How to use the lognormal distribution calculator
- Choose whether you have the mean and standard deviation of ln X (mu and sigma) or the mean and standard deviation of X itself.
- Enter the two numbers. For the second case the page converts them to mu and sigma for you.
- Choose Probability at a value or Value for probability, and enter the value or the probability.
- Read the probability below and above, the density, and the mean, median, mode and standard deviation of the distribution.
Formula
X is lognormal when ln X is normal with mean mu and standard deviation sigma
P(X <= x) = Phi((ln x - mu) / sigma). The value for a probability p is exp(mu + sigma z), with z the normal quantile of p.
Mean = exp(mu + sigma^2 / 2), median = exp(mu), mode = exp(mu - sigma^2)
From a mean m and standard deviation s of X: sigma^2 = ln(1 + s^2 / m^2) and mu = ln m - sigma^2 / 2
The distribution lives on positive numbers and is skewed to the right, with the mean above the median and the median above the mode.
The standard lognormal and a distribution with mean 10
With mu = 0 and sigma = 1, the probability that X is at most e (2.71828) is 0.84134, because ln e = 1 is one standard deviation above the mean of ln X. The mean is 1.6487, the median is 1, the mode is 0.36788 and the standard deviation is 2.1612. The 95th percentile is 5.1803.
For a distribution with a mean of 10 and a standard deviation of 5 on the original scale, the page finds sigma = 0.47238 and mu = 2.1910. The probability of a value at most 12 is 0.73308.
When a lognormal fits
A quantity is often lognormal when it is the product of many small, independent factors, just as a normal quantity is the sum of many small terms. Incomes, particle sizes, the sizes of many biological measurements, the lifetimes of some components and concentrations of pollutants often follow it. A check is to take logs of your data and see whether they look normal.
For values on the normal scale, see the normality skew and kurtosis calculator. For a standard normal score, use the z-score calculator.
Mean versus median
In a skewed distribution the mean and the median differ, and the gap grows with sigma. The median, exp(mu), is the value that half the population falls under. The mean is pulled up by the long tail of large values. Report whichever suits the question, and be clear which one you give.
For another distribution of positive, skewed values, see the gamma distribution calculator, and for waiting times the exponential distribution calculator.
Assumptions and limits
- The values are positive, and the logarithm of the values follows a normal distribution.
- Sigma must be above 0. For the raw scale, the mean and standard deviation must be above 0.
- Probabilities for a value search are between 0 and 1, and not exactly 0 or 1.
- Results use the exact distribution functions and not approximations.
Frequently asked questions
What is a lognormal distribution?
It is the distribution of a positive quantity whose logarithm is normally distributed. It is skewed to the right.
How do I get mu and sigma from a mean and standard deviation?
Use sigma^2 = ln(1 + s^2 / m^2) and mu = ln(m) - sigma^2 / 2. The page does it when you choose the raw scale.
Why is the mean above the median?
The long right tail of large values pulls the mean up, while the median depends only on the middle of the distribution.
Can I find a percentile?
Yes. Choose value for probability and enter, for example, 0.95 for the 95th percentile.