Are mu and sigma the mean and standard deviation of X?

No. They describe the natural logarithm of X. For μ = 0 and σ = 1, the median of X is 1 and its mean is exp(0.5), about 1.6487.

How do I find the probability between two values?

Enter the lower and upper bounds. The calculator subtracts the lower CDF value from the upper CDF value and shades the interval. A lognormal variable has positive support.

How do I calculate a lognormal percentile?

Enter the cumulative probability in the quantile control. For the 95th percentile use 0.95. The page shows the quantile and checks it by passing the result back through the CDF.

Why do the simulated results differ from the formula?

A finite sample has sampling variation. The seed makes a set of draws reproducible; it does not make its sample mean equal to the model mean.

Lognormal Distribution Calculator

Model a positive variable whose natural logarithm is normally distributed. This page opens on Lognormal: enter μ and σ for log X, set the interval and read its probability, or enter a probability to find a quantile. The density curve, moments and seeded sample draws share the same parameters.

Lognormal(μ 0, σ 1)density, 0.5 to 2 shaded

Family

μ, the mean of log X

σ, the SD of log X

μ and σ describe log(X), not X: μ from -10 to 10, σ from 0.05 to 3. The median of X is e to the μ.

Show

Lognormal(μ 0, σ 1) density on 0 to 13.1, which holds all but about half a percent of the distribution. Highest density drawn: 0.657189 at x 0.383. Shaded: 0.5 to 2, probability 0.511783.

Probability by bin
x	model	draws
0 to 1.31	0.607667	-
1.31 to 2.63	0.225409	-
2.63 to 3.94	0.081870	-
3.94 to 5.26	0.036550	-
5.26 to 6.57	0.018632	-
6.57 to 7.89	0.010410	-
7.89 to 9.2	0.006223	-
9.2 to 10.5	0.003919	-
10.5 to 11.8	0.002572	-
11.8 to 13.1	0.001747	-
above 13.1	0.005000	-

results6 decimal places

mean: 1.648721
variance: 4.670774
SD: 2.161197
median: 1.000000
mode: 0.367879
skewness: 6.184877
excess kurtosis: 110.936392
mean of log X: 0
SD of log X: 1

Interval probability

from

P(0.5 < X < 2) = 0.511783

CDF(2) - CDF(0.5) = 0.755891 - 0.244109

Quantile for probability p

x = 1.000000

round trip: CDF(1.000000) = 0.500000, relative gap 0

Simulate

draws

seed

Drawing once the page has loaded.

Mean

1.648721

2.161197

P(0.5 < X < 2)

0.511783

Median

1.000000

Beta(1, 1)mean, variance, CDF at 0.25
mean 1/2 = 0.500000, variance 1/12 = 0.083333, CDF(0.25) = 0.250000
Beta(2, 1)CDF at 0.5, mean, variance
CDF(0.5) = 0.250000, mean 2/3 = 0.666667, variance 1/18 = 0.055556
Lognormal, μ 0, σ 1CDF at 1, median, mean
CDF(1) = 0.500000, median 1.000000, mean exp(0.5) = 1.648721

Accuracy. The parameters describe a mathematical model, not a fit to data or a prediction about real events, and in lognormal mode μ and σ are the mean and standard deviation of log(X), not of X. Probabilities, quantiles and moments are computed numerically, the incomplete beta by a continued fraction, and printed to six decimal places (six significant figures in e-notation below 0.0001 or from a billion up), with each quantile's round trip through the CDF shown; simulated summaries carry sampling error, shown as a standard error, and the same seed gives the same draws. A beta density that is infinite at 0 or 1 is labelled at that end rather than drawn as a finite peak.

Common questions

Are mu and sigma the mean and standard deviation of X?: No. They describe the natural logarithm of X. For μ = 0 and σ = 1, the median of X is 1 and its mean is exp(0.5), about 1.6487.
How do I find the probability between two values?: Enter the lower and upper bounds. The calculator subtracts the lower CDF value from the upper CDF value and shades the interval. A lognormal variable has positive support.
How do I calculate a lognormal percentile?: Enter the cumulative probability in the quantile control. For the 95th percentile use 0.95. The page shows the quantile and checks it by passing the result back through the CDF.
Why do the simulated results differ from the formula?: A finite sample has sampling variation. The seed makes a set of draws reproducible; it does not make its sample mean equal to the model mean.

The parameters describe a mathematical model, not a fit to data or a prediction about real events, and in lognormal mode μ and σ are the mean and standard deviation of log(X), not of X. Probabilities, quantiles and moments are computed numerically, the incomplete beta by a continued fraction, and printed to six decimal places (six significant figures in e-notation below 0.0001 or from a billion up), with each quantile's round trip through the CDF shown; simulated summaries carry sampling error, shown as a standard error, and the same seed gives the same draws. A beta density that is infinite at 0 or 1 is labelled at that end rather than drawn as a finite peak.