Enter test inputs
Example data table
| Scenario | Observed count | Exposure | Null rate | Alternative | Interpretation target |
|---|---|---|---|---|---|
| Customer complaint review | 18 | 12 months | 1.10 | Greater | Check whether complaints increased. |
| Machine failure audit | 6 | 8 weeks | 1.00 | Less | See whether failures dropped. |
| Clinic incident tracking | 14 | 10 shifts | 1.20 | Two-sided | Test any meaningful rate difference. |
Use the table as a template for operational, medical, safety, or quality count data where events arise over measurable exposure.
Formula used
The calculator performs a one-sample Poisson rate test. It compares your observed event count against the count expected under a stated null rate.
Expected count under H₀: λ₀ = r₀ × E
Here, r₀ is the hypothesized rate per exposure unit, and E is total exposure.
Observed rate: r̂ = k / E
The exact one-sided tail probabilities come from the Poisson distribution:
Lower tail: P(X ≤ k | λ₀)
Upper tail: P(X ≥ k | λ₀)
For a two-sided result, this page uses doubled tail probability:
Two-sided p-value = min(1, 2 × min(lower tail, upper tail))
The interval shown for the event rate is the exact Garwood interval. It first estimates limits for the Poisson mean, then divides them by exposure.
Lower mean limit = 0.5 × χ²(α/2, 2k)
Upper mean limit = 0.5 × χ²(1 − α/2, 2(k + 1))
Rate interval = mean interval / E
How to use this calculator
- Enter the observed number of events.
- Enter total exposure, such as hours, days, patients, or units.
- Enter the hypothesized event rate per single exposure unit.
- Choose whether you want a two-sided, upper-tail, or lower-tail test.
- Set your significance level and preferred confidence level.
- Optionally rename the event and exposure labels for clearer reporting.
- Click Run Poisson Test to show results above the form.
- Use the export buttons to save a CSV summary or a PDF report.
FAQs
1. What does this calculator test?
It tests whether an observed event count is consistent with a hypothesized Poisson rate over a stated exposure. You can run lower-tail, upper-tail, or two-sided comparisons and review exact rate limits.
2. When is a Poisson test appropriate?
Use it when events are counts, occur independently, and arise over measurable exposure such as time, distance, area, or population. It works best when the event probability per tiny unit is small.
3. What is exposure in this context?
Exposure is the amount of opportunity for events to happen. Examples include machine hours, clinic days, website sessions, production batches, or road miles. The null rate is always interpreted per one exposure unit.
4. How should I choose the alternative hypothesis?
Choose greater when you only care about an increase, less when you only care about a decrease, and two-sided when either direction matters. Tail selection changes the p-value and the conclusion.
5. Why are both exact and approximate values shown?
The p-value and interval use exact Poisson logic. The approximate z-score is included as a quick directional indicator. For final decisions, rely on the exact p-value and interval instead of the approximation.
6. What does the chart display?
The graph shows the null Poisson distribution for the expected count. It highlights where your observed count sits relative to the distribution center, helping you see whether the result falls in a likely or unlikely region.
7. Does a small p-value prove the rate changed?
No. A small p-value means the observed count would be unusual if the hypothesized rate were true. You still need study design, process knowledge, and data quality checks before claiming a real change.
8. Can I use this for rare-event monitoring?
Yes. It is useful for incidents, defects, failures, infections, arrivals, and other count-based events. Make sure the counts are reasonably independent and measured across a clearly defined exposure amount.