −) most probable number (MPN) assay one could detect anywhere from 0.2 to 10⁵ CFU mL⁻¹ using 6 to 48 reactions (i.e., 8 dilutions × 6 replicates per dilution) depending on the initial concentration of the pathogen and volume sampled.