Lognormal Distribution
Print PageA Lognormal Distribution Calculator (also known as a Log-Normal Probability Calculator, Lognormal CDF & PDF Utility, Geometric Mean Generator, or Black-Scholes Asset Price Probability Analyzer) computes exact Probability Density Function values (f(x) = [1 ÷ (x · σ √[2π])] · exp( -[&ln;(x) - μ]2 ÷ [2σ2] )), Cumulative Distribution Function probabilities (F(x) = Φ[ (&ln;x - μ) ÷ σ ]), expected arithmetic mean (E[X] = eμ + σ2/2), median (x0.5 = eμ), mode (xmode = eμ - σ2), variance, and geometric standard deviation (GSD = eσ) for variables whose natural logarithms follow a normal distribution (Y = &ln;(X) ∼ N[μ, σ2]).
In quantitative finance, stock price movements, income distribution analysis, and environmental aerosol particle sizing, the lognormal distribution models positive random variables that are right-skewed and bounded at zero, where multiplicative growth forces produce log-symmetric curves.
Our free online Lognormal Distribution Calculator provides instant calculations across all continuous log-transformed parameters:
- Probability Density Function (PDF for x > 0):
f(x; μ, σ) = [ 1 ÷ (x · σ √[2π]) ] · e-( &ln;x - μ )2 ÷ ( 2σ2 ). - Cumulative Distribution Function (CDF / P[X ≤ x]):
F(x; μ, σ) = Φ[ ( &ln;x - μ ) ÷ σ ]. - Expected Arithmetic Mean (E[X]):
E[X] = eμ + σ2 ÷ 2. - Median Lifespan / Value (x0.5 = GM):
x0.5 = eμ = Geometric Mean (GM). - Mode / Peak Density Value (xmode):
xmode = eμ - σ2. - Geometric Standard Deviation (GSD):
GSD = eσ. - Variance (Var[X]):
Var[X] = ( eσ2 - 1 ) · e2μ + σ2.
Master Lognormal Reference Table (μ = 4.0 log-$, σ = 0.50 log-SD Stock Price)
The table below displays geometric metrics, modal peaks, arithmetic means, and cumulative price probabilities for a stock asset price model with log-location μ = 4.0 and log-scale σ = 0.50:
| Lognormal Metric Name | Mathematical Formula | Calculated Value (μ=4.0, σ=0.50) | Z-Score Transformation Inactive | Financial & Real-World Meaning |
|---|---|---|---|---|
| Mode (Peak Density Price) | eμ – σ2 = e4.0 – 0.25 | $42.52 per share | Z = -0.50 | Most Frequently Occurring Single Price |
| Median (50th Percentile GM) | eμ = e4.0 | $54.60 per share | Z = 0.00 | 50% Probability Above / 50% Below |
| Arithmetic Mean (Expected E[X]) | eμ + σ2/2 = e4.125 | $61.87 per share | Z = +0.25 | Expected Average Portfolio Value |
| Geometric Standard Deviation (GSD) | GSD = eσ = e0.50 | 1.6487 Factor (× / ÷) | Log Scale Factor | Multiplicative Dispersion Range ($33.12 to $90.02) |
| Lower Risk CDF (P[X ≤ $40.00]) | Φ[ (&ln;40 – 4.0) ÷ 0.50 ] | 26.69% Probability | Z = -0.6222 | Downside Risk Probability Below $40 |
| Upper Gain CDF (P[X > $100.00]) | 1 – Φ[ (&ln;100 – 4.0) ÷ 0.50 ] | 11.31% Probability | Z = +1.2103 | Upside Bull Probability Above $100 |
Step-by-Step Financial Asset Price Calculation
To evaluate a tech stock whose future share price follows a lognormal distribution with log-mean μ = 4.0 and log-standard deviation σ = 0.50:
Step 1 (Calculate Geometric Mean / Median Price): Median = e^4.0 = 54.598 ≈ $54.60
Step 2 (Calculate Mode / Peak Density Price): Mode = e^(4.0 - 0.50^2) = e^(4.0 - 0.25) = e^3.75 = 42.521 ≈ $42.52
Step 3 (Calculate Expected Arithmetic Mean E[X]): E[X] = e^(4.0 + 0.50^2 / 2) = e^(4.0 + 0.125) = e^4.125 = 61.868 ≈ $61.87
Step 4 (Calculate Probability Share Price Drops Below $40.00 P[X ≤ 40]): &ln;(40) = 3.68888 &implies; Z = (3.68888 - 4.0) ÷ 0.50 = -0.31112 ÷ 0.50 = -0.6222 &implies; Φ(-0.6222) = 0.2669 ≈ 26.69%
Step 5 (Calculate Probability Share Price Surpasses $100.00 P[X > 100]): &ln;(100) = 4.60517 &implies; Z = (4.60517 - 4.0) ÷ 0.50 = +0.60517 ÷ 0.50 = +1.2103 &implies; 1 - Φ(1.2103) = 1 - 0.8869 = 0.1131 ≈ 11.31%
Thus, while the median price is $54.60, the right-skewed tail elevates the expected arithmetic mean to $61.87, with an 11.31% chance of reaching $100+.
Continuous Distributions Comparison: Lognormal vs. Normal vs. Weibull
Below is a comparative reference chart detailing when to use the Lognormal distribution versus related continuous probability models:
| Probability Distribution | Domain & Skewness Behavior | Underlying Mechanism | Primary Practical Application |
|---|---|---|---|
| Lognormal Distribution | Strictly Positive (x > 0), Right-Skewed | MULTIPLICATIVE Growth Processes | Stock prices, wealth distribution, particle sizes. |
| Normal (Gaussian) Distribution | All Real Numbers (-&infty; to +&infty;), Symmetric | ADDITIVE Independent Effects (Central Limit) | Measurement errors, adult heights, test scores. |
| Weibull Distribution | Strictly Positive (x ≥ 0), Flexible Skew | Extreme Value Wear-Out Models | Material fatigue, wind speed forecasting, reliability. |
History & Mathematics: 1879 Francis Galton to 1973 Black-Scholes
1879 Francis Galton & Donald McAlister
In 1879, English polymath Francis Galton observed that psychological and biological variations follow logarithmic normal curves, and mathematician Donald McAlister derived the formal lognormal probability density function in Proceedings of the Royal Society.
1931 Robert Gibrat & Law of Proportional Growth
In 1931, French economist Robert Gibrat formulated Gibrat’s Law of Proportional Growth, proving that multiplicative expansion drives corporate firm sizes into lognormal distributions.
1973 Fischer Black & Myron Scholes
In 1973, economists Fischer Black and Myron Scholes incorporated lognormal asset price assumptions into the Nobel Prize-winning Black-Scholes options pricing model.
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Frequently Asked Questions (FAQ)
What is the Lognormal Distribution CDF formula?
The Cumulative Distribution Function is F(x) = Φ[ (&ln;x - μ) ÷ σ ].
How do Arithmetic Mean and Median differ in a Lognormal distribution?
The median is Median = eμ (equal to the Geometric Mean), whereas the arithmetic mean is elevated by variance to E[X] = eμ + σ2/2.
Why are stock prices modeled using a Lognormal distribution?
Stock prices cannot fall below zero (bounded at 0) and compound multiplicatively over time, making lognormal curves far superior to standard normal curves for financial modeling.