Five events among twenty subjects. "Events over everyone" gives 25.0000%; Kaplan-Meier gives a cumulative incidence of 30.1202%. Median survival is never reached, which is why reporting a mean survival time from data like this is badly wrong.
20 subjects, 5 events, 15 censored
Median survival not reached
Kaplan-Meier puts survival at 69.8798% by the end of follow-up, so the cumulative incidence is 30.1202%. Counting events over everyone gives 25.0000% — 5.120 percentage points lower, because it treats every censored subject as a permanent survivor. The curve never reaches 0.5, so there is no median — and a mean survival time computed from these numbers would be meaningless.
Survival at end
69.880%
at t = 30.000
Naive events / n
25.000%
censored counted as survivors
Median survival
not reached
first time S(t) ≤ 0.5
Restricted mean
24.2387
area under the curve to t = 30.000
The survival curve
The life table: at-risk count, events, censored and survival at every observed time
Time
At risk
Events
Censored
Interval survival
S(t)
95% CI
3.0000
20
1
0
0.950000
0.950000
0.8545 – 1.0000
5.0000
19
0
1
1.000000
0.950000
0.8545 – 1.0000
6.0000
18
1
0
0.944444
0.897222
0.7622 – 1.0000
8.0000
17
0
1
1.000000
0.897222
0.7622 – 1.0000
9.0000
16
1
0
0.937500
0.841146
0.6757 – 1.0000
11.0000
15
0
1
1.000000
0.841146
0.6757 – 1.0000
12.0000
14
0
1
1.000000
0.841146
0.6757 – 1.0000
14.0000
13
1
0
0.923077
0.776442
0.5811 – 0.9718
15.0000
12
0
1
1.000000
0.776442
0.5811 – 0.9718
17.0000
11
0
1
1.000000
0.776442
0.5811 – 0.9718
18.0000
10
1
0
0.900000
0.698798
0.4713 – 0.9263
20.0000
9
0
1
1.000000
0.698798
0.4713 – 0.9263
21.0000
8
0
1
1.000000
0.698798
0.4713 – 0.9263
22.0000
7
0
1
1.000000
0.698798
0.4713 – 0.9263
24.0000
6
0
1
1.000000
0.698798
0.4713 – 0.9263
25.0000
5
0
1
1.000000
0.698798
0.4713 – 0.9263
26.0000
4
0
1
1.000000
0.698798
0.4713 – 0.9263
28.0000
3
0
1
1.000000
0.698798
0.4713 – 0.9263
30.0000
2
0
2
1.000000
0.698798
0.4713 – 0.9263
The curve steps down only at EVENT times. A censored subject leaves the at-risk column without moving the curve — which is exactly why the later steps are larger: each one is a share of a smaller risk set.
Censoring must be independent of prognosis Greenwood standard errors, widening as the risk set shrinks A median that is not reached is a result, not a gap
What this tool shows
Five events among twenty subjects with fifteen censored: counting events over everyone gives 25.0000%, and Kaplan-Meier gives a cumulative incidence of 30.1202%. The naive figure understates the risk by 5.12 percentage points, because it treats every subject who left the study event-free as a permanent survivor. On the same data the median is never reached — which is why a mean survival time computed from those numbers would be meaningless.
The Kaplan-Meier product-limit estimate with the full life table
Greenwood standard errors and 95% confidence limits at every step
Median survival, reported as "not reached" when the curve never crosses 0.5
Restricted mean survival time — the area under the curve to a stated horizon
The naive events-over-everyone figure beside it, so the gap is visible
The at-risk column, which is where censoring actually acts
Product-limit Greenwood limits Naive count beside it Median or "not reached"
Not medical advice. Censoring must be unrelated to prognosis.
Updated 13 September 2026 · Works in any browser, no installation
The Kaplan-Meier estimator gives the probability of surviving past each time, using every subject for as long as they were actually observed. A subject who left the study event-free at month 6 contributed six months of information and then stopped; they are neither an event nor a survivor, and handling that correctly is the entire point. The estimate steps down only at event times, and censoring merely removes someone from the at-risk column.
At a glance
Formula shown
S(t) = Π over event times tᵢ ≤ t of (nᵢ − dᵢ)/nᵢ, where nᵢ is the number still at risk just before tᵢ and dᵢ the number of events at it. Censored subjects reduce n for later intervals without contributing a d. Greenwood’s formula gives the variance: Var(S(t)) = S(t)²·Σ dᵢ/(nᵢ(nᵢ − dᵢ)), which is why the confidence band widens as the risk set empties.
Scenario support
Clinical trial survival and time to progression, customer churn and subscription retention, time to equipment failure, employee tenure, time to relapse or readmission, and any outcome measured as "how long until" where some subjects have not had it yet.
Educational estimate
Planning support from the values you enter — not professional advice.
Censoring is not missing data, and it is not a survivor
The reason survival analysis exists as a separate field is that a subject who is still event-free when last seen carries real information that neither obvious summary uses correctly.
Counting events over everyone treats them as permanent survivors. On the shipped preset that gives 25.0000% where the Kaplan-Meier cumulative incidence is 30.1202% — an understatement of 5.12 points, and the gap grows with the amount of censoring.
Discarding them entirely throws away everything they contributed. A subject followed for 28 of 30 months without an event tells you a great deal about the first 28 months, and dropping them biases the estimate the other way.
Kaplan-Meier uses them for exactly as long as they were observed. They sit in the at-risk column until they leave, and their departure moves no step — it only makes the NEXT step larger, because each subsequent event is a bigger share of a smaller risk set.
Which is visible in the table as the interval-survival column. Early events divide by a large risk set and barely move the curve; late events divide by a handful and drop it sharply.
The extreme case makes it unarguable. Five subjects, one event: if the event comes FIRST — 5, then four censored — Kaplan-Meier and the naive count both give 0.2. If the same single event comes LAST — four censored, then 20 — the naive count still says 0.2 and Kaplan-Meier says 1. Identical counts, a fivefold difference, purely from the ordering.
The assumption that does the work: independent censoring
Kaplan-Meier makes almost no distributional assumptions. It makes one structural assumption, and it is untestable from the data.
Censoring must be unrelated to prognosis. A subject censored at month 6 must have the same future risk as a subject still under observation at month 6.
Administrative censoring satisfies it. Everyone still event-free when the study closed is censored for a reason that has nothing to do with how they were doing — which is why the tool’s fourth preset is the least troubling kind.
Dropout usually does not. Patients who feel worse withdraw more often, so they are censored precisely because their risk was higher. The estimate is then optimistic, and no amount of data fixes it.
Nothing in the dataset can detect this, because the whole point is that what happened after censoring is unobserved. The usual response is a sensitivity analysis: recompute assuming every censored subject had the event immediately, and again assuming none of them ever did, and see whether the conclusion survives both.
"Not reached" is a result, and a mean is usually wrong
Two summary numbers get misused here, and the first one is misused by being computed at all.
The mean survival time cannot be estimated from censored data. It needs the whole curve down to zero, and when the curve stops at 0.6987 the area beyond is unknown. Averaging the observed times — censored ones included — is simply wrong, and biased downward.
The median is the first time survival reaches or drops below 0.5. It is a point on the curve, not an average, and it is robust to the unobserved tail because the curve has already crossed it.
When the curve never reaches 0.5 there IS no median, and the tool says “not reached” rather than inventing one. That is informative: more than half the cohort was still event-free at the end of follow-up.
Restricted mean survival time is the principled alternative. It is the area under the curve out to a stated horizon — 24.2387 to month 30 on the shipped preset — and it is well defined whether or not a median exists, provided the horizon is stated.
Reporting survival at fixed timepoints is the other good option. “Survival at 12 months was 82%” is checkable against the table and needs no median at all.
The confidence band widens as the risk set empties
The interval at the right-hand end of a Kaplan-Meier curve is usually far wider than it looks, and Greenwood’s formula says exactly why.
Variance accumulates as a sum over event times of d/(n(n−d)), so each interval adds a term that grows as n shrinks. The band cannot narrow, only widen.
The tail is estimated from very few subjects. A curve that starts with 200 people may have 6 left at the last event, and the survival estimate there is a statement about those six.
Which is why the number-at-risk row belongs under every published curve. Without it a reader cannot tell whether a late separation between two arms rests on a hundred subjects or on three.
The plain Greenwood interval can also run outside 0 to 1, which the tool clips. Log-log transformed intervals avoid that and are the better choice when the estimate is near either boundary.
Comparing two curves is a separate question
Eyeballing two Kaplan-Meier curves is where most misreadings of survival data happen, because the visual gap and the statistical evidence are different things.
The log-rank test is the standard comparison, accumulating observed minus expected events across every event time. It is the most powerful test when the hazard ratio is roughly constant.
And it is close to blind when the curves cross. Early and late differences cancel in the running total, so two arms that differ enormously in opposite directions can produce a p-value near 0.5.
A hazard ratio summarises the comparison as one number, which requires the ratio to be roughly constant over time — the proportional-hazards assumption, and the thing most worth checking before quoting one.
Comparing survival at a single timepoint is a different test again, and choosing that timepoint after seeing the curves is a selection that no p-value accounts for.
Reporting a Kaplan-Meier analysis
Four things, and the first is the one most often missing from a published figure.
Print the number at risk under the curve. It is the only way a reader can judge how much the right-hand end of the plot is worth.
Give the median with its confidence interval, or say it was not reached. A median quoted without an interval implies a precision that a step function rarely has.
Report how many were censored and why. Administrative censoring at study close and dropout are different situations, and only the second threatens the analysis.
And give survival at pre-specified timepoints. They are checkable against the table, they do not depend on a median existing, and picking them in advance avoids choosing the timepoint that flatters the result.
Sources and methodology
References for the product-limit estimator and its variance.
Method. The estimate is the product over event times of (n−d)/n, with censored subjects leaving the risk set between steps rather than contributing one — which is what makes the naive comparison the tool prints beside it meaningful rather than rhetorical: on the shipped preset the two differ by 5.12 percentage points on identical data. Greenwood’s variance is accumulated across intervals and the limits clipped to [0, 1]. Restricted mean survival is the exact area under the step function to the last observed time, computed as a sum of rectangles rather than by approximation. The median is reported as the first time the curve reaches or drops below 0.5, and as “not reached” when it never does — the suite asserts that a cohort whose curve stays above 0.5 returns null rather than the largest observed time. Fewer than two subjects, a negative time and a cohort with no events at all all return no result. That engine is verified on every change against 87 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Log-Rank TestObserved minus expected accumulated at every event time, so the crossing-hazards blind spot is visible: a running total that peaks at +3.2652 and ends at -1.4717 gives p = 0.458065.
Hazard RatioBoth standard formulas from one log-rank output — Peto one-step 3.1899 and O/E 2.6554 — with a note on which of them is guaranteed to agree with the p-value beside it.
Life TableThe actuarial estimator with the effective-at-risk column shown and Kaplan-Meier beside it: ten-unit intervals cost 1.3753 percentage points, one-unit intervals cost exactly zero.
Exponential DistributionTime-until-event probabilities from a rate or a mean, always showing both — with a panel proving elapsed time changes nothing, which is a constant hazard rate and false for anything that wears out.
Weibull DistributionReliability probabilities with the hazard curve drawn beside the density — because the shape parameter is the hazard story: below 1 the risk falls, at 1 it is exactly an exponential, above 1 it rises.
Incidence RateEvents per person-time with the exact Poisson interval, including at zero events: the exact two-sided upper limit is 3.688879 per unit of person-time, not the 3 the one-sided rule of three gives.
An educational tool, not medical advice. The Kaplan-Meier estimate assumes censoring is unrelated to prognosis — an assumption nothing in the data can test, and one that dropout routinely breaks in the optimistic direction. A mean survival time cannot be estimated from censored data, and the tail of any survival curve rests on however few subjects remain at risk there.
Published a product-limit survival tool with the full life table, Greenwood confidence limits, median and restricted mean survival.
Printed the naive events-over-everyone count beside the estimate so the cost of mishandling censoring is a number rather than a warning: five events among twenty subjects give 25.0000% by the plain count and a 30.1202% cumulative incidence by Kaplan-Meier.
Sharpened it to an extreme case in the verification suite and on the page: five subjects and ONE event. If the event comes first the two methods agree at 0.2; if the same single event comes last, the naive count still says 0.2 and Kaplan-Meier says 1. Identical counts, a fivefold difference, purely from the ordering.
Reported 'not reached' when the curve never crosses 0.5 rather than inventing a median, and gave restricted mean survival with its horizon as the alternative — since a mean survival time cannot be estimated from censored data at all.
Verified that with nothing censored every step equals the plain running proportion still alive, and that survival never rises, stays inside [0,1] and keeps its confidence limits there across 120 generated cohorts.
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