LC50 calculator
Median lethal concentration from concentration, deaths and group size, corrected for control mortality.
Median lethal concentration from concentration, deaths and group size, corrected for control mortality.
The LC50, or median lethal concentration, is the concentration of a substance in water or air that kills half of the exposed organisms within a stated exposure time. It is the standard end point of acute aquatic and inhalation toxicity tests: the 96 h LC50 for fish, the 48 h LC50 (or EC50 for immobilisation) for Daphnia, the 4 h LC50 for inhalation in rodents. A lower LC50 means a more toxic substance. The value only has meaning together with the species, the exposure time and the unit.
Mortality rises with concentration along an S-shaped curve. On a log concentration axis that curve is close to a cumulative normal (probit) or logistic (logit) distribution, so the model is a straight line on the transformed scale:
Here probit(p) is the inverse normal of the proportion killed, written without the historical +5 offset. The calculator fits a and b by maximum likelihood on the counts of dead and exposed animals, so each group is weighted by its size and groups with 0 or 100 percent mortality are used rather than dropped. Other points on the curve follow from the same line: log₁₀(LCp) = (probit(p) − a) / b, which gives the LC10 and LC90 shown beside the result. Confidence intervals come from the delta method on the log scale.
Some control animals usually die for reasons unrelated to the test substance. Abbott's formula removes that background from each treated group:
Enter the control group as a row with concentration 0. The calculator pools all control rows, shows the corrected mortality for each concentration, and builds the background rate into the fitted model, P = c + (1 − c) × F(a + b × log₁₀ concentration), where c is the control mortality. The LC50 is then the concentration at which the substance itself kills half of the animals that would otherwise have survived. The Spearman–Kärber estimate uses the corrected proportions directly.
Worked example. A 96 h fish test with 20 animals per tank: control 1 dead, then 0.5 mg/L 1 dead, 1 mg/L 2, 2 mg/L 5, 4 mg/L 10, 8 mg/L 16 and 16 mg/L 20 dead. Control mortality is 1/20 = 5 %. At 4 mg/L the observed mortality is 50 % and the corrected mortality is (0.50 − 0.05) / 0.95 = 47.4 %. The probit fit gives a 96 h LC50 of3.92 mg/L (95 % CI 3.01 – 5.11) with a slope of 3.07, and Spearman–Kärber gives 3.93 mg/L. Without the control row the probit LC50 would be 3.42 mg/L, because background deaths would be counted as toxic effect.
| Method | Assumes | Use when |
|---|---|---|
| Probit | Log-normal distribution of tolerances | The default in ecotoxicology guidelines; two or more partial mortalities |
| Logit | Logistic distribution of tolerances | Almost the same LC50 as probit, slightly wider tails for LC10 and LC90 |
| Spearman–Kärber | Only that mortality rises with concentration | Few or no partial mortalities, but the series runs from 0 to 100 percent |
Acute tests with a steep response often give one partial mortality or none. A parametric fit is then poorly determined and Spearman–Kärber is the usual choice, provided the lowest concentration killed nothing (after correction) and the highest killed everything.
For dilute aqueous solutions 1 mg/L equals 1 ppm by mass and 1 µg/L equals 1 ppb. For gases, ppm is a volume ratio and converts to mg/m³ through the molar mass: mg/m³ = ppm × M / 24.45 at 25 °C and 1 atm. The unit selector only labels the result; enter every concentration in the same unit. To convert between mass and molar concentrations use themolarity calculator. LC50 falls as exposure lengthens, so a 24 h and a 96 h LC50 of the same substance are different numbers and the time must be reported with the value.
For a dose per body weight rather than a concentration, use theLD50 calculator. For a graded, non-lethal response use theEC50 calculator or theIC50 calculator, and for virus titres by end-point dilution the TCID50 calculator.
Expose groups of organisms to a series of concentrations, count the dead in each group after a fixed time, and fit a curve of mortality against log concentration. The LC50 is the concentration where the fitted mortality is 50 percent. This calculator fits a probit or logit model by maximum likelihood, or uses the non-parametric Spearman-Karber method, and gives a 95 percent confidence interval.
Abbott's formula removes background deaths that are not caused by the test substance: corrected mortality = (observed − control) / (1 − control), with mortalities as proportions. If 5 percent of the control animals die and 50 percent of a treated group die, the corrected mortality is (0.50 − 0.05) / 0.95 = 47.4 percent.
Most acute toxicity guidelines, including OECD 203 for fish and OECD 202 for Daphnia, require control mortality of 10 percent or less for the test to be valid. Abbott's correction is meant for small background losses; it cannot rescue a test in which a large share of the controls died.
Aquatic LC50 values are reported in mg/L or µg/L of water, which for dilute solutions equal ppm and ppb. Inhalation LC50 values are in mg/m³, mg/L or ppm of air. The exposure time is always part of the result, as in 96 h LC50 for fish or 48 h LC50 for Daphnia, because a longer exposure gives a lower LC50.
LC50 is the concentration in the surrounding water or air that kills half the organisms over a stated exposure time. LD50 is an administered dose per unit body weight, usually mg/kg. The statistics are the same; LC50 is used when the dose actually taken up cannot be measured, as with fish in a tank or animals breathing a vapour.