Calculator Inputs
Enter event data, exposure, and a null rate. The page calculates one-sided or two-sided exact probabilities and exact intervals.
Example Data Table
These examples show how the calculator behaves with different observed counts, exposures, null rates, and hypothesis directions.
| Observed Events | Exposure | Null Rate | Alternative | Observed Rate | Rate Ratio | P-Value |
|---|---|---|---|---|---|---|
| 14 | 10 | 1.0000 | Greater | 1.4000 | 1.4000 | 0.1355 |
| 3 | 10 | 0.5000 | Less | 0.3000 | 0.6000 | 0.2650 |
| 9 | 6 | 1.2000 | Two-sided | 1.5000 | 1.2500 | 0.4528 |
| 0 | 4 | 0.8000 | Less | 0.0000 | 0.0000 | 0.0408 |
Formula Used
The page treats the observed count as a Poisson random variable with mean equal to rate multiplied by exposure.
How to Use This Calculator
- Enter the observed number of events as a whole number.
- Enter the total exposure corresponding to the count.
- Enter the null rate expected per one exposure unit.
- Select whether your test is less, greater, or two-sided.
- Set alpha to match your desired significance threshold.
- Choose output precision and run the exact test.
- Review the result table, confidence interval, and decision.
- Download the results as CSV or PDF when needed.
Interpretation Notes
- Use this test for independent counts observed over fixed exposure.
- The null rate should reflect the expected baseline intensity.
- Small p-values indicate the observed count is unlikely under H0.
- The exact interval estimates plausible rates supported by the data.
- Two-sided values use a discrete probability ordering approach.
Frequently Asked Questions
1. What does this calculator test?
It tests whether an observed event count is consistent with a specified Poisson rate after accounting for the exposure amount. It returns exact probabilities rather than large-sample approximations.
2. When should I use a Poisson exact test?
Use it for count data with independent events, stable exposure, and a meaningful baseline rate. It is especially useful when counts are small and normal approximations are unreliable.
3. What is exposure in this context?
Exposure is the denominator linked to opportunity for events. It can be time, distance, population, machine hours, patient days, transactions, or any measurable observation base.
4. Why does the calculator show three p-values?
Lower-tail, upper-tail, and two-sided probabilities are all displayed for transparency. The selected alternative hypothesis determines which p-value is used for the formal test decision.
5. How is the exact confidence interval computed?
The interval is found by numerically solving Poisson tail equations for the mean count. Those mean limits are then divided by exposure to produce an exact rate interval.
6. What does the rate ratio mean?
The rate ratio compares the observed rate with the null rate. Values above one suggest an elevated rate, while values below one suggest a reduced rate.
7. Can I use this page for zero events?
Yes. Zero counts are valid in Poisson models. The page still computes the exact p-values and an upper confidence bound, while the lower bound remains zero.
8. Does a non-significant result prove rates are equal?
No. A non-significant result only means the data do not provide enough evidence against the null rate at the chosen alpha. It does not prove equality.