Lognormal Distribution Calculator
Get lognormal probabilities below, above or between values, or the x for a given probability. Enter μ and σ of ln X, or the mean and standard deviation of X itself and the calculator converts them; mean, median, mode and skewness come with a shaded chart.
The mean of ln X, any real number
The standard deviation of ln X, positive
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What the lognormal distribution is
A variable X is lognormal when its natural logarithm ln X follows a normal distribution. That makes X positive and skewed to the right, which fits incomes, house prices, stock prices over time, particle sizes and repair times, where effects multiply rather than add.
f(x) = 1 / (x · σ · √(2π)) · exp(−(ln x − μ)² / (2σ²)), for x > 0
P(X ≤ x) = Φ((ln x − μ) / σ)
Median = e^μ Mean = e^(μ + σ²/2) Mode = e^(μ − σ²)
Variance = (e^(σ²) − 1) · e^(2μ + σ²) Skewness = (e^(σ²) + 2) · √(e^(σ²) − 1)
The order mode < median < mean always holds, and the gap grows with σ. μ and σ are the mean and standard deviation of ln X, not of X: a common mistake is to enter the average of the raw data as μ.
Starting from the mean and SD of your data
If you know the mean m and standard deviation s of X itself, choose "Mean and SD of X itself". The calculator converts them exactly:
σ² = ln(1 + s²/m²)
μ = ln(m) − σ²/2
For m = 10 and s = 5 this gives σ² = ln 1.25 = 0.2231 and μ = 2.1910, and the resulting distribution has exactly the mean 10 and standard deviation 5 you started with.
Worked example
Suppose ln X has mean μ = 3 and standard deviation σ = 0.5. What is P(X ≥ 30)? Standardize the logarithm: z = (ln 30 − 3)/0.5 = 0.8024, and the area to the right of z under the standard normal curve is 0.211162. Load example shows that value together with a mean of 22.7599, a median of 20.0855 (= e³) and a mode of 15.6426 (= e2.75).
Software equivalents
| Software | Cumulative probability | Parameters |
|---|---|---|
| Excel / Sheets | LOGNORM.DIST(x, μ, σ, TRUE) | μ and σ of ln X |
| R | plnorm(x, meanlog = μ, sdlog = σ) | μ and σ of ln X |
| Python (SciPy) | scipy.stats.lognorm.cdf(x, s=σ, scale=exp(μ)) | s = σ, scale = e^μ |
| MATLAB | logncdf(x, mu, sigma) | μ and σ of ln X |
Related guides and calculators
A variable is lognormal when its logarithm follows the normal distribution. Other right-skewed models are the gamma, Weibull and exponential distributions, and the skewness and kurtosis calculator measures how skewed a sample is. Read probability distributions for an overview.
Frequently Asked Questions
What are μ and σ in a lognormal distribution?
They are the mean and standard deviation of the natural logarithm of X, not of X. The mean of X is e^(μ + σ²/2) and its median is e^μ. To start from the mean and SD of X, choose that option in the calculator.
How do I convert a mean and standard deviation into lognormal parameters?
Compute σ² = ln(1 + s²/m²) and then μ = ln(m) − σ²/2, where m and s are the mean and SD of X. The calculator does this when you select Mean and SD of X itself.
Why is the mean of a lognormal larger than its median?
The distribution has a long right tail, and a few very large values pull the mean up. The median is e^μ, the mean is e^(μ + σ²/2), so the mean is always larger by the factor e^(σ²/2).
How do I calculate a lognormal probability in Excel?
Use LOGNORM.DIST(x, mean, standard_dev, TRUE) with the mean and standard deviation of ln X. The inverse is LOGNORM.INV(probability, mean, standard_dev).
Can the lognormal distribution take negative values?
No. X is always positive because it is e raised to a normal variable. If your data include zero or negative values, a lognormal model does not apply without a shift.
How do I test whether my data are lognormal?
Take the natural logarithm of every value and check whether the logs look normal, for example with a normal Q-Q plot or the Shapiro-Wilk test. If they do, use the mean and SD of the logs as μ and σ.
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