Counts and Rates
Calculate an exact confidence interval for a Poisson event rate using the Garwood method. Enter the observed number of events, total exposure, and desired confidence level. The calculation runs entirely in your browser.
Let x be the observed number of events and T be the total exposure or time at risk. The maximum likelihood estimate of the Poisson rate is λ̂ = x/T.
This calculator uses the exact two-sided Garwood confidence interval. Rather than relying on a normal approximation, the interval is obtained from the relationship between the Poisson and chi-square distributions.
Here, 1 − α is the requested confidence level, and χ2p,ν denotes the p-th quantile of a chi-square distribution with ν degrees of freedom. When no events are observed (x = 0), the lower confidence limit is defined as zero.
Poisson event counts are discrete, so the normal approximation can perform poorly when the observed event count is small. The Garwood interval is based directly on the Poisson distribution and provides conservative coverage for the nominal confidence level. It also remains well-defined when the observed event count is zero.
The estimated rate is the number of events divided by the total exposure. The units of the resulting rate are determined entirely by the units used for exposure. For example, 14 events over 400 person-years gives an estimated rate of 0.035 events per person-year.
Suppose 14 events are observed during 400 person-years of exposure and a 95% confidence interval is required. The estimated rate is 14 / 400 = 0.035 events per person-year. The exact Garwood interval is approximately 0.019135 to 0.058724 events per person-year.
Garwood, F. (1936). Fiducial Limits for the Poisson Distribution. Biometrika, 28(3/4), 437–442. DOI: 10.1093/biomet/28.3-4.437.
The worked example used for validation is also independently documented by StatsDirect: 14 events over 400 person-years, producing an exact 95% confidence interval of 0.019135 to 0.058724 events per person-year.