Calculator
Example data table
This sample shows daily support tickets received by a team. The mean count estimates λ, and the Poisson standard deviation is the square root of that mean.
| Day | Observed tickets | Notes |
|---|---|---|
| Monday | 2 | Lower traffic |
| Tuesday | 4 | Normal workload |
| Wednesday | 3 | Routine pattern |
| Thursday | 5 | Promotional bump |
| Friday | 6 | Peak weekday demand |
| Saturday | 4 | Steady volume |
| Sunday | 3 | Reduced operations |
Formula used
For a Poisson random variable X ~ Poisson(λ), the variance equals the mean:
Var(X) = λ
The standard deviation is the square root of the variance:
σ = √λ
When you scale the observation window, the adjusted mean becomes:
λadjusted = λbase × interval multiplier
For datasets, this page estimates λ with the sample mean. For count-frequency tables, it uses the weighted mean.
How to use this calculator
- Choose whether you know λ directly or need it estimated.
- Enter a known mean, a dataset, or count-frequency pairs.
- Set the interval multiplier to scale the event window.
- Pick a confidence multiplier for an approximate range.
- Adjust decimal places and chart length if needed.
- Submit the form to show results above the calculator.
- Review the chart, summary table, and probabilities.
- Download the result set as CSV or PDF.
FAQs
1. What does this calculator measure?
It measures the Poisson standard deviation, which describes typical spread around the expected event count. It also shows related values like variance, zero-event probability, and an approximate range.
2. When should I use a Poisson model?
Use it for count data representing independent events over fixed intervals, especially when events are relatively rare and the average rate stays roughly stable.
3. Why is the standard deviation the square root of λ?
In a Poisson distribution, the variance equals λ by definition. Since standard deviation is the square root of variance, the result becomes √λ.
4. Can I estimate λ from real observations?
Yes. Enter raw counts or a count-frequency table. The calculator estimates λ using the sample mean, then computes the related spread measures.
5. What does the interval multiplier do?
It rescales the expected count to a longer or shorter window. If your base mean is per hour and you analyze three hours, multiply λ by 3.
6. Is the displayed range an exact confidence interval?
No. It is a normal-style approximation using λ ± zσ. It is useful for quick planning, but exact Poisson intervals may differ, especially for small means.
7. Why does the chart use whole numbers only?
Poisson variables count discrete events, so only nonnegative integers are possible outcomes. The chart plots exact probabilities for each count value.
8. What if my data has overdispersion?
If the observed spread is much larger than the mean, a simple Poisson model may understate variability. In that case, consider alternatives such as a negative binomial model.