Lognormal distribution calculator

The formulas

f(x) = 1 / (xσ√(2π)) · e^{−(ln x − μ)² / (2σ²)}, x > 0 F(x) = Φ((ln x − μ) / σ) x_p = e^{μ + σ z_p}, z_p = Φ⁻¹(p)

mean = e^{μ + σ²/2} median = e^μ mode = e^{μ − σ²} variance = (e^σ² − 1) e^{2μ + σ²}

Every lognormal probability is a normal probability in disguise: take the log of x, standardise it with μ and σ, and look up Φ.

Worked example

Repair times at a service desk are lognormal with μ = 2 and σ = 0.5 (log-hours). What share of repairs finish within 10 hours?

z = (ln 10 − 2) / 0.5 = (2.30259 − 2) / 0.5 = 0.60517 P(X ≤ 10) = Φ(0.60517) = 0.727467

About 72.7% of repairs take 10 hours or less, and 27.3% take longer. The typical repair (the median) takes e² = 7.389 hours, the average is e^2.125 = 8.3729 hours, and the standard deviation is 4.46225 hours.

For a service-level target, ask for the 95th percentile instead: z_0.95 = 1.64485, so x = e^{2 + 0.5 × 1.64485} = e^2.82243 = 16.8176 hours. One repair in twenty takes longer than that.

When the lognormal fits

A quick check on real data: take the natural log of every value and look at the result. If the logs look roughly symmetric and bell-shaped, the lognormal is a reasonable model, and the mean and standard deviation of the logs are your estimates of μ and σ. A ratio of mean to median well above 1 in the raw data is another hint; for a lognormal it equals e^σ²/2, which is 1.133 in the example above. If the data include zeros, the lognormal cannot fit them as they stand.

Reading the result

On the chart the peak sits well left of the mean and the right tail runs on for a long way. That shape is why reports of lognormal data should quote the median, or the geometric mean and geometric SD, rather than the arithmetic mean ± SD. Mean ± 2 SD here would be 8.37 ± 8.92, which runs below zero even though no repair can take negative time.

Common mistakes

• Entering the mean and SD of X as μ and σ. They are parameters of ln X; use the second input mode if you have the raw-scale figures.
• Using log base 10. The formulas use the natural log. If your μ and σ come from log₁₀ data, multiply both by ln 10 = 2.302585.
• Assuming e^μ is the mean. It is the median; the mean is larger by a factor of e^σ²/2.
• Entering x ≤ 0. A lognormal variable is always positive, so P(X ≤ 0) = 0.