Use this lognormal distribution calculator to compute distribution moments, point probabilities, interval probability, and quantiles. Enter parameters of the underlying normal distribution: μ and σ for ln(X).
Example: μ = 0 means the median of X is exp(0) = 1.
Example: 0.95 gives the 95th percentile.
What is a lognormal distribution?
A random variable X is lognormally distributed if ln(X) is normally distributed. This means values of X are always positive and usually right-skewed: many small-to-moderate values with a long tail of larger outcomes. In real life, lognormal models often appear when outcomes are formed by multiplicative effects.
Common examples include income components, asset prices over short periods (under certain assumptions), biological growth processes, particle sizes, repair times, and environmental concentrations.
How to use this lognormal distribution calculator
1) Enter μ and σ for ln(X)
These are not the arithmetic mean and standard deviation of X itself. They describe the normal distribution in log space.
- μ shifts the distribution left or right on the log scale.
- σ controls spread and skewness; larger σ means heavier right tail.
2) Enter an x value
The calculator returns PDF and CDF at x:
- PDF f(x): local density at x (not a probability by itself).
- CDF P(X ≤ x): cumulative probability up to x.
- Tail P(X > x): complement of CDF.
3) Enter interval bounds a and b
Get the probability that X lands between two positive values: P(a ≤ X ≤ b).
4) Enter percentile p
For any probability level p between 0 and 1, the calculator returns the quantile xp such that P(X ≤ xp) = p.
Core formulas used by the calculator
If ln(X) ~ N(μ, σ²), then:
Mean: E[X] = exp(μ + σ²/2)
Median: exp(μ)
Mode: exp(μ - σ²)
Variance: (exp(σ²) - 1) * exp(2μ + σ²)
PDF: f(x) = (1 / (xσ√(2π))) * exp(-((ln x - μ)² / (2σ²))), x > 0
CDF: F(x) = Φ((ln x - μ)/σ)
Quantile: x_p = exp(μ + σΦ⁻¹(p))
Interpretation tips
- Median vs Mean: For lognormal distributions, the mean is usually larger than the median due to right skew.
- Scale matters: Changes in μ and σ can produce large changes in the upper tail.
- Units stay positive: This model is ideal for quantities that cannot be negative.
- Multiplicative uncertainty: If factors multiply together, lognormal is often a strong candidate.
Practical example
Suppose processing time X (in minutes) is lognormal with μ = 0.2 and σ = 0.6. You can use this page to answer questions like:
- What is the chance a job finishes within 2 minutes?
- What fraction of jobs fall between 1 and 3 minutes?
- What time covers 95% of jobs (95th percentile)?
This is exactly what operations teams, finance analysts, reliability engineers, and researchers need when planning buffers and risk thresholds.
Common mistakes to avoid
- Entering σ ≤ 0 (invalid).
- Using x, a, or b values that are ≤ 0 (lognormal support is strictly positive).
- Confusing μ and σ (log scale parameters) with mean and standard deviation of X.
- Interpreting PDF as a direct probability for a single point.
Final note
This lognormal distribution calculator is designed for quick analysis and intuition building. For high-stakes modeling, pair these results with sensitivity analysis, domain assumptions, and empirical data checks.