Evaluate exact Poisson count tests with reliable statistics. Compare observed events, expected rates, and evidence using simple inputs and clear interpretation.
| Observed Count | Exposure | Null Rate | Alpha | Alternative |
|---|---|---|---|---|
| 12 | 10 | 1.0 | 0.05 | Two-sided |
| 4 | 8 | 0.9 | 0.05 | Less |
| 19 | 12 | 1.1 | 0.01 | Greater |
This sample shows how observed event totals, exposure, and a benchmark rate combine in an exact Poisson significance test.
The calculator tests whether an observed count is consistent with a Poisson model under a null rate.
Expected count under the null: μ0 = λ0 × t
Poisson probability: P(X = x) = e-μ μx / x!
Lower-tail test: P(X ≤ x)
Upper-tail test: P(X ≥ x)
Two-sided exact test: sum probabilities for counts with probability less than or equal to the observed probability under the null model.
An exact test for Poisson distribution helps analysts evaluate count data without relying only on normal approximations. This is valuable when observed events are rare, samples are small, or exposure differs across studies. Many operational and scientific datasets record counts such as defects, calls, incidents, arrivals, or failures. A Poisson model is often the starting point for those processes.
This calculator compares an observed event count against a null event rate. It converts the null rate and exposure into an expected count under the hypothesis. Then it computes the exact probability structure from the Poisson distribution. The output includes lower-tail, upper-tail, and exact two-sided evidence, depending on the alternative hypothesis you choose.
Use this method when your response variable is a nonnegative count and events are assumed independent over a defined exposure window. It is helpful in data science, reliability monitoring, queue analysis, quality control, public health tracking, and digital operations reporting. Exact methods are also useful when counts are low and approximation error can distort the decision.
A small p-value suggests the observed count is unlikely under the null rate. If the p-value is below alpha, the calculator recommends rejecting the null hypothesis. If it is larger, the result does not provide strong enough evidence for rejection. The expected count helps show the benchmark level, while the sample rate helps compare observed performance to the target rate.
Always define exposure carefully. Exposure may represent hours, visits, machine cycles, user sessions, or geographical units. Rate interpretation depends on that choice. Two-sided testing checks for any difference, while one-sided testing checks only for increases or decreases. Combine the statistical result with context, data quality review, and domain assumptions before making decisions.
It tests whether an observed count is consistent with a Poisson process under a specified null rate and exposure value using exact probability calculations.
Use it for discrete count data such as defects, incidents, arrivals, or failures when events are independent and measured over a known exposure period or area.
Exact testing is more reliable when counts are small or moderate. Approximate z methods can be less accurate in sparse event settings.
Exposure is the observation base. It may be time, population, distance, area, transactions, sessions, or another unit linked to event opportunity.
The null rate is the benchmark event frequency per one unit of exposure. The calculator multiplies it by exposure to get the expected count.
It sums probabilities for all counts that are at least as unlikely as the observed count under the null Poisson distribution.
It means the sample does not provide enough evidence against the null rate at your chosen alpha level. It does not prove the null is true.
Yes. Use the CSV button to save tabular output. Use the PDF button to open a print-friendly view and save it as a PDF.