Lognormal Distribution Calculator
Free lognormal distribution calculator. PDF, CDF, mean, median, mode, variance, percentiles. Income,
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Lognormal — PDF, CDF for ln(X) ~ N(μ,σ²)
Models income, stock prices, particle sizes. Right-skewed, positive-only. Mean, median, mode, percentiles.
Real-World Scenarios — Click to Load
Inputs
PDF Curve
CDF Curve
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • If ln(X) ~ N(μ, σ²), then X follows a lognormal distribution — the logarithm of X is normal
- • Lognormal is right-skewed and positive-only — ideal for income, prices, sizes, durations
- • Mean = exp(μ + σ²/2), Median = exp(μ), Mode = exp(μ − σ²) — mean > median > mode for σ > 0
- • P(a ≤ X ≤ b) = Φ((ln(b)−μ)/σ) − Φ((ln(a)−μ)/σ) — use standard normal CDF
- • Percentile q: x_q = exp(μ + σ × Φ⁻¹(q)) — inverse normal gives the value
Did You Know?
How It Works
1. The Log-Normal Connection
X is lognormal iff ln(X) is normal. So μ and σ are the mean and SD of ln(X), not of X itself.
2. The PDF Shape
Right-skewed, starts at 0, peaks at the mode, and has a long right tail — no negative values.
3. Mean vs Median
Mean > median because of the right tail. High outliers pull the mean up more than the median.
4. CDF via Standard Normal
F(x) = Φ((ln(x)−μ)/σ). Transform to z-score in log-space, then use normal CDF.
5. Percentiles
x_q = exp(μ + σ × Φ⁻¹(q)). Inverse normal gives z, then exponentiate to get the value.
Expert Tips
When to Use Lognormal
Positive data, right-skewed, multiplicative effects (e.g., growth rates)
Parameter Interpretation
μ and σ are for ln(X). Median = exp(μ). Larger σ = more skew.
Lognormal vs Normal
Lognormal is for positive multiplicative data; normal for additive symmetric data
Fitting from Data
Estimate μ = mean(ln(x)), σ = sd(ln(x)) from your sample
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | Excel | R | Manual |
|---|---|---|---|---|
| PDF + CDF charts | ✅ | ⚠️ Requires chart | ⚠️ Requires plot | ❌ |
| Shaded P(a≤X≤b) | ✅ | ❌ | ❌ | ⚠️ Manual |
| Percentiles | ✅ | ✅ LOGNORM.INV | ✅ qlnorm | ⚠️ Complex |
| Mean, median, mode | ✅ | ❌ Manual | ❌ Manual | ✅ |
| 7 presets | ✅ | ❌ | ❌ | ❌ |
Frequently Asked Questions
When is data lognormally distributed?
When the logarithm of the data is normally distributed. Common for positive, right-skewed data: income, prices, sizes, durations, particle diameters.
What is the difference between μ and the mean of X?
μ is the mean of ln(X). The mean of X is exp(μ + σ²/2), which is always greater than exp(μ) = median when σ > 0.
Why is the lognormal distribution right-skewed?
Because it is the exponential of a normal. The exponential stretches the right tail — a few large values can be very large.
How do I fit a lognormal to my data?
Take the natural log of each observation. Compute mean and SD of ln(x). Those are μ and σ. Check normality of ln(x) with Q-Q plot.
What is the relationship to the normal distribution?
X ~ Lognormal(μ,σ²) iff ln(X) ~ Normal(μ,σ²). The lognormal is the exponentiation of a normal.
Can the lognormal have zero or negative values?
No. The support is (0, ∞). The PDF is zero for x ≤ 0.
When should I use lognormal vs exponential?
Lognormal has a peak and long tail; exponential decays from a peak at 0. Use lognormal for sizes, prices; exponential for memoryless waiting times.
How does σ affect the shape?
Larger σ increases skewness and variance. Small σ makes the lognormal look more like a normal (after log transform).
Lognormal by the Numbers
Official Data Sources
Disclaimer: This calculator uses standard normal CDF/inverse approximations. Results are for educational and professional reference. For critical applications (financial risk, reliability), verify against established statistical software.
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