How this calculator works
Take natural logarithms before calculating the supplied mean and SD. The raw measurement is positive and generally right-skewed. The median is exp(μ), while the mean is larger because of the upper tail. Log parameters depend on the measurement unit, so keep that unit fixed throughout the calculation.
Formula
ln(X) ~ N(μ,σ²); F(x) = Φ((ln(x)-μ)/σ); E[X] = exp(μ+σ²/2); Var[X] = (exp(σ²)-1)exp(2μ+σ²).
Worked example
With μ = 0 and σ = 1, the median is 1, the mean is 1.6487213 and the probability below 1 is 50%.
Advertisement
Frequently asked questions
Which logarithm is used?
The natural logarithm, base e.
Can X equal zero?
The continuous support is strictly positive. Evaluation at zero returns zero cumulative probability.
Can I enter the arithmetic mean as μ?
No. μ is the mean after taking natural logarithms.