Lognormal Distribution Calculator

Compute PDF, CDF, and key statistics for the lognormal distribution given mu and sigma.

📉 What is the Lognormal Distribution?

The lognormal distribution is a continuous probability distribution for a random variable X whose natural logarithm follows a normal distribution. In other words, if Y = ln(X) is normally distributed with mean mu and standard deviation sigma, then X is lognormally distributed. The key consequence is that X is always positive, never zero or negative, making this distribution ideal for quantities that cannot go below zero.

Lognormal distributions appear throughout science, finance, and engineering. Stock prices are commonly modeled as lognormal because percentage changes (not absolute changes) are symmetric and normally distributed. Insurance claim amounts, city population sizes, internet file sizes, blood pressure readings in large populations, and the latency of web requests all tend to follow lognormal or near-lognormal patterns. Any time a variable is the product of many independent multiplicative factors, the central limit theorem on the logarithms guarantees approximate lognormality.

A common misconception is that mu is the mean of X. It is not. Mu is the mean of ln(X). The actual mean of X is exp(mu + sigma squared divided by 2), which always exceeds exp(mu). Similarly, sigma is not the standard deviation of X but of ln(X). This distinction matters greatly: a dataset with mu = 0 and sigma = 1 has a mean of approximately 1.65 and a standard deviation of approximately 2.16, illustrating how the lognormal distribution stretches the upper tail relative to a simple normal distribution centered at 1.

This calculator handles both the probability query use case (given x, find P(X at or below x), the PDF height, and the survival probability) and the pure statistics use case (given mu and sigma alone, find mean, variance, skewness, and kurtosis). Toggle between Probability and Stats mode using the tabs above.

📐 Formula

f(x) = (1 ÷ (x ⋅ σ ⋅ √(2π))) ⋅ exp(−(ln x − μ)² ÷ (2σ²))

x = query value (must be > 0)

μ = mean of ln(X), the log-scale location parameter

σ = standard deviation of ln(X), the log-scale scale parameter (> 0)

CDF = Φ((ln x − μ) ÷ σ), where Φ is the standard normal CDF

Mean = exp(μ + σ² / 2)

Median = exp(μ)

Mode = exp(μ − σ²)

Variance = (exp(σ²) − 1) ⋅ exp(2μ + σ²)

Skewness = (exp(σ²) + 2) ⋅ √(exp(σ²) − 1)

Excess Kurtosis = exp(4σ²) + 2⋅exp(3σ²) + 3⋅exp(2σ²) − 6

Example: μ = 0, σ = 1, x = 1 gives CDF = 50%, PDF ≈ 0.3989, Mean ≈ 1.6487

📖 How to Use This Calculator

Steps

Choose a calculation mode — Select Probability to find CDF, PDF, and survival for a specific x value, or select Stats to explore distribution-wide statistics like mean, variance, and skewness.

Enter mu and sigma — Type the log-scale mean (mu) and log-scale standard deviation (sigma, must be positive) into the input boxes or drag the sliders.

Enter x (Probability mode) — In Probability mode, enter the query value x (must be greater than 0) for which you want to compute the CDF and PDF.

Click Calculate — Press Calculate to instantly see the CDF, PDF, survival function, and key moments. Results update automatically as you adjust sliders.

💡 Example Calculations

Example 1 — Standard Lognormal LN(0, 1) at x = 1

Find P(X ≤ 1) for a standard lognormal distribution

Parameters: mu = 0, sigma = 1.

CDF = Φ((ln 1 − 0) / 1) = Φ(0) = 50%.

PDF at x = 1: f(1) = 1/(1 ⋅ 1 ⋅ √(2π)) ≈ 0.3989.

Mean = exp(0 + 0.5) ≈ 1.6487; Median = exp(0) = 1; Mode = exp(0 − 1) ≈ 0.3679.

CDF = 50.00% | PDF = 0.3989 | Mean = 1.6487

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Example 2 — Standard Lognormal LN(0, 1) at x = 2

Find the probability that X is at most 2 under LN(0, 1)

Parameters: mu = 0, sigma = 1, x = 2.

CDF = Φ(ln(2) / 1) = Φ(0.6931) ≈ 75.58%.

Survival P(X > 2) = 1 − 0.7558 ≈ 24.42%.

CDF = 75.58% | Survival = 24.42%

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Example 3 — Asset Price Model LN(2, 0.5) at x = 10

Stock price model with log-mean 2 and log-volatility 0.5, threshold $10

Parameters: mu = 2, sigma = 0.5, x = 10.

CDF = Φ((ln 10 − 2) / 0.5) = Φ((2.3026 − 2) / 0.5) = Φ(0.6052) ≈ 72.73%.

Mean of X = exp(2 + 0.125) = exp(2.125) ≈ 8.368. The median = exp(2) ≈ 7.389.

CDF = 72.73% | Mean = 8.368 | Median = 7.389

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❓ Frequently Asked Questions

What is the lognormal distribution and when should I use it?+

The lognormal distribution models positive-valued quantities whose logarithm is normally distributed. Use it when your data is always positive, right-skewed, and arises from multiplicative processes. Examples include stock prices, income data, insurance claims, particle sizes, and system latencies.

How is mu different from the mean of the lognormal distribution?+

Mu is the mean of ln(X), not the mean of X itself. The actual mean of X is exp(mu + sigma^2 / 2). For mu = 0 and sigma = 1 this equals approximately 1.6487, not 1. As sigma grows, the mean increases even if mu stays fixed because the right tail pulls the average upward.

What is the lognormal CDF formula?+

The CDF is F(x) = Phi((ln x - mu) / sigma), where Phi is the standard normal cumulative distribution function. This works for any x greater than 0. For x at or below 0 the CDF is 0 because lognormal variables are strictly positive.

How do I find the median of a lognormal distribution?+

The median is simply exp(mu). This is because setting the CDF equal to 0.5 gives Phi(0) = 0.5, which corresponds to (ln(median) - mu) / sigma = 0, so ln(median) = mu and median = exp(mu). The median is always less than the mean for sigma greater than 0.

What is the mode of a lognormal distribution and how is it calculated?+

The mode is exp(mu - sigma^2). It is always less than both the mean and median. For large sigma the mode can be very close to zero even when the mean is large, illustrating the heavily right-skewed nature of the distribution at high spread values.

Why is the lognormal distribution used in finance and stock modeling?+

Stock prices cannot go below zero, and percentage changes (log-returns) tend to be approximately normally distributed. If log-returns are normally distributed, the price at any future time follows a lognormal distribution. The Black-Scholes options pricing model relies on exactly this assumption.

How do I estimate mu and sigma from data?+

Take the natural logarithm of every observation. Then compute the sample mean and sample standard deviation of these log-transformed values. Those sample statistics are maximum-likelihood estimators of mu and sigma respectively.

What is the skewness formula for a lognormal distribution?+

Skewness equals (exp(sigma^2) + 2) * sqrt(exp(sigma^2) - 1). It is always positive (right-skewed). For sigma = 0.5 the skewness is approximately 1.75; for sigma = 1 it is approximately 6.18. The skewness grows rapidly with sigma.

What is the variance of a lognormal distribution?+

The variance is (exp(sigma^2) - 1) * exp(2*mu + sigma^2). This can be very large relative to the mean when sigma is large. For mu = 0 and sigma = 1, the variance is approximately 4.67, giving a coefficient of variation (std dev / mean) greater than 1.

How does changing sigma affect the shape of the lognormal distribution?+

Larger sigma spreads the distribution more. The mean increases, the mode decreases, and the skewness and kurtosis rise dramatically. For sigma near 0 the distribution is nearly symmetric and bell-shaped (resembling a normal). For sigma around 1 or greater the distribution is strongly right-skewed with a heavy upper tail.

What is the difference between lognormal and exponential distributions?+

Both are positive-valued, but they have different shapes. The exponential distribution has a monotonically decreasing PDF starting from x = 0 with a constant hazard rate. The lognormal PDF rises from zero, peaks at the mode, then falls, and its hazard rate is not constant. Lognormal is better for data that first increases then decreases in hazard, such as certain failure-time models.

Can the lognormal distribution model environmental or biological data?+

Yes. Environmental concentrations of pollutants, particle sizes in aerosols, bacterial colony counts, latency periods of diseases, and body weight distributions in animal populations all frequently fit lognormal distributions. Regulatory agencies like the EPA often use lognormal models when setting safe exposure thresholds.

🔗 Related Calculators

What is the lognormal distribution used for in practice?

The lognormal distribution models quantities that are always positive and multiplicatively influenced by many small random factors. Common uses include stock prices, insurance claim sizes, income distributions, particle diameters in aerosols, and latency measurements in computer systems.

What is the difference between mu and the actual mean of X?

Mu is the mean of ln(X), the logarithm of the random variable. The actual mean of X is exp(mu + sigma squared divided by 2), which is always larger than exp(mu). Only when sigma is very small do the two coincide approximately.

How do I calculate lognormal CDF?

The CDF at x equals the standard normal CDF evaluated at (ln(x) minus mu) divided by sigma. In notation: F(x) = Phi((ln x - mu) / sigma), where Phi is the standard normal CDF.

What is the lognormal PDF formula?

The PDF is f(x) = (1 / (x * sigma * sqrt(2*pi))) * exp(-(ln(x) - mu)^2 / (2 * sigma^2)) for x greater than 0. It equals zero for x at or below zero.

How do I find the median of a lognormal distribution?

The median equals exp(mu). This follows because P(X less than or equal to exp(mu)) = Phi((ln(exp(mu)) - mu) / sigma) = Phi(0) = 0.5. So exactly half the distribution lies below exp(mu).

How is the mode of a lognormal distribution calculated?

The mode (peak of the PDF) is exp(mu minus sigma squared). For sigma greater than 1, the mode is less than 1 even when mu equals 0, illustrating how larger spread pulls the peak leftward.

What does skewness tell me about a lognormal distribution?

Skewness measures asymmetry. The lognormal skewness equals (exp(sigma squared) + 2) * sqrt(exp(sigma squared) - 1). It is always positive, confirming the distribution has a right tail. Higher sigma produces more skewness.

What is excess kurtosis for a lognormal distribution?

Excess kurtosis equals exp(4*sigma^2) + 2*exp(3*sigma^2) + 3*exp(2*sigma^2) - 6. For sigma = 1 this is approximately 110.94, far above the normal value of 0, indicating extremely heavy tails.

Can I use this calculator for financial modeling?

Yes. Lognormal distributions are foundational in the Black-Scholes options pricing model, where log-returns are assumed normally distributed. Enter the drift (mu) and volatility (sigma) to find probabilities that an asset price exceeds a target value x.

What happens when sigma approaches zero?

As sigma approaches zero the distribution collapses to a point mass at exp(mu). The PDF becomes infinitely tall and narrow, the mean approaches the median and mode (all equal exp(mu)), and skewness and kurtosis approach their normal distribution limits.

How is the lognormal related to the normal distribution?

If X follows a lognormal distribution with parameters mu and sigma, then ln(X) follows a normal distribution with mean mu and standard deviation sigma. Every lognormal probability can therefore be computed by transforming to the standard normal.

What is the survival function in the lognormal context?

The survival function (also called the complementary CDF) equals 1 minus F(x), giving the probability that X is greater than x. It is widely used in reliability engineering to express the probability that a component survives beyond time x.

📌 Quick Tips

💡Mu (mean of ln X) and sigma (std dev of ln X) are parameters of the underlying normal distribution, not the mean and std dev of X itself.

💡If your data is always positive and right-skewed, the lognormal distribution is often a good model for it.

💡The median of a lognormal is simply exp(mu), making it easy to back-calculate mu from a known median.

💡Larger sigma values create heavier right tails and higher skewness, which is common in income and stock-price data.

💡Use the Stats mode to explore how changing mu and sigma shifts the mean, mode, and spread without fixing a query value.