Log Normal Survival Calculator

Calculator Form

Example Data Table

Example inputs use μ = 3.1 and σ = 0.45.

What This Calculator Does

A log normal survival model fits positive time data where the logarithm of time follows a normal distribution. This pattern often appears in lifetimes, response durations, income data, maintenance delays, and right-skewed operational measures. The calculator estimates survival probability at a chosen time and also builds a range table for interpretation.

Survival analysis asks a direct question: what is the probability that an event has not happened by time t? In this model, that quantity is the survival function S(t). The page also computes failure probability F(t), density f(t), hazard h(t), and cumulative hazard H(t). These measures describe different viewpoints of the same distribution.

The log mean μ shifts the lifetime scale, while the log standard deviation σ controls spread and skewness. Larger values of σ typically create heavier right tails and wider dispersion. The percentile tool is useful when you need a time threshold such as the 90th percentile for planning, reliability targets, or risk buffers.

The range section helps compare time points instead of checking only one value. This makes the calculator useful for inspection intervals, warranty analysis, service contracts, stockout timing, replacement policies, and threshold studies. The chart makes pattern reading easier because survival, failure probability, hazard, and density change at different rates across time.

The export tools support quick reporting. CSV is useful for spreadsheets and model checks. PDF is useful for sharing a clean result snapshot with tables and the graph area. Everything stays in one page, so the workflow remains simple for analysts, students, and operational teams.

Formula Used

For a positive time value t, let Z = (ln(t) - μ) / σ.

• Failure Probability: F(t) = Φ(Z)
• Survival Function: S(t) = 1 - Φ(Z)
• Density: f(t) = exp(-(ln(t)-μ)^2 / (2σ^2)) / (tσ√(2π))
• Hazard: h(t) = f(t) / S(t)
• Cumulative Hazard: H(t) = -ln(S(t))
• Mean: exp(μ + σ²/2)
• Median: exp(μ)
• Mode: exp(μ - σ²)
• Percentile: x(p) = exp(μ + σΦ⁻¹(p))

Here, Φ is the standard normal cumulative distribution function and Φ⁻¹ is its inverse.

How to Use This Calculator

1. Enter the analysis time where you want the survival estimate.
2. Provide the log mean μ and log standard deviation σ.
3. Enter a percentile probability, such as 0.90 for the 90th percentile.
4. Set range start, range end, and range step for the table and chart.
5. Press Calculate Survival to show results above the form.
6. Review summary metrics, the range table, and the Plotly chart.
7. Use CSV for spreadsheet work and PDF for sharing results.

Frequently Asked Questions

1) When should I use a log normal survival model?

Use it when time or size data are positive and strongly right-skewed. It is common in reliability, waiting times, biological measures, and financial duration studies.

2) What does survival probability mean here?

It is the probability that the event has not happened by time t. A higher value means more observations are expected to remain beyond that time.

3) Why must time be positive?

The model uses the natural logarithm of time. Since logarithms require positive inputs, zero and negative values are not valid for log normal calculations.

4) What is the difference between survival and failure probability?

They are complements. Failure probability is F(t), while survival is S(t) = 1 - F(t). Together they describe event status by time t.

5) What does the hazard output show?

Hazard estimates the event rate at a specific time, conditional on surviving up to that point. It helps compare how instantaneous risk changes across the timeline.

6) Why does the calculator ask for μ and σ?

Those are the parameters of the normal distribution after taking logs. They control location, spread, skewness, and all derived survival metrics.

7) What is the percentile result used for?

It gives the time below which a chosen proportion of outcomes falls. This is helpful for design limits, service targets, and planning thresholds.

8) Can I export the calculated results?

Yes. CSV exports tables for spreadsheet analysis, while PDF saves a report-style snapshot of the calculated section for sharing or documentation.