Calculator Inputs
This calculator assumes natural logarithms. Enter μ and σ for ln(X), then evaluate density, cumulative chance, tail chance, interval probability, and percentiles.
Example Data Table
The following example uses μ = 1.1, σ = 0.45, x = 3.5, interval [2, 5], and percentile p = 90%.
| Metric | Example Value | Meaning |
|---|---|---|
| PDF at x = 3.5 | 0.239114 | Relative density around 3.5. |
| CDF at x = 3.5 | 0.632873 | Probability that X is 3.5 or less. |
| Right-tail probability | 0.367127 | Probability that X exceeds 3.5. |
| Range probability [2, 5] | 0.688234 | Probability that X falls between 2 and 5. |
| 90th percentile | 5.347869 | Value below which 90% of outcomes lie. |
| Mean | 3.324270 | Expected value of the lognormal distribution. |
| Median | 3.004166 | Middle point of the distribution. |
| Mode | 2.453462 | Most likely location of the density peak. |
Formula Used
Let Y = ln(X). If Y ~ N(μ, σ²), then X follows a lognormal distribution.
1) Probability density function
f(x) = [1 / (xσ√(2π))] × exp(- (ln(x) - μ)² / (2σ²)), for x > 0
2) Cumulative distribution function
F(x) = Φ((ln(x) - μ) / σ)
3) Right-tail probability
P(X > x) = 1 - F(x)
4) Interval probability
P(a ≤ X ≤ b) = F(b) - F(a)
5) Percentile or quantile
xp = exp(μ + σΦ-1(p))
6) Summary measures
Mean = exp(μ + σ² / 2)
Median = exp(μ)
Mode = exp(μ - σ²)
Variance = (exp(σ²) - 1) × exp(2μ + σ²)
How to Use This Calculator
- Enter the location parameter μ for the normal distribution of ln(X).
- Enter the scale parameter σ. It must be positive.
- Provide an x value to measure density, cumulative chance, and tail probability.
- Enter lower and upper bounds to evaluate interval probability.
- Enter a percentile in percent form, such as 90 or 95.
- Choose the number of decimal places for your displayed outputs.
- Press the calculate button to show results directly above the form.
- Use the CSV and PDF buttons to export your final results.
Frequently Asked Questions
1) What is a lognormal distribution?
A lognormal distribution models positive values whose natural logarithms are normally distributed. It appears in growth processes, waiting times, prices, sizes, and other right-skewed data.
2) When should I use a lognormal model?
Use it when the variable cannot be negative and the data are strongly right-skewed. It is useful for multiplicative effects, compounding, and values that vary across several scales.
3) What do μ and σ represent here?
They belong to the normal distribution of ln(X), not to X itself. μ controls location on the log scale, while σ controls spread and skewness.
4) Why must x be greater than zero?
The lognormal distribution is only defined for positive values because it uses the natural logarithm of x. Zero or negative inputs are outside the model.
5) What is the difference between PDF and CDF?
The PDF shows relative density at one point. The CDF gives the cumulative probability that the variable is less than or equal to a chosen value.
6) What does the percentile output mean?
The percentile output gives the x value below which a chosen proportion of outcomes fall. The 90th percentile means 90% of values are at or below it.
7) Can I use percentages like 90 instead of 0.90?
Yes. This calculator accepts percentile entries in percent form. If you enter 90, it automatically converts the value to 0.90 for quantile calculations.
8) Why are the mean, median, and mode different?
A lognormal distribution is skewed to the right. That skew pulls the mean highest, keeps the median in the middle, and places the mode nearer the density peak.