Interactive module guide

Log-Normal Distribution Calculator & Guide

This lognormal distribution calculator lets you calculate probabilities for positive, right-skewed values and explore how its log-scale parameters change the curve.

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The log-normal distribution models a strictly positive variable whose natural logarithm is normally distributed.

It is often plausible when many independent effects multiply rather than add, producing a right-skewed scale with occasional large values.

Use μ and σ as parameters of log(X), not of X itself, and do not select the model simply because a histogram has a long right tail.

Is this the right module for my question?

Use it when

Use a log-normal model for positive measurements generated by multiplicative growth, proportional error, or products of several positive factors. Examples can include particle sizes, some biological concentrations, project durations, and financial quantities under restricted modeling assumptions.
It is useful when taking natural logarithms yields an approximately symmetric, stable distribution and when zero or negative values are not part of the generating mechanism. The family supports probability questions about exceeding positive thresholds and spans a wide range of right-skewness.
The explorer helps distinguish median-scale behavior from arithmetic averages. In skewed data the mean can be pulled far above the typical observation, so multiple summaries should be reported.

Choose another method when

Avoid the standard log-normal distribution when zero or negative values are genuine outcomes. Adding an arbitrary constant to force positivity changes the modeled question and should not be done without scientific justification.
Do not use it for censored lifetimes without methods that handle censoring, or for heavy-tail risk solely because it permits large values. Its tail may be too light or too heavy for a particular process, and mixtures can imitate log-normal shape.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

Multiplicative completion time

Inputs: ln(X) ~ Normal(ln(20), 0.4); evaluate completion beyond 30 days.
Representative result: P(X>30) ≈ 0.155.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:59 min

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Read the complete transcript

The Log-Normal Distribution. Why can a 20-day median coexist with a much longer right tail?

The log-normal distribution models a strictly positive variable whose natural logarithm is normally distributed. It is often plausible when many independent effects multiply rather than add, producing a right-skewed scale with occasional large values. Use mu and sigma as parameters of log X, not of X itself, and do not select the model simply because a histogram has a long right tail.

The 1 divided by x factor in the density arises from transforming a normal variable back to the original positive scale. Equal distances on the log scale correspond to ratios on the original scale, so the distribution is asymmetric in X even though the natural log of X is symmetric. Multiplying X by a positive constant adds a constant to mu without changing sigma. This makes the family coherent under unit changes, provided the transformation is handled consistently. Reporting the original unit remains essential. The density peak is not the mean and generally not the median. Use the exponential of mu minus sigma squared for the mode, the exponential of mu for the median, and the exponential of mu plus sigma squared divided by 2 for the mean. Name the summary rather than calling all of them an “average.”

Suppose project completion time X in days is modeled as log-normal with mu equals the natural log of 20 and sigma equals 0.4. The median is therefore 20 days. A planner wants the modeled probability that completion takes more than 30 days. Set mu approximately 2.9957 and sigma equals 0.4, then evaluate the upper tail at x equals 30. Standardize the log threshold, z equals the natural log of 30 minus the natural log of 20 divided by 0.4 approximately 1.014. The normal upper-tail probability beyond z approximately 1.014 is about 0.155. The same value should appear as 1 minus F 30 in the log-normal explorer. Compute the arithmetic mean the exponential of mu plus sigma squared divided by 2 equals 20 the exponential of 0.08 approximately 21.67 days. It exceeds the 20-day median because large durations stretch the right side of the distribution.

Under the chosen log-normal model, about 15.5 percent of projects exceed 30 days. This estimate describes the modeled population, not a guaranteed delivery rate. Dependencies between project stages, schedule truncation, scope changes, and parameter uncertainty can materially alter the tail.

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How to read the result

The density peak is not the mean and generally not the median. Use exp(μ−σ²) for the mode, exp(μ) for the median, and exp(μ+σ²/2) for the mean. Name the summary rather than calling all of them an “average.”

An upper-tail result describes the modeled fraction above a positive threshold. It does not establish the probability of a future financial loss or delivery delay unless the log-normal process and parameter estimates are justified for that setting.

Inputs and parameter meaning

Enter μ as the mean of ln(X), not the arithmetic mean of X. The median of X is exp(μ), while the mean is exp(μ+σ²/2). This difference grows as σ increases and is central to interpreting the model.

Enter σ as the positive standard deviation of ln(X). It is dimensionless after a reference unit is chosen for the logarithm and controls multiplicative spread. The geometric standard deviation is exp(σ), which describes proportional rather than additive variation.

Threshold x must be positive. The CDF can be computed by converting ln(x) to a normal z-score: [ln(x)−μ]/σ. This link makes parameter checks possible with an independent normal CDF.

Formula and available parameters

Density or probability mass

f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)

Mean

e^{\mu + \sigma^2/2}

Variance

e^{2\mu + \sigma^2}(e^{\sigma^2} - 1)

Location parameter (μ): interactive range -2 to 2; default 0.
Scale parameter (σ): interactive range 0.1 to 2; default 1.

What the method is doing

The 1/x factor in the density arises from transforming a normal variable back to the original positive scale. Equal distances on the log scale correspond to ratios on the original scale, so the distribution is asymmetric in X even though ln(X) is symmetric.

Multiplying X by a positive constant adds a constant to μ without changing σ. This makes the family coherent under unit changes, provided the transformation is handled consistently. Reporting the original unit remains essential.

Worked example: a multiplicative completion time

Suppose project completion time X in days is modeled as log-normal with μ=ln(20) and σ=0.4. The median is therefore 20 days. A planner wants the modeled probability that completion takes more than 30 days.

1Set μ≈2.9957 and σ=0.4, then evaluate the upper tail at x=30. Standardize the log threshold: z=[ln(30)−ln(20)]/0.4≈1.014.
2The normal upper-tail probability beyond z≈1.014 is about 0.155. The same value should appear as 1−F(30) in the log-normal explorer.
3Compute the arithmetic mean exp(μ+σ²/2)=20exp(0.08)≈21.67 days. It exceeds the 20-day median because large durations stretch the right side of the distribution.

Interpretation

Under the chosen log-normal model, about 15.5% of projects exceed 30 days. This estimate describes the modeled population, not a guaranteed delivery rate. Dependencies between project stages, schedule truncation, scope changes, and parameter uncertainty can materially alter the tail.

Common mistakes

Entering the original-scale mean as μ is wrong. μ belongs to ln(X), and the arithmetic mean includes the σ²/2 adjustment.
Confusing natural logarithms with base-10 logarithms changes parameters. State the log base; the formulas on this page use the natural logarithm.
A straight-looking log-scale histogram is not a fit assessment. Inspect a log-scale quantile plot, residual structure, collection mechanism, and alternative positive families.

Limits and independent validation

The explorer does not estimate parameters, adjust for censoring, or quantify parameter uncertainty. It also cannot determine whether multiplicative causation is present.

Long-tail predictions can be unstable when σ is estimated from limited data. Extrapolating beyond the observed range requires sensitivity checks and domain review.

Before using the result

Confirm all observations are genuinely positive and review ln(X) for symmetry, outliers, subgroup structure, and time trends. Compare log-normal implications with gamma or Weibull alternatives when tails matter.
Verify one CDF value by transforming the threshold to a normal z-score. Check the displayed mean, median, and variance against the stated μ and σ before reporting.
Run a unit-sensitivity check by converting the original measurements to another legitimate positive unit and transforming μ consistently. The resulting probability for the same physical threshold should not change. If it does, a log parameter, threshold, or logarithm base has been converted incorrectly.

normal distribution →Explore the distribution followed by ln(X) and verify transformed probabilities.gamma distribution →Compare a different positive right-skewed family with distinct parameter meaning.data fitting →Assess supported candidate families against observed positive measurements.

← Weibull Distribution Continuous Uniform Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.