# Lognormal Distribution ## The Idea in Brief A lognormal distribution describes outcomes that can't go below zero but have no upper limit — and where the logarithm of the outcome follows a normal distribution. Task durations, income distributions, stock prices, and city populations all follow this pattern. The key practical insight: the median (middle value) is lower than the mean (average), often significantly so. This asymmetry explains why intuitive estimates systematically underpredict outcomes. --- ## Key Concepts ### 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 For lognormal distributions, the mean is always higher than the median. The relationship depends on the variance, but a rough rule of thumb: - **Mean ≈ 1.6 × Median** for moderate variability This means if someone gives you a "most likely" estimate (the median), the expected average outcome is roughly 60% higher. ### Why Task Estimates Follow Lognormal 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. The asymmetry is structural: - Early completions are bounded (you can only save so much time) - Delays compound (one problem often reveals another) - Unknowns are discovered, not eliminated --- ## Implications **Estimates are medians, expectations are means.** When someone estimates a task at two weeks, they're giving you the median — the most likely duration. But if you're planning across many tasks, you need the mean. The gap between them is where schedules slip. **The long tail isn't a planning failure.** Tasks that run 2-3× over estimate aren't anomalies — they're part of the distribution. Treating them as failures to be eliminated misunderstands the shape of uncertainty. **Different stakeholders need different percentiles.** The person doing the work thinks about the median (the realistic target). The manager needs the mean (what to expect on average). The executive making commitments needs a higher percentile (80th-90th) to avoid overcommitting. --- ## Sources - Statistical foundations in probability theory - Practical applications in software estimation (PERT, Monte Carlo simulation) - Financial modelling (stock price movements, option pricing) --- ## See in Notes - [What Estimates Mean](https://www.anishpatel.co/what-estimates-mean/) — Why "two weeks" becomes three: the gap between median estimates and mean expectations