Lognormal Distribution Calculator

About the Lognormal Distribution Calculator

The lognormal distribution calculator computes probabilities, percentiles, and statistical properties for the lognormal distribution — a continuous distribution whose logarithm is normally distributed. If X is lognormal with parameters μ and σ, then ln(X) ~ Normal(μ, σ²).

The lognormal distribution naturally models quantities that arise from multiplicative processes: stock prices, income distributions, particle sizes, failure times, biological measurements, and file sizes. Its characteristic right skew means the mean exceeds the median, which exceeds the mode.

Enter the location parameter μ (log-scale mean) and scale parameter σ (log-scale standard deviation) to compute PDF, CDF, interval probabilities, a visual density curve, a detailed percentile table, and a properties reference card. The same parameter pair also makes it easy to compare the median, mean, and mode, which is useful when the data is strongly right-skewed.

When This Page Helps

The lognormal distribution appears in finance, biology, environmental science, reliability engineering, and anywhere positive, right-skewed data is produced by multiplicative effects. This calculator gives you PDF/CDF evaluation, percentiles, a density curve, and a properties table without making you do the logarithms by hand.

It is a practical fit when the mean, median, and mode are all meaningfully different and the raw values cannot go below zero.

How to Use the Inputs

1. Enter μ (mu, the mean of the natural log of X) and σ (sigma, the standard deviation of ln(X)).
2. Enter x for point PDF and CDF calculations.
3. Set x₁ and x₂ for interval probability P(x₁ ≤ X ≤ x₂).
4. Review the density curve — the shaded region shows the interval probability.
5. Examine the percentile table for key quantiles.
6. Use presets to explore common applications.

Example Calculation

Tips & Best Practices

• The median (e^μ) is always less than the mean (e^(μ+σ²/2)) for the lognormal — the right tail pulls the mean higher.
• If your data is right-skewed and positive, plotting the log-transformed data should look approximately normal if it's truly lognormal.
• Stock prices are often modeled as exp(drift + σ×Z), making returns lognormal — this is the basis of Black-Scholes option pricing.
• A product of many independent positive random variables tends toward lognormal (central limit theorem for products).
• The coefficient of variation √(e^σ² − 1) depends only on σ, not μ — useful for comparing variability across different scales.
• For reliability analysis, the lognormal models failure times where early failures and wear-out both contribute to the distribution shape.

Multiplicative Central Limit Theorem

Just as the normal distribution emerges from sums of independent variables, the lognormal emerges from products. If X = Y₁ × Y₂ × ... × Yₙ where Yᵢ are positive independent variables, then ln(X) = ln(Y₁) + ... + ln(Yₙ), which by the CLT tends to normal. Hence X tends to lognormal.

Black-Scholes and Financial Modeling

The Black-Scholes option pricing model assumes stock prices follow geometric Brownian motion, making future prices lognormally distributed. The parameters μ (drift) and σ (volatility) determine expected returns and risk. Understanding lognormal percentiles is crucial for Value at Risk (VaR) calculations.

Environmental and Health Applications

Pollutant concentrations, bacterial colony counts, and particle size distributions often follow lognormal models. Environmental regulations frequently use geometric mean and geometric standard deviation rather than arithmetic ones, reflecting the underlying lognormal nature of the data.

Frequently Asked Questions

• What's the difference between μ/σ and the real mean/std dev?

  μ and σ are parameters of the underlying normal distribution of ln(X). The actual mean of X is e^(μ+σ²/2), which is larger than e^μ (the median). Similarly, the actual std dev depends on both μ and σ in a complex way.

• How do I estimate μ and σ from data?

  Take the natural log of all data points, then compute the sample mean and sample standard deviation of the logged values. Those are your estimates of μ and σ.

• Why is the lognormal used for income distributions?

  Income results from multiplicative factors — education, experience, industry, etc. The central limit theorem for products leads to lognormal distributions, explaining the characteristic right skew of income data.

• Can the lognormal distribution take negative values?

  No, the lognormal is strictly positive (X > 0) since it's the exponential of a normal variable. This makes it appropriate for inherently positive quantities like prices, times, and sizes.

• How does it compare to the normal distribution?

  The normal is symmetric and can take any real value. The lognormal is right-skewed and strictly positive. If ln(X) is normal, then X is lognormal. For small σ, the lognormal approaches normality near its peak.

• What is the geometric mean of a lognormal?

  The geometric mean equals the median: e^μ. This makes the geometric mean a natural "typical value" for lognormal data, more representative than the arithmetic mean.