Calculator Inputs
Example Data Table
| Scenario | Mean λ per interval | Target intervals | Expected count | Variance | Standard deviation |
|---|---|---|---|---|---|
| Website errors per hour | 2.50 | 1 | 2.50 | 2.50 | 1.5811 |
| Support tickets per day | 8.00 | 1 | 8.00 | 8.00 | 2.8284 |
| Machine faults over 3 shifts | 1.20 | 3 | 3.60 | 3.60 | 1.8974 |
| Customer arrivals in 4 hours | 5.00 | 4 | 20.00 | 20.00 | 4.4721 |
Formula Used
For a Poisson random variable X with rate λ, the expected value and variance are both λ.
Mean = λ
Variance = λ
Standard deviation = √λ
If you combine t equal intervals, then the new mean becomes λ × t.
Variance across t intervals = λ × t
P(X = k) = e-λ × λk / k!
λ ± z × standard error, where the standard error depends on the estimation mode selected.
How to Use This Calculator
- Choose whether you know λ directly, want to estimate it from observed counts, or want to use a sample mean.
- Enter the base event information such as λ, observed events with exposure, or sample mean with sample size.
- Set how many intervals you want to combine. This scales the expected count and the variance together.
- Enter a reference count to evaluate exact and cumulative probabilities for the scaled interval.
- Adjust confidence level, chart range, decimals, and interval label for clearer output.
- Press the calculate button. The result block appears above the form with summary metrics, a table, export buttons, and a Plotly chart.
Frequently Asked Questions
1) What is Poisson variance?
For a Poisson variable, variance equals the mean λ. If the average event count is 6 per interval, the theoretical variance is also 6.
2) Why does the variance equal the mean?
That equality is a defining property of the Poisson model. It holds when events occur independently and at a stable average rate within equal intervals.
3) When should I use observed events mode?
Use it when you have a total count over several intervals and want to estimate λ per base interval. The tool divides observed events by exposure intervals.
4) What does target interval scaling do?
It combines equal intervals. If λ is 3 per hour and you scale to 4 hours, the expected count becomes 12 and the variance also becomes 12.
5) What is the Fano factor here?
The Fano factor is variance divided by mean. For an ideal Poisson process, it equals 1. Values far above 1 in real data may suggest overdispersion.
6) What does the probability output represent?
The exact probability shows the chance of observing exactly the selected reference count. The cumulative probability shows the chance of observing that count or fewer.
7) Is the confidence interval exact?
No. This page uses a normal approximation for quick interpretation. It is useful for estimates, but exact interval methods may differ, especially with very small counts.
8) Can I use this for overdispersed data?
You can inspect the outputs, but pure Poisson assumptions may not fit overdispersed data well. In those cases, negative binomial models are often more appropriate.