Understand the relationship
The reasoning behind the result
From counts to survivorship
lᵢ = nᵢ/n₀
Dividing the number alive at each observation by the original cohort count gives the survivorship fraction. Baseline survivorship is one. The curve cannot rise for a completely observed closed cohort, so increasing counts are rejected.
This model requires no new entrants and no missing or censored outcomes. If disappearance might be failed detection or loss to follow-up, a reduction in recorded count cannot automatically be treated as mortality.
Interval fractions use the interval's starting survivors
qᵢ = (nᵢ − nᵢ₊₁)/nᵢ; pᵢ = nᵢ₊₁/nᵢ
The loss fraction q refers to individuals alive at the start of that interval, not to the original cohort. Its complement p is conditional survival through the interval. Both are undefined when the starting count is zero.
For unequal intervals, these fractions cover different durations. A ten-day fraction should not be compared with a one-day fraction as if they were rates per day.
Integrating the curve requires an interval assumption
Restricted mean duration ≈ ΣLᵢ/n₀
The trapezoid uses the average of the starting and ending counts multiplied by the interval length. It assumes a linear decline between observations, equivalent to using midpoint timing for the interval losses in this summary.
Summing these cohort-time areas and dividing by baseline count gives an approximate mean duration alive during the observed window. If survivors remain at the final time, this is explicitly restricted follow-up duration; no unobserved tail is invented.
Remaining observed-window time is conditional
Tᵢ = Σⱼ≥ᵢLⱼ; eᵢ,restricted = Tᵢ/nᵢ
T adds the recorded cohort-time from the start of an interval to the observation horizon. Dividing by that interval's starting survivor count gives conditional mean time alive within the remaining observed window, under the same trapezoidal assumption.
This is a retrospective cohort summary, not an individual life expectancy or prognosis. The end of observation remains a boundary even when the table uses traditional life-table symbols.
Follow the numbers
A three-day complete-count record
- Counts at days 0, 1, 2 and 3 are 100, 80, 50 and 0. Survivorship is 1, 0.8, 0.5 and 0.
- Trapezoidal cohort-time is 90 + 65 + 25 = 180 individual-days.
- Dividing by 100 original individuals gives 1.8 days of mean duration alive over the recorded window.
The 1.8-day estimate uses midpoint timing between observations; exact individual loss times were not entered.
Quick guide
How to use this calculator
- Enter complete counts from one closed cohort, starting at baseline time zero.
- Use the actual observation times; intervals need not be equal.
- Inspect the interval loss and survival columns, including undefined states after extinction.
- Read integrated duration as a trapezoidal approximation restricted to the last observation, not a future lifespan forecast.
Calculation method
Calculation and interpretation
Explain how a cohort's repeated complete counts form a survivorship curve and interval life table.
lᵢ = nᵢ/n₀; dᵢ = nᵢ − nᵢ₊₁; qᵢ = dᵢ/nᵢ; interval cohort-time Lᵢ = Δtᵢ(nᵢ + nᵢ₊₁)/2
Worked example
A three-day complete-count record
The 1.8-day estimate uses midpoint timing between observations; exact individual loss times were not entered.
lᵢ = nᵢ/n₀; dᵢ = nᵢ − nᵢ₊₁; qᵢ = dᵢ/nᵢ; interval cohort-time Lᵢ = Δtᵢ(nᵢ + nᵢ₊₁)/2
Supported inputs
Precision and limits
Descriptive cohort records
A recorded fraction does not establish a biological cause or predict an individual's outcome. Keep counting definitions, observation period and missing-status rules consistent. No treatment, husbandry or management action is prescribed.
Continue calculating
Related calculators
