How this calculation works
An observed event count needs an exposure denominator before it becomes a rate. Zero events in limited exposure also leaves uncertainty about the underlying rate. This page reports an exact Poisson count interval scaled by entered exposure. The interval uses equal-tail Garwood limits rather than a symmetric normal approximation that can produce negative rate bounds. Counts and exposure remain separate inputs so their units can be checked.
Rate = events/exposure; Garwood bounds use chi square quantiles with 2k and 2(k+1) degrees of freedomA worked example
Zero events over one hundred exposure units gives an observed rate of zero. At ninety-five-percent equal-tail coverage, the upper endpoint is −ln(0.025)/100, about 0.0368888 per unit. Doubling exposure with the same zero count halves that upper rate endpoint. The additional observation changes what rates remain compatible with the experiment even though the point estimate stays zero.
Good to know
Overdispersion, clustering and time-varying rate are outside this calculation. An observed count rate is not automatically a per-mission reliability probability or a lifetime-distribution parameter. No causal explanation, safety threshold or automatic decision is supplied. A zero event count should never be interpreted as proof that future events are impossible.
Reference: McMaster University — exact Poisson confidence limits (opens in a new tab) ↗
What this calculation answers
An observed event count needs an exposure denominator before it becomes a rate. Zero events in limited exposure also leaves uncertainty about the underlying rate. This page reports an exact Poisson count interval scaled by entered exposure.
The interval uses equal-tail Garwood limits rather than a symmetric normal approximation that can produce negative rate bounds. Counts and exposure remain separate inputs so their units can be checked.
The Zero failure reliability demonstration sample plan calculator counts complete independent mission trials. Exposure-rate inference uses a different denominator and cannot treat partial time as a completed mission.
Use the right input conventions
Events must be a whole nonnegative count. Exposure can be fractional but must be positive, such as total device-hours or distance traveled. Use one consistent exposure definition across the complete count.
The model assumes a constant rate and a Poisson count. Several correlated events from the same incident or changing operating conditions may not behave like independent increments with one rate.
Interpret the result
The observed rate is count divided by exposure. Interval endpoints use chi-square quantiles, with a zero lower endpoint when no events were seen. A positive upper endpoint is still required in that case.
The confidence interval describes a procedure under the model, not a posterior probability that this particular fixed rate lies within the displayed endpoints. Exposure uncertainty is not included.
Work through the example
Zero events over one hundred exposure units gives an observed rate of zero. At ninety-five-percent equal-tail coverage, the upper endpoint is −ln(0.025)/100, about 0.0368888 per unit.
Doubling exposure with the same zero count halves that upper rate endpoint. The additional observation changes what rates remain compatible with the experiment even though the point estimate stays zero.
Where the model stops
Overdispersion, clustering and time-varying rate are outside this calculation. An observed count rate is not automatically a per-mission reliability probability or a lifetime-distribution parameter.
No causal explanation, safety threshold or automatic decision is supplied. A zero event count should never be interpreted as proof that future events are impossible.
A few questions, answered
Why is the upper bound positive with no observed events?
Finite exposure can occur without events even when the underlying rate is positive.
What changes in the comparison?
Double zero-event exposure and inspect the halved upper rate bound.