# Lognormal distribution A lognormal distribution describes outcomes that can't go below zero but have no upper limit - and where many small multiplicative factors drive the result. Task durations, project costs, company valuations. The practical insight: the median is lower than the mean, often significantly so. Estimates based on "most likely" systematically underpredict the average. --- ## The shape **Bounded below, unbounded above.** Values can't go negative, but there's no ceiling. **Right-skewed.** A long tail of high values pulls the mean above the median. **Multiplicative effects.** Outcomes driven by many small multiplicative factors tend toward lognormal. The median-mean gap matters most in practice. A rough rule of thumb: **mean ≈ 1.6 × median** for moderate variability. If someone gives you a "most likely" estimate (the median), the expected average outcome is roughly 60% higher. Tasks can finish slightly early but can run very late. You can't complete a two-week task in negative time, but you can absolutely complete it in six weeks. Early completions are bounded. Delays compound - one problem often reveals another. --- [[Estimates]] builds on this directly. The essay's core argument is that estimates and expectations are different things that require different numbers. The person doing the work gives you a median - their best guess of the most likely outcome. The manager planning across many tasks needs the mean - what to expect on average. The executive making external commitments needs a high percentile. The lognormal shape explains why these diverge and by how much. [[Business maths]] uses the distribution as part of its quantitative foundations. Once you accept that most business variables are lognormally distributed, several common planning failures become predictable: schedules that always slip (because you planned to the median), budgets that always overrun (same reason), and "stretch targets" that are hit less often than leaders expect (because the right tail is longer than intuition suggests). [[Confidence]] extends this to the presentation of numbers. Giving a point estimate for a lognormally distributed variable is actively misleading - it hides the asymmetry. State ranges and be explicit about which percentile you're quoting.