Poisson Confidence Interval Calculator

This calculator gives a confidence interval for a Poisson count, the range within which the true average rate of events is likely to lie, based on an observed number of events. The Poisson distribution models counts of events that happen independently at a constant average rate: the number of customers arriving in an hour, defects per batch, accidents per month, decay events per second, or rare occurrences over time or space. When you observe such a count, it is only an estimate of the true underlying rate, and a confidence interval expresses how precise that estimate is. Because Poisson counts have a known relationship between their mean and their variability, the interval can be built directly from the observed count. This tool does it. You enter the observed number of events and choose a confidence level, and the calculator returns the point estimate, which is simply the observed count, along with the lower and upper bounds of the confidence interval and the margin of error. The results update as you type. Use it to put error bars on a count, to report a rate with its uncertainty, for quality control and reliability, or for statistics study. It uses the normal approximation, where the interval is the count plus or minus a z-value for the confidence level times the square root of the count, since the variance of a Poisson count equals its mean. A wider confidence level, such as 99 percent, gives a wider interval. A key insight the calculator shows is that the relative precision improves as counts grow: because the margin depends on the square root of the count, a larger count has a proportionally narrower interval. The normal approximation works well for counts that are not too small, roughly 20 or more; for very small counts, exact methods based on the chi-square distribution are more accurate, so treat small-count intervals with care.

Normal approximation: count plus or minus z x square root of count, since a Poisson variance equals its mean. Reliable for counts of about 20 or more; use exact methods for small counts.