The formula
A variable X is lognormal when ln(X) is normal. Take the normal PDF, substitute the log, and carry the 1/x from the chain rule:
f(x) = 1/(x·σ·√(2π)) · exp( −(ln x − μ)² / (2σ²) ) for x > 0
Here μ and σ are the mean and standard deviation of the logarithm, not of x itself. That distinction is where most mistakes start.
The cumulative distribution is just the normal CDF run on ln x:
F(x) = Φ( (ln x − μ) / σ )
so the fraction failed by time x is "how many log-sigmas is ln x above μ," read straight off a z-table.
Because the shape lives in log space, the useful summary numbers split into two camps. The ones that are clean in log space:
median = e^μ (also the geometric mean)
mode = e^(μ − σ²)
And the ones that carry the skew, in linear space:
mean = e^(μ + σ²/2)
variance = ( e^(σ²) − 1 ) · e^(2μ + σ²)
The mean always sits above the median, and the gap grows fast with σ. For σ = 0.8, the mean is e^(0.32) = 1.377 times the median — a 38% upward bias if you confuse the two.
The multiplicative reading, which is the one worth keeping:
σ* = e^σ (geometric standard deviation, a factor, not an offset)
A one-sigma interval isn't median ± something, it's median ×/ σ* — divide on the low side, multiply on the high side. With a median life of 1000 h and σ = 0.8, the 68% band runs 1000/2.23 to 1000·2.23, i.e. roughly 449 h to 2226 h. The 95% band (±1.96σ) runs about 208 h to 4797 h. Verified numerically.
Where you meet it
- At the fatigue test stand. Run twenty identical coupons at the same stress amplitude and they don't fail at the same cycle count — they scatter over a factor of five or ten, skewed long. Cycles-to-crack is a multiplicative story (a small flaw grows a small percentage each cycle), so the log of life plots straight and the median-life plus
σsummary is standard S-N practice. - At the reliability review, arguing repair time. Mean-time-to-repair hides a long tail: most fixes are quick, a few drag out because the part is buried or the tech has to fair a harness. Time-to-repair is classically lognormal, and MIL-HDBK-style maintainability math is built on it. Quoting an arithmetic mean MTTR without the
σ*factor undersells how bad the 90th-percentile job really is. - In semiconductor life prediction. Electromigration, corrosion, and diffusion-driven wearout all advance by increments proportional to how far the damage has already gone — the textbook multiplicative-degradation setup — so the failure times come out lognormal with a median
T50and shapeσ. Accelerated-life models (Black's equation, Arrhenius) shiftT50;σusually rides along nearly constant. - In an RF link margin. Shadowing — the slow fade from terrain and buildings blocking a path — is lognormal in linear power, which is the same as saying it's normal in dB. That's why coverage margins get budgeted in dB with a Gaussian standard deviation: the "log" in lognormal is the dB axis.
How it works
The engine is the multiplicative central limit theorem. Add many small independent random effects and the sum tends to normal; multiply them and the product tends to lognormal, because the log of a product is a sum of logs and that sum goes normal [1][3]. Any process built from proportional steps — percentage growth, damage that compounds, gains cascaded down a chain — lands here by default, the same way additive noise lands on the Gaussian.
That single idea explains the distribution's whole personality. It's strictly positive (a product of positive factors can't cross zero), it's right-skewed (a run of above-average multipliers pushes the tail out far while a run of below-average ones only asymptotes toward zero), and it's closed under multiplication: multiply two independent lognormals and you get another lognormal, with σ_total = √(σ₁² + σ₂²) in log space. That is the exact RSS rule from tolerance stacking, just relocated to the log axis.
Now the gotchas, in the order they bite.
μ and σ are log-space parameters. e^μ is the median, not the mean. Fit a lognormal, read μ, exponentiate, and call it the average — and you've reported the median while thinking you reported the mean, low by a factor of e^(σ²/2). For a repair-time study with σ = 0.8 that error is 38%.
Back-transforming the mean of the logs isn't the mean. The clean way to estimate parameters is to take logs, average them, take the SD, and back-transform. But exp(mean of logs) gives you the geometric mean — the median — not the arithmetic mean of the raw data. If a contract pays against arithmetic mean throughput, you owe the bias correction × e^(σ²/2). Skipping it is the single most common lognormal accounting error.
Lognormal and Weibull impersonate each other on small samples. Both are positive, skewed, two-parameter life models, and with 15 or 20 failures you usually can't tell them apart from a fit — the NIST reliability handbook says as much [2]. They diverge exactly where you can't afford it: far out in the tails, and in the hazard rate. That's the next trap.
The hazard rate is not monotonic. Weibull can give you a hazard that rises forever (classic wearout). The lognormal hazard rate rises, peaks, and then falls — it eventually decreases with age. Physically that says "if a unit has already survived well past the median, its instantaneous failure rate is dropping." For a genuine wearout mechanism that's wrong, and picking lognormal purely for a clean fit can hand you an optimistic end-of-life prediction. Match the distribution to the physics, not just to the plot.
The tail is a factor, not an offset. Because scatter is multiplicative, the same σ that gives a tidy ×/1.5 band at a median of 100 hours gives a ×/1.5 band at a median of 100,000 hours — a 50,000-hour spread. Intuition calibrated on additive ± error badly underestimates the reach of a lognormal tail. Sanity-check with a lognormal probability plot: straight line means the model holds; curvature at the ends means it doesn't.
History
The distribution was born from a complaint about averages. In 1879 Francis Galton — the English polymath who also gave statistics regression and correlation — argued in a short note to the Royal Society that for many biological and social quantities the geometric mean, not the arithmetic mean, was the natural measure of the typical value, because those quantities combine by multiplication rather than addition [1][3][4]. His paper "The geometric mean, in vital and social statistics" ran in the Society's Proceedings, and immediately after it, on the same pages, came the mathematical follow-through [3][4].
That follow-through was Donald McAlister's "The law of the geometric mean," Proceedings of the Royal Society volume 29, 1879 [1][3][4]. McAlister worked out the density, the median, and the dispersion measures of the distribution you get when the logarithm is normal — the first real mathematical treatment. To this day the median-and-shape parameterization engineers reach for traces back to his memoir [3].
The mechanical intuition came from an astronomer. Jacobus Kapteyn — the Dutch stellar-statistics pioneer better known for Kapteyn's Star — published Skew Frequency Curves in Biology and Statistics in 1903 and built a physical machine, a modified Galton board with triangular pins whose widths grow across the board, so that each bounce multiplies a particle's position instead of nudging it, and the piles settle into a lognormal instead of a bell [3]. It made the multiplicative mechanism something you could watch.
The economists gave it a law. Robert Gibrat, in his 1931 Les Inégalités économiques, showed that if things grow by a percentage independent of their current size — the "law of proportionate effect" — their sizes end up lognormal [1][5]. Firm sizes, city sizes, and incomes all fell in line, and Gibrat's law is still how economists explain the shape. The engineering canon was fixed later by Aitchison and Brown's 1957 monograph The Lognormal Distribution, which pulled the theory together and is the citation reliability texts still lean on [1][3].
Related tools
- /tools/bearing-life-l10 — bearing L10 rating life is the classic Weibull application (Lundberg-Palmgren, ISO 281); the lognormal-vs-Weibull trap in the gotchas above is exactly the choice that bites here
- /tools/arrhenius-rate — accelerated-life testing shifts the lognormal median
T50with temperature while the shapeσholds roughly constant - /tools/half-life-decay — the neighboring model for proportional, multiplicative change over time