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# Kaplan-Meier curve

- Concept

A step curve showing what share of people remain free of the event as time passes. It drops one step at each event.

Updated Nov 5, 2026

## Key points

- S(t) = product of (1 − events / at risk) at each event time.
- The curve drops only at an event; censored people reduce the number at risk and are shown with ‘|’.
- Median survival: the first time at which S(t) ≤ 0.5.
- At the tail of the curve few people remain at risk and each step is large and imprecise, which is why the numbers-at-risk table is shown.
- Two curves are compared with the log-rank test; the size of the difference with the HR.
- A higher curve means better survival; the gap between curves at a given time is the absolute difference.

## What it is

The Kaplan-Meier (product-limit) estimator calculates the probability of remaining free of the event over time. At each time an event occurs, the probability of getting past it is (at risk − events) / at risk, and cumulative survival is the product of those probabilities. Censored people leave the at-risk group without lowering the curve and are usually shown as small tick marks.

## Worked example

- Data: 10 patients. Deaths at 2, 4, 5, 8 and 11 months; censored at 3, 6, 9, 12 and 12 months.
- Month 2: 10 at risk, 1 death: S = 9/10 = 0.900.
- Month 3: one censored (no step); 8 remain. Month 4: 1 death: S = 0.900 × 7/8 = 0.788.
- Month 5: 7 at risk, 1 death: S = 0.788 × 6/7 = 0.675.
- Month 6: one censored; 5 remain. Month 8: 1 death: S = 0.675 × 4/5 = 0.540.
- Month 9: one censored; 3 remain. Month 11: 1 death: S = 0.540 × 2/3 = 0.360.
- Median: first time with S ≤ 0.5, that is 11 months. Estimated survival at 10 months = 54%.
- If another group had 75% at 10 months, its curve would run above; the log-rank test would say whether the gap exceeds chance.
- Reading: each step reflects one death among those still at risk; the last steps rest on very few patients.

## Markers

- S(t)
- product limit
- |

## Common pitfalls

- Reading the right-hand end of the curve as precise when only two or three people remain at risk.
- Comparing curves at a single time point chosen after looking at them.
- Dropping the curve at a censoring time (censoring is not an event).

## Memory aid

‘A step for a death, a tick for a loss’: events lower the curve; censoring only marks it.

## Test yourself

Think of the answer, then tap the question to see it.

## How is Kaplan-Meier cumulative survival calculated?

By multiplying, at each event time, the probability of getting past it: (at risk − events) / at risk.

## What happens to the curve when a patient is censored?

It does not drop; the patient leaves the at-risk group and is usually marked with a small vertical tick.

## How is median survival read from the curve?

It is the first time at which the curve reaches or drops below 50%.

## Why is the right-hand end of a Kaplan-Meier curve unreliable?

Because few people remain at risk and each event makes a large step with a wide confidence interval.

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## Connections

## Related 4

- [Survival analysis and censoring](https://durgical.com/en/epidemiology/survival-analysis-and-censoring/)
- [Log-rank test](https://durgical.com/en/epidemiology/log-rank-test/)
- [Hazard ratio and the Cox model](https://durgical.com/en/epidemiology/hazard-ratio-and-the-cox-model/)
- [Actuarial life tables](https://durgical.com/en/epidemiology/actuarial-life-tables/)

[← All of epidemiology](https://durgical.com/en/epidemiology/)

Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.

Sources: [Empirical model fitting - distribution free (Kaplan-Meier) approach](https://www.itl.nist.gov/div898/handbook/apr/section2/apr215.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [Censoring](https://www.itl.nist.gov/div898/handbook/apr/section1/apr131.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [How do you compare reliability between two or more populations?](https://www.itl.nist.gov/div898/handbook/apr/section4/apr44.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).

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Representative model, for educational purposes only.

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Source: https://durgical.com/en/epidemiology/kaplan-meier-curve/
License: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Cite as: DurgiPedia (durgical.com), https://durgical.com/en/epidemiology/kaplan-meier-curve/