Understand the relationship
The reasoning behind the result
A positive tube is an occurrence observation
P(no growth unit) = exp(−λm)
Under the Poisson model, λ is growth-unit concentration per gram or millilitre of original sample, and m is the amount of original sample represented in one tube. A tube is positive when it contains at least one detectable growth unit.
This assumes independent tubes, randomly distributed growth units and a response whenever at least one such unit is present. Clumps may behave as growth units rather than individual cells. Method sensitivity, inhibition, contamination and misclassification are not corrected automatically.
Original-sample equivalent is the exposure
The total liquid in a tube may contain medium and diluted material. Only the original-sample-equivalent amount belongs in λm. For a liquid stock, 1 mL of a 100-fold dilution represents 0.01 mL of original sample.
For homogenized solid material, derive gram equivalents from the actual preparation and dilution record. Changing all exposure amounts by a factor changes the inferred concentration inversely; an additional dilution correction must not then be applied twice.
The estimate maximizes the outcome likelihood
Σ[gⱼmⱼ/(exp(λmⱼ) − 1)] = Σ[(tⱼ − gⱼ)mⱼ]
Each group contributes its positive and negative counts. For a mixed outcome with at least one positive and one negative tube, the tool solves the likelihood score numerically. It supports unequal tube counts and arbitrary positive exposure amounts rather than selecting a table row by appearance.
With one exposure amount and a mixed outcome, the result reduces to −ln(negative fraction)/m. The table compares observed positives with expected positives at the fitted concentration; it does not automatically discard an unusual dilution group.
Boundary outcomes need different uncertainty statements
Mixed outcomes use a large-sample log-concentration interval from observed likelihood curvature. The displayed 95% interval is approximate, can be poor for sparse or unusual outcomes, and is not the exact interval method of a published MPN lookup table.
For all-negative tubes, the maximum-likelihood point is zero, but a one-sided 95% upper bound is −ln(0.05) divided by total original-sample exposure. For all-positive tubes, no finite maximum-likelihood estimate exists; a one-sided lower bound is obtained from the probability of all tubes being positive. These limits do not certify absence, sterility or safety.
Follow the numbers
Five positives among ten equivalent tubes
- Each tube represents 0.1 mL of original sample; five of ten are negative, giving negative fraction 0.5.
- For one dilution group, λ = −ln(0.5)/0.1 = 6.93147 growth units/mL.
- At that concentration, predicted positive probability is 1 − exp(−6.93147 × 0.1) = 0.5, corresponding to five expected positives among ten tubes.
The estimate matches the entered occurrence pattern under the model; it is not a direct cell count or an assay-validation result.
Quick guide
How to use this calculator
- Choose original sample mass or volume as the concentration basis.
- Enter the original-sample-equivalent amount per tube, not total medium volume.
- Enter total conclusively tested and positive tubes for each group.
- Review the pattern, fitted expectation and labelled interval method; a numerical result does not validate the assay.
Calculation method
Calculation and interpretation
Connect a supplied dilution-test outcome to its likelihood-based concentration estimate and visible model assumptions.
P(positive in tube j) = 1 − exp(−λmⱼ); maximize Π[1 − exp(−λmⱼ)]ᵍʲ exp[−λmⱼ(tⱼ − gⱼ)]
Worked example
Five positives among ten equivalent tubes
The estimate matches the entered occurrence pattern under the model; it is not a direct cell count or an assay-validation result.
P(positive in tube j) = 1 − exp(−λmⱼ); maximize Π[1 − exp(−λmⱼ)]ᵍʲ exp[−λmⱼ(tⱼ − gⱼ)]
Supported inputs
Precision and limits
Poisson growth-unit scenario
Independent, randomly distributed detectable growth units; no inhibition, false results, confirmation-subsampling adjustment or organism-specific method is inferred.
Uncertainty method
Mixed outcomes use an approximate 95% interval on log concentration from observed information. Boundary outcomes use one-sided 95% probability bounds. No regulatory acceptance or sample-safety conclusion is provided.
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