Lognormal Distribution Calculator

Distributions and Statistical Models

Enter the parameters of the lognormal distribution to calculate its properties.

Examples

Explore some common scenarios for the lognormal distribution.

Standard Lognormal Distribution

Standard Case

A standard lognormal distribution with μ=0 and σ=1.

σ: 1, μ: 0

x: 1.5

Stock Price Analysis

Financial Modeling

Modeling a stock price with a given mean and volatility.

σ: 0.4, μ: 2

x: 10

Reliability Analysis

Engineering

Analyzing the time-to-failure of a component.

σ: 0.75, μ: 5

x: 100

Real Estate Price Modeling

Real Estate

Modeling real estate prices in a specific area.

σ: 0.2, μ: 12

x: 200000

Other Titles
Understanding Lognormal Distribution: A Comprehensive Guide
An in-depth look at the properties, applications, and calculations of the lognormal distribution.

What is the Lognormal Distribution?

  • Core Concept
  • Key Parameters
  • Relationship to Normal Distribution
The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. In other words, if a random variable X is lognormally distributed, then Y = ln(X) has a normal distribution. This distribution is widely used to model continuous random quantities that are always positive and have skewed distributions with long right tails.
Key Parameters
A lognormal distribution is defined by two parameters: the location parameter (μ, or log-scale) and the scale parameter (σ, or shape). These are the mean and standard deviation of the variable's logarithm, not the variable itself. Changing μ shifts the entire distribution, while changing σ alters its shape.
Relationship to Normal Distribution
The link between the lognormal and normal distributions is fundamental. If you take the natural logarithm of a lognormally distributed dataset, the resulting data will be normally distributed. This property is crucial for analysis and is the reason for the distribution's name.

Step-by-Step Guide to Using the Lognormal Distribution Calculator

  • Inputting Parameters
  • Interpreting the Results
  • Using the Examples
Inputting Parameters
To use the calculator, you need to provide three values: Scale (σ), Location (μ), and the x-value. 'Scale' is the standard deviation of the log data and must be positive. 'Location' is the mean of the log data. 'x-value' is the specific point at which you want to calculate the probability functions and must be non-negative.
Interpreting the Results
The calculator provides six key outputs: PDF (the probability of the variable taking a specific value x), CDF (the probability of the variable being less than or equal to x), and the distribution's Mean, Median, Mode, and Variance. These metrics provide a full picture of the distribution's characteristics.

Real-World Applications of Lognormal Distribution

  • Finance and Economics
  • Engineering and Reliability
  • Biology and Medicine
Finance and Economics
In finance, lognormal distributions are famously used to model stock prices, as a stock price cannot be negative and its changes are often multiplicative. It is a cornerstone of the Black-Scholes model for option pricing.
Engineering and Reliability
In reliability engineering, the time it takes for a material to fail under stress often follows a lognormal distribution. This is used to predict product lifespans and maintenance schedules.
Biology and Medicine
Many biological processes result in lognormally distributed quantities. Examples include the size of living tissue, the number of hairs on a person's head, and the latency period of infectious diseases.

Common Misconceptions and Correct Methods

  • Confusing Parameters
  • Interpreting the Mean
  • Assumption of Symmetry
Confusing Parameters (μ and σ)
A common mistake is to interpret μ and σ as the direct mean and standard deviation of the data. They are the mean and standard deviation of the logarithm of the data. The actual mean and variance of the distribution are calculated from these parameters, as shown in the results.
Assumption of Symmetry
Unlike the normal distribution, the lognormal distribution is not symmetric. It is positively skewed, with a long tail to the right. The mean, median, and mode are all different values, which is a key characteristic of its shape.

Mathematical Derivation and Formulas

  • Probability Density Function (PDF)
  • Cumulative Distribution Function (CDF)
  • Key Statistical Properties
Probability Density Function (PDF)
The formula for the PDF of a lognormal distribution is: f(x) = (1 / (x σ sqrt(2π))) exp(-(ln(x) - μ)² / (2 σ²)) for x > 0.
Cumulative Distribution Function (CDF)
The CDF is given by: F(x) = Φ((ln(x) - μ) / σ), where Φ is the CDF of the standard normal distribution.
Key Statistical Properties

Formulas:

  • Mean: E[X] = exp(μ + σ²/2)
  • Median: Med[X] = exp(μ)
  • Mode: Mode[X] = exp(μ - σ²)
  • Variance: Var[X] = (exp(σ²) - 1) * exp(2μ + σ²)