Ten-unit intervals on heavily censored data: the actuarial estimate is 0.712551 against Kaplan-Meier’s 0.698798, a gap of 1.3753 percentage points. Half the withdrawals in each interval are being credited with five units of exposure they may not have had.
4 intervals, 20 subjects
Survival at the end: 71.2551%
Kaplan-Meier on the same data gives 69.8798%, a difference of 1.3753 percentage points. That gap is the cost of the half-withdrawal assumption: the actuarial method credits every withdrawal with half an interval of exposure, and Kaplan-Meier uses the exact time instead.
Actuarial survival
71.255%
4 intervals
Kaplan-Meier
69.880%
exact times, no grouping
Gap
1.3753 pp
the assumption's cost
Median survival
not reached
to the end of an interval
The actuarial life table, with the effective number at risk in each interval
Interval
Entering
Events
Withdrawn
Effective at risk
q
p
S(t)
0.000 – 10.000
20
3
2
19.000
0.157895
0.842105
0.842105
10.000 – 20.000
15
2
4
13.000
0.153846
0.846154
0.712551
20.000 – 30.000
9
0
7
5.500
0.000000
1.000000
0.712551
30.000 – 40.000
2
0
2
1.000
0.000000
1.000000
0.712551
The effective at-risk column is where the assumption lives: entering minus HALF the withdrawals. A subject who left on the first day of an interval and one who left on the last are treated identically, and the wider the interval the more that costs.
Kaplan-Meier on the same records uses 19 distinct event and censoring times rather than 4intervals, and needs every individual time to do it — which is exactly what a grouped registry table does not have.
Works on grouped data, which KM cannot Withdrawals credited with half an interval Narrower intervals converge to Kaplan-Meier
What this tool shows
On one dataset, ten-unit intervals put survival at 0.712551 where Kaplan-Meier puts it at 0.698798 — a gap of 1.3753 percentage points. Two-unit intervals bring it to 0.697876, a gap of 0.0922. One-unit intervals — narrow enough that no interval holds more than one distinct observed time — give exactly Kaplan-Meier. The gap IS the half-withdrawal assumption, and it can be measured rather than warned about.
The actuarial life table with the effective-at-risk column shown explicitly
Conditional death and survival probabilities per interval, and the cumulative curve
Kaplan-Meier on the same records, with the gap between them as a number
Interval boundaries you choose, so the effect of width is visible
The median, reported to the end of the interval in which the curve crosses 0.5
The one situation the actuarial method handles and Kaplan-Meier cannot
Interval-based Effective at risk Gap from KM measured Works on grouped data
Use Kaplan-Meier when you have the individual times.
Updated 13 September 2026 · Works in any browser, no installation
The actuarial life table estimates survival by dividing follow-up into intervals and computing, for each, the chance of surviving it given you entered it. Because events and withdrawals are pooled within an interval, it needs a rule for how much exposure a withdrawal contributed — and the standard rule credits half. That is the one assumption Kaplan-Meier does not need, and the reason the two estimates differ.
At a glance
Formula shown
For each interval: effective at risk n' = entering − withdrawn/2; conditional death q = events/n'; conditional survival p = 1 − q; and cumulative survival S = Π p over the intervals so far. The withdrawn/2 is the whole assumption — a subject who left on the first day of an interval and one who left on the last are treated identically, so the wider the interval the more that costs.
Scenario support
Registry and census data reported as counts per year, insurance and pension mortality tables, historical cohort studies where only grouped counts survive, product warranty returns by month, and any source that reports events and withdrawals per period without individual times.
Educational estimate
Planning support from the values you enter — not professional advice.
The half-withdrawal rule, and what it costs
Within an interval the method cannot know when a withdrawal happened. It assumes the middle, and the tool puts a number on that assumption by computing Kaplan-Meier alongside.
Ten-unit intervals on the shipped data: 0.712551 against 0.698798. A gap of 1.3753 percentage points, and the actuarial estimate is the optimistic one because it credits withdrawals with more exposure than several of them had.
Two-unit intervals: 0.697876, a gap of 0.0922 points. One-unit intervals, which on integer data means no interval holds more than one distinct time: a gap of exactly zero.
The actuarial estimator IS Kaplan-Meier once the intervals are narrow enough. That is not an approximation claim — with at most one distinct time per interval, withdrawn/2 has nothing to act on and the two formulas coincide.
The convergence is not monotone, though. Five-unit intervals give a gap of 1.4045 points, slightly WORSE than ten-unit. Which times fall either side of a boundary matters as much as the width, so narrowing helps on average rather than at every step.
When to use it rather than Kaplan-Meier
With individual times in hand, Kaplan-Meier is strictly better. The actuarial method exists for everything else.
Grouped data is the main case. A registry that publishes deaths and withdrawals per year, a historical cohort whose records survive only as annual counts, a warranty system reporting monthly totals — none of those can produce a Kaplan-Meier curve at all.
Very large cohorts are the other. A Kaplan-Meier table with 200,000 distinct event times is unusable as a table, and grouping into years loses almost nothing when each interval contains thousands of events.
Standard mortality tables are actuarial by construction, which is why insurance and pension work uses this form — and why the qₓ column in a published life table is exactly the conditional death probability this tool computes.
Where individual times exist, grouping is a loss with no compensating gain. On the tool’s fourth preset it costs 11.1440 percentage points, which is a great deal to pay for a tidier table.
Choosing interval boundaries changes the answer
The boundaries are an analyst decision that the output depends on, and they should be chosen before looking at the survival pattern.
Narrower is more accurate and less stable. An interval containing two subjects and one event gives a conditional survival of 0.5 on almost no information, and a chain of such intervals produces a jagged curve.
Wider is smoother and more biased. The half-withdrawal assumption has more room, and the cumulative estimate drifts — upward when withdrawals cluster early in intervals, downward when they cluster late.
The usual guidance is enough events per interval to be stable — five or ten — with equal widths so the table reads as a rate.
Choosing boundaries after seeing where the events fell is the trap. It is the same selection problem as choosing a histogram bin width to make a peak appear, and it has no p-value to correct.
Three assumptions, and only one is unique to this method
Two of them it shares with every survival estimator. The third is the one this page is about.
Censoring must be unrelated to prognosis. Shared with Kaplan-Meier, untestable from the data, and broken by informative dropout in the optimistic direction.
The hazard must be roughly constant within an interval. Shared with any grouped method. A ten-year interval spanning a period when risk changed sharply describes an average that nobody experienced.
Withdrawals must be uniform within the interval. Unique to this method, and the one the effective-at-risk column implements. Administrative censoring at a study close violates it badly — everyone withdraws on the same day, which is never the middle.
That last one has a fix when the timing is known: use the actual person-time instead of entering minus half the withdrawals, which is the exact version of the same idea and is what an incidence rate does.
The median is only as precise as the interval
A life table cannot locate the median more precisely than its own grid, which is a resolution limit rather than an estimation error.
The tool reports the END of the interval in which cumulative survival first reaches 0.5. With ten-unit intervals that is a ten-unit window, not a point.
Linear interpolation within the interval is the common refinement, and it assumes survival falls linearly inside it — which is a second assumption stacked on the first.
Kaplan-Meier gives an exact median because it steps at actual event times, so the crossing point is one of the observed times rather than a bracket.
And a curve that never reaches 0.5 has no median either way, which the tool reports as “not reached” rather than as the largest boundary.
Reporting a life table
Four things, and the second is what lets a reader recompute anything.
Give the interval boundaries. The estimate depends on them, so a table without them is not reproducible.
Give entering, events and withdrawn per interval. Three integers a row, and every other column follows from them.
Say why intervals were used. Grouped source data is a good reason; tidiness when individual times exist is not, and costs real accuracy.
And report the median as an interval, not a point, unless you interpolated — in which case say so, because that is an extra assumption about what happens inside the window.
Method. Each interval’s effective at-risk count is entering minus half the withdrawals, and the conditional survival is accumulated multiplicatively — which is the classical Berkson-Gage form rather than a rearrangement of it. Kaplan-Meier is computed from the same records by the same code that powers the Kaplan-Meier page, so the gap the tool prints is a like-for-like comparison rather than two implementations being contrasted. That comparison is what makes the central claim exact rather than cautionary: ten-unit intervals on the shipped data give 0.712551 against 0.698798 (+1.3753 points), two-unit intervals 0.697876 (−0.0922), and one-unit intervals — below the spacing of the observed times — give exactly Kaplan-Meier, because withdrawn/2 has nothing to act on when no interval holds more than one distinct time. The suite asserts that identity rather than describing it. Fewer than two subjects, fewer than two boundaries, and negative times 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:
Kaplan-MeierSurvival with censoring handled, and the naive count printed beside it: five events in twenty subjects give 25.0000% by the plain count and a 30.1202% cumulative incidence by Kaplan-Meier.
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.
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.
Standardized Mortality RatioSMR with the expected count built stratum by stratum, and the small-count disagreement shown: 5 observed against 2.07 expected gives p = 0.0417 and an exact interval of 0.7843 to 5.6369.
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.
An educational tool, not medical or actuarial advice. The actuarial estimator assumes withdrawals are spread uniformly within each interval — administrative censoring at a study close breaks that badly, since everyone withdraws on the same day. Where individual follow-up times exist, Kaplan-Meier needs no such assumption and should be preferred.
Published the actuarial estimator with the effective-at-risk column shown explicitly, since entering-minus-half-the-withdrawals is where the method's only unique assumption lives.
Priced that assumption instead of describing it, by computing Kaplan-Meier from the same records through the same code the Kaplan-Meier page uses: ten-unit intervals give 0.712551 against 0.698798, a gap of 1.3753 percentage points; two-unit intervals give 0.697876, a gap of 0.0922.
Established the exact convergence: one-unit intervals — narrow enough that no interval holds more than one distinct observed time — reproduce Kaplan-Meier to machine precision, because withdrawn/2 then has nothing to act on. The suite asserts that identity rather than the page describing it.
Recorded honestly that the convergence is NOT monotone: five-unit intervals give a gap of 1.4045 points, slightly worse than ten-unit, because which times fall either side of a boundary matters as much as the width.
Set out the one situation where this method is the only option — data that arrives already grouped, with no individual times — and measured what grouping costs when it is not: 11.1440 percentage points on the shipped registry-style preset.
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