Poisson Confidence Interval Calculator
When you observe a count of rare events, such as defects, accidents or disease cases, the exact Poisson confidence interval gives a range for the underlying rate that is valid even for very small counts. This calculator uses the Garwood method based on chi-square quantiles, which never produces a negative lower limit and handles a count of zero correctly. Enter the observed count and your confidence level to get the lower limit, upper limit and interval width for the expected number of events. The chi-square quantiles are computed by inverting the regularized incomplete gamma function.
Exact Poisson interval formula
alpha = 1 - confidence level / 100
Lower = 0.5 * chi-square quantile(alpha/2, 2k); zero when k = 0
Upper = 0.5 * chi-square quantile(1 - alpha/2, 2k + 2)
Width = Upper - Lower
chi-square quantile(p, df) = 2 * inverse regularized gamma(df/2, p)
The count k must be a non-negative whole number; the confidence level must be between 0 and 100.
Poisson interval context
- The exact method is preferred over the normal approximation for small counts.
- It never returns a negative lower limit, unlike the count plus or minus normal approach.
- For k = 0 the lower limit is exactly zero and the upper limit is about 3.689 at 95 percent.
- For k = 10 the 95 percent interval is approximately 4.80 to 18.39.
- Divide both limits by your exposure to express the interval as a rate per unit time or area.
Poisson confidence interval: frequently asked questions
What is an exact Poisson confidence interval?
Given an observed count k of events, the exact (Garwood) confidence interval for the underlying Poisson rate uses chi-square quantiles. The lower limit is half the chi-square quantile at alpha/2 with 2k degrees of freedom, and the upper limit is half the chi-square quantile at 1 minus alpha/2 with 2k plus 2 degrees of freedom.
Why use the exact method instead of a normal approximation?
The normal approximation, count plus or minus a multiple of the square root of the count, is poor for small counts and can give negative lower limits. The exact chi-square method is valid for any count, including zero, and never produces a negative rate, so it is preferred when counts are small.
What happens when the observed count is zero?
When k is zero the lower limit is exactly zero, because you cannot have a negative rate. The upper limit is half the chi-square quantile at 1 minus alpha/2 with 2 degrees of freedom, which for 95 percent confidence is about 3.689.
How do I set the confidence level?
Enter the confidence level as a percentage, such as 95. The calculator converts this to alpha equals 1 minus level/100 and splits it equally between the two tails. Higher confidence levels produce wider intervals.
Can I turn the interval into a rate per unit time?
Yes. This calculator gives the interval for the expected count. If your count was observed over a known exposure such as person-years or area, divide both limits by that exposure to express the confidence interval as a rate per unit.
Official sources
- U.S. NIST/SEMATECH e-Handbook of Statistical Methods: Engineering Statistics Handbook, Poisson and chi-square distributions.
- NIST Digital Library of Mathematical Functions: Incomplete Gamma Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.