Log-Normal Distribution Mean, Median, Mode, Skewness & 95th Percentile Calculator

Compute Log-Normal mean e^(μ+σ²/2), median e^μ, mode e^(μ−σ²), and tail percentiles.

Log-Normal Expected Mean E[X]
39.646 units
Median / Geometric Mean (e^μ) 33.116 units
Distribution Mode (e^(μ − σ²)) 23.104 units
95th Percentile (e^(μ + 1.645σ)) 88.847 units
Inputs
⚡ Scenarios:
μ
Location parameter μ (mean of ln X).
σ
Scale parameter σ (standard deviation of ln X, >0).
Results & Graph
Log-Normal Expected Mean E[X]
39.646 units
Median / Geometric Mean (e^μ)
33.116 units
Distribution Mode (e^(μ − σ²))
23.104 units
95th Percentile (e^(μ + 1.645σ))
88.847 units

Log-Normal Expected Mean E[X]

Graph 1 • Log-Normal Distribution • Domain Architecture & Live Telemetry Maps Log Scale Mean Mu, Log Scale Std Dev Sigma through Log-Normal Distribution Formula.
Log-Normal Expected Mean E[X]: 39.646 units Log-Normal Distribution • Invariants & Metrics Log-Normal Expected Mean E[X 39.646 units Median / Geometric Mean (e^μ 33.116 units Distribution Mode (e^(μ − σ² 23.104 units 95th Percentile (e^(μ + 1.64 88.847 units
Graph 2 • Log-Normal Expected Mean E[X] Across Log Scale Mean Mu Sweep (40% → 160%) Evaluates Median = e^μ; Mean = e^(μ + σ²/2); Mode = e^(μ − σ²); Variance = ( across 7 operating points (current input highlighted).
0 units 178.1 units 356.1 units 4.86 1.40μ 9.78 2.10μ 19.69 2.80μ 39.65 3.50μ 79.84 4.20μ 160.8 4.90μ 323.8 5.60μ
Graph 3 • Joint Sensitivity: Log Scale Mean Mu × Log Scale Std Dev Sigma Compares Log-Normal Expected Mean E[X] when Log Scale Std Dev Sigma is shifted +25% (solid) vs. −25% (dashed).
0 units 68.95 units 137.9 units --- Log Scale Std Dev Sigma @ 75% 15.35 -30% 25.95 -15% 43.87 +0% 74.16 +15% 125.4 +30%
Summary: For ln(X) ~ N(3.5, 0.6²), geometric median is 33.115, mode is 23.104, arithmetic mean is 39.646, and 95th percentile is 88.846.
Summary: For ln(X) ~ N(3.5, 0.6²), geometric median is 33.115, mode is 23.104, arithmetic mean is 39.646, and 95th percentile is 88.846.
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Formula & Mathematical Methodology: Log-Normal Distribution Formula

Median = e^μ
Mean = eμ + σ²/2
Mode = eμ − σ²
Variance = (eσ² − 1) × e2μ + σ²
Median = e^μ; Mean = e^(μ + σ²/2); Mode = e^(μ − σ²); Variance = (e^(σ²) − 1) × e^(2μ + σ²)
  • logScaleMeanMu: Log Scale Mean Mu (μ)
  • logScaleStdDevSigma: Log Scale Std Dev Sigma (σ)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Log-Normal Distribution

Compute Log-Normal mean e^(μ+σ²/2), median e^μ, mode e^(μ−σ²), and tail percentiles.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: Median = e^μ; Mean = e^(μ + σ²/2); Mode = e^(μ − σ²); Variance = (e^(σ²) − 1) × e^(2μ + σ²).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Log-Normal Distribution via Median = e^μ; Mean = e^(μ + σ²/2); Mode = e^(μ − σ²); Variance = (e^(σ²) − 1) × e^(2μ + σ²).

Frequently Asked Questions

How does the Log-Normal Distribution Mean, Median, Mode, Skewness & 95th Percentile Calculator compute the solution?

It evaluates Median = e^μ; Mean = e^(μ + σ²/2); Mode = e^(μ − σ²); Variance = (e^(σ²) − 1) × e^(2μ + σ²) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Log-Normal Distribution Mean, Median, Mode, Skewness & 95th Percentile Calculator: inputs, method, and how to read the result

Compute Log-Normal mean e^(μ+σ²/2), median e^μ, mode e^(μ−σ²), and tail percentiles.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Log Scale Mean Mu: Location parameter μ (mean of ln X).
  • Log Scale Std Dev Sigma: Scale parameter σ (standard deviation of ln X, >0).

What the calculation includes

With the default values shown, Log-Normal Expected Mean E[X] is 39.646 units. The calculation follows Log-Normal Distribution Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Log Scale Mean Mu from 3.5 μ to 3.85 μ changes Log-Normal Expected Mean E[X] from 39.646 units to 56.261 units. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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