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Lognormal Distribution Calculator

The lognormal distribution models phenomena where the logarithm of a variable follows a normal distribution. It appears in reliability engineering, finance, environmental science, and network analysis. Use this calculator to compute probability densities, cumulative probabilities, quantiles, and key statistics (mean, median, variance, skewness) for any lognormal parameter set.

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Creators Wojciech Sas, PhD
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What is the Lognormal Distribution?

A lognormal distribution describes a continuous random variable whose natural logarithm is normally distributed. If Y ~ N(μ, σ²), then X = exp(Y) follows a lognormal distribution. This transformation ensures X takes only positive values, making it ideal for modeling quantities that cannot be negative: component lifespans, financial asset prices, rainfall amounts, and biological measurements.

The distribution is heavily right-skewed, with a long tail extending toward larger values. This asymmetry reflects real-world processes where most observations cluster near lower values but occasional extremes occur. The parameters μ (scale) and σ (shape) control the distribution's location and spread on the logarithmic scale, not on the original scale of X.

Lognormal Distribution Formulas

The probability density function and key descriptive statistics depend on the two parameters μ and σ, which represent the mean and standard deviation of ln(X) respectively. Below are the essential formulas for working with lognormal distributions.

f(x) = 1 / (x × σ × √(2π)) × exp(−(ln(x) − μ)² / (2σ²))

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]

  • x — The argument value at which to evaluate the probability density function
  • μ — Scale parameter; the mean of the natural logarithm of X
  • σ — Shape parameter; the standard deviation of the natural logarithm of X
  • p — Quantile level (0 to 1) for which to find the corresponding X value

Calculator Modes and Usage

This tool offers six distinct calculation modes to handle different analytical needs:

  • Probability density function (PDF): Enter μ, σ, and an x-value to find f(x), the height of the distribution curve at that point.
  • Cumulative distribution function (CDF): Calculate F(x), the probability that a random observation falls at or below a given value.
  • Quantile: Find the x-value corresponding to a specified cumulative probability (e.g., the 95th percentile).
  • Descriptive statistics: Compute mean, median, mode, variance, standard deviation, and skewness for your parameter set.
  • Tail probabilities: Determine P(X ≤ x), P(X < x), P(X ≥ x), or P(X > x) for risk and reliability assessments.

Begin by selecting your desired mode, entering μ and σ, then provide any additional arguments your mode requires.

Real-World Applications

The lognormal distribution excels at modeling inherently positive, skewed phenomena across diverse fields. In engineering and reliability, it represents component failure times and equipment time-to-failure in wear-out regimes. Financial analysts use it to model stock prices, returns, and asset valuations, since prices cannot be negative and exhibit proportional growth.

Environmental and hydrological data frequently follow lognormal patterns: daily rainfall accumulation, pollutant concentrations, and river discharge rates. In biology and medicine, lognormal distributions describe bacterial colony sizes, enzyme concentrations, and survival times post-treatment. Internet and network systems show lognormal characteristics in message sizes, download times, and request intervals. Industrial quality control leverages it for particle size distributions and defect frequencies.

Common Pitfalls and Practical Tips

Avoid these frequent mistakes when working with lognormal distributions.

  1. Confusing μ and σ with X statistics — Remember that μ and σ are the mean and standard deviation of ln(X), not of X itself. The actual mean of X is exp(μ + σ²/2), which is always larger than exp(μ). Confusing these leads to incorrect parameter estimates and wrong probability calculations.
  2. Assuming mode always exists — The mode exists only when σ > 1. For σ ≤ 1, the distribution is monotonically decreasing and has no mode. Always check your shape parameter before interpreting modal behaviour.
  3. Ignoring the right tail in risk assessment — The lognormal distribution's heavy right tail means extreme values carry non-negligible probability. When modeling financial losses or system failures, standard deviations and means can underestimate tail risk. Use quantiles (e.g., 99th percentile) for safety-critical applications.
  4. Parameter fitting from small samples — Estimating μ and σ from limited data introduces substantial uncertainty. Use at least 30-50 observations, and consider confidence intervals around your parameter estimates when making consequential decisions.

Frequently Asked Questions

What does μ represent in the lognormal distribution?

μ is the scale parameter—specifically, the mean of ln(X). It shifts the distribution left or right on the logarithmic scale. A larger μ moves the entire distribution toward higher values. Importantly, μ is not the mean of X itself. The mean of X equals exp(μ + σ²/2), which incorporates both parameters. Understanding this distinction is critical when fitting the distribution to data.

How do I fit a lognormal distribution to my data?

Calculate the natural logarithm of each observation, then compute the mean and standard deviation of those log-transformed values. These become your estimates for μ and σ respectively. This method works because ln(X) must be normally distributed for X to be lognormal. Verify normality of the log-transformed data using a Q-Q plot or Shapiro-Wilk test before accepting the fit.

When should I use lognormal instead of normal distribution?

Use lognormal when your data is strictly positive, right-skewed, and exhibits proportional rather than additive growth. If plotting ln(X) produces a bell curve but X itself is highly skewed, lognormal is appropriate. Examples include financial returns, failure times, and particle sizes. The normal distribution works for symmetric phenomena like measurement errors or temperature variations.

What is the difference between mean and median in a lognormal distribution?

The median is exp(μ), simple and parameter-free. The mean, exp(μ + σ²/2), is always larger than the median because of right-skewness. As σ increases, this gap widens significantly. For decision-making, the median better represents a typical observation, while the mean better captures the distribution's expected value for calculations involving sums or aggregates.

Can X be negative in a lognormal distribution?

No. By definition, X = exp(Y) where Y is normal. Since the exponential function always returns positive values, X is strictly positive for any real μ and σ. This is why lognormal distributions are ideal for non-negative quantities like prices, lifespans, and amounts. If your data contains zeros or negative values, lognormal is unsuitable.

How does increasing σ affect the distribution shape?

A larger σ increases variability and right-skewness. Small σ (e.g., 0.1) produces a distribution tightly concentrated near the median with light tails. Large σ (e.g., 1.5 or higher) creates heavy right-tail behaviour with frequent extreme values. For σ > 1, the mode becomes exp(μ − σ²), which can be much smaller than the mean, emphasizing the asymmetry.

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