Live model
Place exact uncertainty bounds around an observed Poisson rate
Observed count divided by exposure is only a point estimate. This page inverts Poisson tails to produce a two-sided Garwood interval, including the special zero-count lower boundary, then scales every rate consistently for reporting.
| Estimate | Count / quantile basis | Tail probability | Rate per unit | Rate per report scale |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
alpha=1-C; L=0.5chi^2(alpha/2,2k)/T; U=0.5chi^2(1-alpha/2,2(k+1))/T; L=0 when k=0
The implementation evaluates chi-square quantiles by inverting the regularized gamma CDF with bisection. It does not use a symmetric rate +/- standard-error shortcut, which can behave poorly for sparse counts and at zero.
Use the calculator
Five steps for a defensible rate interval
- Freeze the event definition. Count only events that satisfy the same inclusion rule.
- Enter measured exposure. A rate interval is invalid without the opportunity denominator.
- Select coverage before seeing bounds. Avoid choosing confidence merely to obtain a preferred width.
- Pick a readable report scale. Scaling changes presentation, never confidence or evidence.
- Export point and interval together. Preserve counts, exposure, method, exact endpoints, and assumptions.
Five foundations
Understanding exact Poisson rate uncertainty
1. Count and exposure are inseparable
Fourteen events in twenty hours and fourteen in twenty years imply radically different rates and intervals.
2. Exact refers to coverage construction
Garwood endpoints come from Poisson-tail inversion rather than a large-sample normal approximation. Model assumptions still remain.
3. Sparse intervals are asymmetric
The non-negative rate boundary and skewed count likelihood make the upper distance larger than the lower distance.
4. Zero events does not prove zero rate
The point estimate and lower bound are zero, but a positive upper bound quantifies rates still compatible with an event-free exposure.
5. Confidence is a procedure property
A 95% interval method covers the true rate in at least about 95% of repeated samples under its assumptions; it is not a posterior probability.
Calculation anatomy
Symbols, units, and default tail inversion
| Symbol | Meaning | Domain |
|---|---|---|
| k | Observed event count | whole, non-negative |
| T | Observed exposure | strictly positive |
| C | Confidence coefficient | 0.50-0.999 |
| alpha | Total non-coverage probability | 1-C |
| chi^2(p,nu) | Chi-square p-quantile with nu degrees of freedom | non-negative |
| S | Reporting scale | positive exposure units |
At defaults, the point rate is (14/20)x100. The lower endpoint uses the 2.5% chi-square quantile with 28 degrees of freedom; the upper uses the 97.5% quantile with 30 degrees of freedom. Both are divided by exposure before multiplying by 100.
Deep analysis
Three interval choices that affect communication
Exact versus Wald
A Wald interval centers rate +/- zxSE and can cross zero. Garwood avoids negative rates and has conservative discrete-data coverage.
Confidence versus prediction
This interval concerns the underlying rate. A future count prediction also includes Poisson process variation and is wider in a different sense.
Scale versus precision
Reporting per 100 may be easier to read than per one, but multiplying by 100 does not add information or reduce uncertainty.
Decision cases
Observed and zero-count examples
Machine stoppage rate
Fourteen stoppages over twenty machine-months are reported per 100 machine-months. The plant keeps the asymmetric interval beside the point rate when comparing lines.
Event-free pilot
A pilot records zero defects over positive exposure. The lower endpoint is zero, but the exact upper endpoint prevents "no observed defects" from becoming "zero possible defect rate."
Terms
Confidence-interval vocabulary
- Point rate
- Observed events divided by exposure.
- Garwood interval
- Poisson interval obtained from chi-square tail inversion.
- Coverage
- Long-run fraction of constructed intervals containing the true rate.
- Alpha
- Total probability allocated outside a two-sided interval.
- Reporting scale
- Positive multiplier used to express rates per convenient exposure.
- Zero-count boundary
- Rule setting the exact lower endpoint to zero when k=0.
FAQ
Questions about exact event-rate intervals
What does exact Garwood mean?
It inverts Poisson tails with chi-square quantiles rather than using a symmetric approximation.
Why is the interval asymmetric?
Rates cannot be negative and sparse count evidence is skewed.
What if no events occur?
The lower endpoint is zero and the upper remains positive.
Is 95% a probability that this rate is inside?
Not under frequentist interpretation; it is the method's repeated-sampling coverage.
What does reporting scale do?
It expresses the same rate per a convenient number of exposure units.
Does exact mean assumption-free?
No. Constant rate, independence, and exposure accuracy are still required.
Limits and evidence
Interpretation boundaries
- The interval conditions on accurately measured exposure and a constant Poisson rate.
- Garwood coverage is conservative for discrete counts; actual coverage can exceed the nominal level.
- It describes rate uncertainty, not a future-count prediction interval.
- Multiple comparisons and data-driven confidence choices are not adjusted.
- Reporting-scale multiplication changes units only.
Sources and related tools