How to use the poisson rate confidence interval calculator
- Enter the number of events observed, a whole number from 0 up.
- Enter the time or area observed. Use 1 if you only want the interval for the count.
- Optionally name the rate, for example calls per hour, and choose the confidence level.
- Read the exact interval for the count and the interval for the rate.
Formula
Lower limit = chi-square(alpha/2; 2k) / 2, and it is 0 when k = 0
Upper limit = chi-square(1 - alpha/2; 2k + 2) / 2
Rate interval = the count interval divided by the time or area observed
Here k is the observed count and chi-square(q; df) is the chi-square point with a share q below it
This is the exact (Garwood) interval. It is built to cover the true value at least as often as the stated level, so it is a little wide.
Five events, twelve calls in eight hours, and zero events
Observing 5 events gives a 95 percent interval for the underlying count of 1.6235 to 11.668. Twelve calls in 8 hours gives a count interval of 6.2006 to 20.962 and a rate of 0.77507 to 2.6202 calls per hour.
Observing no events in 30 days gives an interval from 0 to 3.6889 for the count, so the rate is at most 0.12296 per day with 95 percent confidence. This matches the rule of three, which says that after seeing nothing the upper bound is about 3 divided by the time.
Exact or approximate
The common shortcut of the count plus or minus 1.96 times its square root works well when the count is large, but it fails for small counts and gives a lower limit below zero for counts under about 4. The exact interval handles any count, including zero. For small counts it is noticeably wider and lopsided, which is the honest picture of the uncertainty.
To find the chance of a number of events at a known rate, use the Poisson distribution calculator. For the waiting time until several events, see the gamma distribution calculator.
Checking the assumptions
The Poisson model needs events to happen independently at a steady rate. Clusters, rush hours and contagion break that, and the interval is then too narrow. When exposure differs between groups, compare rates and not counts. For an interval on a mean or a proportion instead, see the confidence interval calculator.
Assumptions and limits
- Events happen independently at a constant rate during the whole observation.
- The count is a whole number, and the time or area is above 0.
- The interval is exact and two-sided, with equal probability in each tail.
- The rate interval assumes the exposure is known exactly.
Frequently asked questions
How do I find a confidence interval for a Poisson count?
Use chi-square points: half the lower point for 2k degrees of freedom and half the upper point for 2k + 2. For 5 events at 95 percent it is 1.6235 to 11.668.
What if I observed zero events?
The lower limit is 0 and the upper limit is about 3.69 for 95 percent confidence, so the rate is at most about 3.69 divided by the time observed.
How is this different from the normal approximation?
The exact interval is valid for small counts. The normal approximation is poor when the count is small.
Can I get a rate?
Yes. Enter the time or area observed, and the page divides the interval by it.