Calculator inputs
Example data table
This example estimates the mean from a frequency table of event counts observed across 44 equal intervals.
| Event count x | Observed frequency f | x × f |
|---|---|---|
| 0 | 10 | 0 |
| 1 | 16 | 16 |
| 2 | 11 | 22 |
| 3 | 5 | 15 |
| 4 | 2 | 8 |
| Total | 44 | 61 |
Estimated mean λ = 61 ÷ 44 = 1.386364
Formula used
1. Estimated Poisson mean:
λ̂ = Σ(x × f) ÷ Σf or λ̂ = Σxᵢ ÷ n
2. Poisson probability mass function:
P(X = x) = e-λ × λx ÷ x!
3. Poisson variance and standard deviation:
Variance = λ, Standard deviation = √λ
4. Cumulative probability:
P(X ≤ k) = Σ P(X = x), for x from 0 to k
5. Approximate mean interval:
λ̂ ± z × √(λ̂ ÷ n)
The interval above is a normal approximation. It works best when the sample is reasonably large and intervals are consistent.
How to use this calculator
1. Choose Frequency table if you already grouped counts into x and f pairs.
2. Choose Raw sample observations if you have one count for each interval.
3. Enter a target x value to calculate an exact event-count probability.
4. Enter a cumulative limit to find the chance of observing up to that count.
5. Set the future interval count to project expected frequencies for planning.
6. Click Calculate Poisson Mean to show results above the form.
7. Use the CSV and PDF buttons to export your summary and distribution table.
What this calculator helps you analyze
FAQs
1. What does the Poisson mean represent?
It represents the average number of events expected in one fixed interval, area, unit, or record when events occur independently at a stable rate.
2. When should I use a Poisson mean calculator?
Use it when you analyze count data such as arrivals, defects, incidents, or requests measured over equal time periods or equally sized observation units.
3. Can I estimate the mean from raw counts?
Yes. Enter each observed count in raw sample mode. The calculator computes the sample average, which is the standard estimator for the Poisson mean.
4. What is the difference between observed and expected counts?
Observed counts come from your data. Expected counts are generated from the fitted Poisson model and show what the distribution predicts at each event level.
5. Why does the variance matter?
In a true Poisson process, the variance is close to the mean. Large differences may indicate clustering, excess zeros, missing predictors, or inconsistent intervals.
6. What does overdispersion mean?
Overdispersion means the sample variance is noticeably larger than the estimated mean. It suggests the data may be more spread out than a basic Poisson model assumes.
7. Is the confidence interval exact?
No. This version uses a normal approximation for quick interpretation. It is practical for larger samples, but exact intervals can differ slightly.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet-friendly output and the PDF button for a formatted summary and probability table you can share or archive.